Obligationes

TableofContents 

Preface 7 

LiborBěhounek 

FuzzyLogicsInterpretedasLogicsofResources 9 

FrancescoBerto 

StrongParaconsistencyandExclusionNegation 23 

CatarinaDutilhNovaes 

MedievalObligationesasaRegimentationof 

‘theGameofGivingandAskingforReasons’ 35 

ChristianG.Fermüller 

TruthValueIntervals,Bets,andDialogueGames 51 

BjørnJespersen&MarieDuˇzí 

ProceduralSemanticsforMathematicalConstants 65 

KoheiKishida 

NeighborhoodIncompatibilitySemanticsforModalLogic 79 

VojtěchKolman 

WhatdoGödelTheoremsTellusabout 

Hilbert’sSolvabilityThesis? 91 

TimmLampert 

WittgensteinonPseudo-Irrationals, 

DiagonalNumbersandDecidability 103 

RosenLutskanov 

WhatistheDefinitionof‘LogicalConstant’? 119 

OndrejMajer&MichalPeliˇs 

EpistemicLogicwithRelevantAgents 131 

PeterMilne 

BettingonFuzzyandMany-valuedPropositions 145 

JaroslavPeregrin 

InferentializingConsequence 155

MartinPleitz 

MeaningandCompatibility: 

BrandomandCarnaponPropositions 169 

DagPrawitz 

InferenceandKnowledge 183 

CarolineSemmling&HeinrichWansing 

ASoundandCompleteAxiomaticSystemof 

bdi–stitLogic 201 

SebastianSequoiah-Grayson 

AProceduralInterpretationofSplitNegation 219 

StewartShapiro 

ReferencetoIndiscernibleObjects 231 

PeterSchroeder-Heister 

SequentCalculiandBidirectionalNaturalDeduction: 

OntheProperBasisofProof-theoreticSemantics 245 

Vítězslav ˇ Svejdar 

RelativesofRobinsonArithmetic 261 

LucaTranchini 

TheRoleofNegationinProof-theoreticSemantics: 

aProposal 273 

MichaelvonBoguslawski 

OivaKetonen’sLogicalDiscovery 289

Preface 

TheinternationalsymposiumLogicaorganizedbytheInstituteofPhilosophyoftheAcademyofSciencesoftheCzechRepublichasarelativelylong 

history. Itbeganin1987whenthefirstconferenceofserieswasheldin 

LibliceChateau.Inthebeginningtheconferenceswereofmostlylocalimportance,butovertheyearstheyhaveacquiredthestatusofaprestigious 

internationalconferencewithamultidisciplinaryflavour. 

Theannualsymposiacoverabroadfieldoflogicaltopicsandaimto 

promotedialoguebetweenvariousbranchesoflogic. Logicahostedmany 

presentationsbytopspecialistsinmathematicalandphilosophicallogicas 

wellasanalyticalphilosophyandlinguistics. Theprofessionalorientation 

oftheconferencecanbeillustratedbymentioningseveralnamesofscholars 

thattheconferencewelcomedastheinvitedspeakers:NuelBelnap,Simon 

Blackburn,RobertBrandom,MelvinFitting,YuriGurevich,PetrHájek, 

RomHarré,JaakkoHintikka,WilfridHodges,DavidLewis,PerMartin-Löf, 

BarbaraPartee,GrahamPriest,GregRestall,GabrielSandu,andStewart 

Shapiro. 

Amongthecentralpointsofthe‘publicationpolicy’behindtheproceedingsoftheconferenceistherulethatthevolumehastoappearbeforebeginningofthenextyear’sconference.This‘rush’,however,doesnotaffect,webelieve,thequalityofthepreparationofthevolumes. 

Untillast 

yeartheywerepublishedbytheInstitute’spublishinghouseFilosofia;the 

presentvolumeisthefirstpreparedforCollegePublications.Itcontainsa 

majorityofthepaperspresentedatthesymposiumLogica2008,whichtook 

placefromJune16to20intheformerFranciscanmonasteryofHejnice,the 

CzechRepublic,wheretheparticipantsspentfivedaysnotonlyinthelecturinghallbutalsoinmanyinformaldiscussionsduringthebreaks,lunches 

andsocialevents.AsyoucanseeTheLogicaYearbook2008bringstogether 

varioustextsfrommathematicalandphilosophicallogic,historyandphilosophyoflogic,andnaturallanguageanalysis. 

Ashasbecomeusualfor 

theYearbookseries,thearticleshavenotbeensortedbysubject—they 

areorderedalphabeticallybyauthoranditisuptothereadertopickand 

choose. 

Editor’sacknowledgements 

BoththeLogicasymposiumandthisbookseriesaretheresultofajoint 

effortofmanypeople,whodeservemydeepthanks.Amongthemarethe 

mainorganizersVladimírSvobodaandTimothyChildersfromtheDepartmentofLogicoftheInstituteofPhilosophyandPavelBaran,thedirector 

oftheInstituteofPhilosophy. Theconferencewaspromotedalsobythe 

GrantAgencyoftheCzechRepublic,whichprovidedsignificantsupportby 

financingthegrantprojectno.401/07/0904. Wearefurtherindebtedto 

MarieVučková,HeadoftheForeignRelationsDepartmentoftheInstitute,

8 TableofContents 

fororganizationalsupport.Theorganizationwouldbeimpossiblewithout 

thehelpofPetraIvaničováduringandbeforetheconference.Ourstayin 

HejniceMonasterywasmadepleasantthroughtheeffortsofFatherMiloˇs 

Rabanandthestaffofthemonastery.SpecialthanksalsogototheBernard 

FamilyBreweryofHumpolec,traditionalsponsorofthesocialprogramme 

ofthesymposium.IwouldalsoliketothanktoMarieBenediktováforthe 

layoutofthisvolume.ManythanksgotoCollegePublicationsanditsmanagingdirectorJaneSpurr.Lastbutnotleastwewouldliketothankallthe 

conferenceparticipantsandtoauthorsofthearticlesfortheiroutstanding 

cooperationduringtheeditorialprocess. 

Prague,May2009 MichalPeliˇs

Fuzzy Logics Interpreted as Logics of Resources 

Libor Běhounek ∗ 

Girard’slinearlogic(1987)isofteninterpretedasthelogicofresources, 

whileformalfuzzylogics(seeesp.Hájek,1998)areusuallyunderstoodas 

logicsofpartialtruth. Iwillarguethatdeductivefuzzylogicscanbeinterpretedintermsofresourcesaswell,andthatundermostcircumstances 

theyactuallycaptureresource-awarereasoningmoreaccuratelythanlinear 

logic.Theresource-basedinterpretationthenprovidesanalternativemotivationforformalfuzzylogics,andgivesanexplanationofthemeaningof 

theirintermediarytruthvaluesthatcanbejustifiedmoreeasilythantheir 

traditionalmotivationbasedonpartialtruth. 

1 Linearandsubstructurallogics 

Recallthatlinearlogicanditsvariantsarerepresentativesofbasicsubstructurallogics(see,e.g.,Restall,2000,Paoli,2002,Ono,2003),i.e.,logicsthat 

resultfromdiscardingsomeofthestructuralrulesfromtheGentzen-style 

calculiLKandLJforclassicalandintuitionisticlogic.Inparticular,linear 

logic LLdiscardstherulesofcontraction (C) 

andweakening (W) 

Γ,A,A,∆ =⇒ Σ 

Γ,A,∆ =⇒ Σ 

Γ =⇒ Σ 

A,Γ =⇒ Σ 

Γ =⇒ Σ,A,A,Π 

Γ =⇒ Σ,A,Π 

Γ =⇒ Σ 

Γ =⇒ Σ,A 

fromthecalculusLKforclassicallogic.Intuitionisticlinearlogic ILLdiscardsthesamerules 

(C,W)fromthecalculusLJforintuitionisticlogic. 

Affinelinearlogic ALLandintuitionisticaffinelinearlogic IALLdiscard 

onlytheruleofcontraction (C)fromthecalculiLKandLJ,respectively, 

butretaintheruleofweakening (W). 

∗ TheworkwassupportedbyGrantNo.IAA900090703“Dynamicformalsystems”ofthe 

GrantAgencyoftheAcademyofSciencesoftheCzechRepublicandbyInstitutional 

ResearchPlanAV0Z10300504.

10 LiborBěhounek 

Recallfurtherthatsubstructurallogicsworkingeneralwithtwoconjunctions: 

thelatticeconjunction ∧(alsocalledweak,additive,orextensionalconjunction)andfusion 

&(alsocalledgroup,strong,multiplicative, 

orintensionalconjunction).Similarlythereareingeneraltwodisjunctions 

(lattice ∨andstrong)aswellastwoimplications,twonegations,etc.,but 

thelattersplitconnectiveswillnotplayasignificantroleinouraccount, 

asweshallmainlydealwithintuitionisticsubstructurallogics(whichlack 

strongdisjunction)andcommutativefusion(thenbothimplicationscoincide).Sinceinsuchsubstructurallogicsimplicationinternalizesthesequent 

sign =⇒and &thecommaontheleft-handsideofsequents(cf.Ono,2003), 

thevalidityofthesequent A1,... ,An =⇒ Bisequivalenttothevalidity 

oftheformula A1 & ... & An → B. Consequently,theruleofcontraction 

correspondstothevalidityof A → A & Aandtheruleofweakeningtothe 

validityof A & B → A. 

Thealgebraicsemanticsofsubstructurallogicsisthatofresiduatedlattices(see,e.g.,Jipsen&Tsinakis,2002;Ono,2003;Galatos,Jipsen,Kowalski,&Ono,2007),i.e.,latticesendowedwithanadditionalmonoidaloperation 

∗(representing &)monotonew.r.t.thelatticeorder ≤,anditstwo 

residuals /, \(representingimplications)thatsatisfytheresiduationlaw 

x ∗ y ≤ z iff y ≤ x\z iff x ≤ z/y. 

If ∗iscommutative,thetworesiduals /, \coincideandareusuallydenoted 

by ⇒.Thesetofdesignatedelementsis {x | x ≥ 1},where 1istheneutral 

elementofthemonoidaloperation ∗.Ifconvenient,residuatedlatticesmay 

beexpanded(toOno’s FL-algebras)byaconstant 0forfalsity,whichmakes 

itpossibletodefinenegationas x ⇒ 0. 

Thetermsubstructurallogicswillinthispaperdenotelogicsofclasses 

ofresiduatedlattices,followingthestipulativedefinitionbyOno(2003). 

Inparticular,(affine)intuitionisticlinearlogicisthelogicofall(bounded 

integral)commutativeresiduatedlattices, 1 and(affine)linearlogicisthe 

logicofthosethatfurthermoresatisfythelawofdoublenegation. 

2 Linearlogicasthelogicofresources 

Thereasonwhylinearlogichasbeenregardedasthelogicofresourcesis 

illustratedbyGirard’s(1995)well-known‘Marlboro–Camels’example: 

1 Aresiduatedlatticeiscalledcommutativeifitsmonoidaloperation ∗iscommutative; 

itiscalledboundedintegralif 0 ≤ x ≤ 1forallelements x.Weshallusuallyworkwith 

commutativeresiduatedlatticesonly.


Considerthepropositions 

Thenthesequent 

expressingtheinference 

D =“Ipay$1.”, 

M =“IgetapackofMarlboro.”, 

C =“IgetapackofCamels.” 

D → M,D → C =⇒ D → M & C 

IfIpay$1,IgetapackofMarlboro 

IfIpay$1,IgetapackofCamels 

∴IfIpay$1,IgetapackofMarlboroandIgetapackofCamels 

isderivablebytherulesofclassicalaswellasintuitionisticlogic.Theinferenceis,however,viewedascounter-intuitive,iftheconclusionisstraightforwardlyunderstoodasgettingbothpacks.Thedisputablesequentisnot 

derivableinlinearlogic,though:linearlogiconlyderivesthesequent 

D → M,D → C =⇒ D & D → M & C 

whichunderasimilarinterpretationcapturesthefactthatIneedtopay 

twodollarstogetbothpacksofcigarettes. 

Inthissense,linearlogicissaidtoregardformulaeas‘resources’,which 

are‘spent’whenusedaspremisesofimplications(intheMarlboro–Camels 

example,thepremise Disspentbybeingdetachedfrom D → Mtoobtain 

M,andcannotbeusedagainfor D → Ctoobtain M & C).Moreformally, 

sincepremisescannotinlinearlogicbecontracted(duetothelackoftherule 

(C)),theyactastokensfor‘resources’neededtosupporttheconclusion:a 

sequentisvalidinlinearlogiconlyifithastheneededamountsofpremises 

requiredforarrivingattheconclusion. 2 Inotherwords,linearlogic‘counts’ 

premisesofsequentsasiftheyrepresentedresourcesneededfor‘buying’the 

conclusion(wheredifferentpropositionalletterswouldrepresentdifferent 

typesofresources,whiletheiroccurrencesinthesequentwouldrepresent 

tokensorunitsofthattype). 

Nevertheless,thisfeatureoflinearlogicisduesolelytotheabsenceof 

therule (C)ofcontraction,andthereforeiscommontoallcontractionfreesubstructurallogics. 

Itisnotclearwhyexactlylinearlogicshould 

2 Exactlytheneededamountsin LLor ILL;atleasttheneededamountsintheiraffine 

versions(itbeinganeffectofweakeningthatweneednotspendallpremises).Inlogics 

withboth(C)and(W),e.g.,classicalorintuitionisticlogic,eachpremiserequiredfor 

arrivingattheconclusiononlyneedstobepresentatleastonce.


bemoreadequateasalogicofresourcesthananyothercontraction-free 

logic. Rather,itistobeexpectedthatdifferentcontraction-freelogics 

willcorrespondtodifferentassumptionsonthestructureofresources. In 

thefollowingsectionsIwillarguethatlinearlogicsareinfactadequate 

onlyforverygeneralstructuresofresources,whileundermostcommon 

circumstances,strongerlogicsareappropriate. 

3 Thestructureofresources 

Asafirsttask,weneedtorefineourconceptionofresources.Sinceweaim 

ataninformalsemanticexplanationofcertainlogics,insteadofgivinga 

formaldefinitionweshalljustlistafewexamplesindicatingwhatkindof 

resourceswehaveinmind,andspecifythemathematicalpropertiesthey 

areassumedtosatisfy. 

Ournotionofaresourcewillberatherbroad:itcanincludeanykind 

ofthingsthatcanbecountedormeasured,thatcanbeacquiredandexpended,orusedforanypurpose. 

Amongtheresourcesweconsiderare, 

e.g.:money(costs,prices,debts,etc.);goods(packsofcigarettes,clothes, 

cars,etc.);industrialmaterials(chemicals,naturalrawmaterials,machine 

components,etc.);cookingingredients(flour,salt,potatoes,etc.);computer 

resources(diskspace,computationtime,etc.);penalties(whichcanberegardedasakindofcostsincurred);sets,multisets,orsequences(tuplesor 

vectors)oftheabove;etc. 

Itcanbeobservedthatallofthese(aswellasmanyother)kindsof 

resourcesexhibitthestructureofaresiduatedlattice. Inparticular,there 

is: 

•Apartialorder �comparingtheamountsoftheresources. Forinstance,300gofflourismorethan200gofflour;twopensandthree 

pencilsaremorethanonepenandthreepencils;etc.Forthesakeof 

compatibilitywithfurtherdefinitions,weshallunderstand x � yas 

“theresource xislargerthanorequalto y.”Theorderneednotbe 

linear,asforinstancetwopensarenotcomparablewiththreepencils 

(ifdifferentitemsarecountedseparately).However,itcanbeassumed 

that �isalatticeorder,asthisistrueforallprototypicalcases:by 

definition,itamountstosupposingthatforanytworesources x,y(for 

instance: x=2pensand3pencils; y=1penand4pencils),there 

istheleastresourcethatisatleastaslargeasboth(inthiscase, 

2pensand4pencils)andthelargestresourcethatisatmostaslarge 

asboth(here,1penand3pencils). Eventhoughtheremayexist 

resourcesthatdonotsatisfythisassumption,weleavethemasidein 

ourconsiderations.


•Amonoidaloperation ∗ofcomposition(orfusion)ofresources. For 

example,300gofflourand200gofflouris500gofflour;2pensand 

3pencilsplus1penand3pencilsare3pensand6pencils;etc.Putting 

theresourcestogethercanbeassumedtobeassociative(i.e.,wepresumethatthetotalsumdoesnotdependontheorderofsummation). 

Thekindsofresourcesweconsideralwayshaveaneutralelement e, 

theemptyresource,whichdoesnotchangetheamountwhenadded 

toanotherresource:e.g.,0gofflour;0pensand0pencils;etc.Even 

thoughcompositionofresourcesneednotbecommutative(consider, 

e.g.,theorderofaddingingredientswhencooking),forthesakeof 

simplicityofexpositionweshallonlyconsidercommutative ∗here 

(generalizationtonon-commutative ∗isalwaysstraightforward). 

•Finally,resourcesofalltypicalkindscanbe‘subtracted’or‘evened 

up’,i.e.,theircompositionhastheresidualoperation ⇒expressingthe 

remainder,orthedifferenceofamounts: x ⇒ yistheleastresource 

tobeaddedto xinordertogetaresourceatleastaslargeas y. 3 For 

example,if x=200gofflourand y=300gofflour,then x ⇒ yis 

100gofflour,asoneneedstoadd100gofflourto200gofflourto 

getatleast300g;whileif x=2pensand3pencils,and y=1pen 

and3pencils,then x ⇒ yis0pensand0pencils(i.e.theempty 

resource e),asweneednotaddanythingto xtogetatleast y. 

Allkindsofresourcesweconsiderthushavethestructureofa(commutative)residuatedlattice 

L = (L, ∧, ∨, ∗, ⇒,e). Particularkindsofresources 

canhaveadditionalproperties:forexample,mostusualkindsofresources 

satisfytheso-calleddivisibilitycondition x ∗ (x ⇒ y) = x ∧ y. 

Sinceweaimatasimpleresource-basedinterpretationofexistinglogical 

calculiratherthandevelopmentofanexpressivelyrichlogicofresources 

forcomputerscience,wedonotconsidersuchphenomenaas,e.g.,resource 

dynamicsorpossiblenon-totalityof ∗(whicharemodeledbysuchsystemsas 

thelogicofbunchedimplications,computationlogics,orsynchronousand 

asynchronouscalculi—see,e.g.,Pym&Tofts,2006forreferences),but 

onlyreconstructandrefinetheassumptionsonresourcesthatareadopted 

bylinearlogic. 

4 Formulaeasresources 

Thereareatleasttwopossiblerepresentationsofresource-basedsemantics 

ofsubstructurallogics. Oneofthemtakesresources(i.e.elementsofthe 

3 I.e. x ⇒ y = sup{z | z ∗ x � y},whichisanequivalentformulationoftheresiduation 

lawincompletelattices. Forincompletelattices,amorecautiousformulationbasedon 

Dedekind–MacNeillecutsisdue,namely {z | z ∗ x � y} = {z | z � x ⇒ y},whichisa 

generalequivalentoftheresiduationlaw.


residuatedlattice LdescribedinSection3)directlyassemanticvaluesassignedtopropositionalformulae. 

Recallthatalogicalcalculuscanhave 

interpretationsotherthanpropositional:cf.,e.g.,theinterpretationofthe 

Lambekcalculusasthecategorialgrammar(wherethesemanticvaluesof 

formulaearegrammaticalcategories),ortheCurry–Howardcombinatorial 

interpretationoftheimplicationalfragmentofintuitionisticlogic(where 

formulaeareinterpretedastypesandproofsasprograms). Inasimilar 

vein,wecaninterpretthealgebraicsemanticsofsubstructurallogicsunder 

the“formulae-as-resources”paradigmasfollows: 

•Thesemanticvalueofaformula ϕisaresource �ϕ� ∈ L. 

•TheTarskicondition �1� = eofthealgebraicsemanticsinterpretsthe 

formula 1astheemptyresource(or‘beingforfree’). 

•Similarly,theclause �ϕ & ψ� = �ϕ� ∗ �ψ�saysthatconjunctionrepresentsthefusionofresources. 

•Thevalueofimplication, �ϕ → ψ� = �ϕ� ⇒ �ψ�,istheresource 

neededtogetatleast �ψ�,giventheresource �ϕ�. 

•Finally,thelatticeconnectives ∧, ∨representthemeetandjoinof 

resources(withrespecttothesizeorder �ofresources). 

Theformula ϕisregardedasvalidunderagivenevaluationiff e � �ϕ�,i.e., 

iffitrepresentsaresourcethatisforfreeorevencheaper. 

5 Resourcesaspossibleworlds 

Anotherwayhowtointerpretsubstructurallogicsintermsofresources 

(cf.Pym&Tofts,2006)istoregardthestructure Lofresourcesasa 

Kripkeframe (L, �)endowedwithamonoidalstructure (∗,e).Unlikeinthe 

“formulae-as-resources”paradigm,formulaearehereinterpretedaspropositions,andresourcesonlyserveasindicesthatmay(ormaynot)validate 

them. Theforcingrelation r � ϕ,“theresource r ∈ Lsupportstheformula 

ϕ,”isrequiredtosatisfythefollowingconditions: 

• e � 1, 

• r � ϕ & ψiff ∃s,t ∈ L: r � s ∗ tand s � ϕand t � ψ, 

• r � ϕ → ψiff ∀s ∈ L: if s � ϕ,then r ∗ s � ψ, 

• r � ϕ ∧ ψiff r � ϕand r � ψ(“sharedresources”—contrastthe 

clausefor &),


• r � ϕ ∨ ψiff ∃s,t ∈ L: r � s ∨ tand (s � ϕor s � ψ)and (t � 

ϕor t � ψ), 

andtheconditionofpersistence(if r � sand s � ϕ,then r � ϕ),expressing 

that“largerresourcessufficeaswell.”Theformula ϕisdefinedtobevalid 

under �iff e � ϕ,i.e.,iffsupportedevenbytheemptyresource. 4 

6 Theroleoftautologies 

Intheabovesemantics,tautologiesw.r.t.aclass Kof(commutative)residuatedlatticesaredefinedastheformulae 

ϕthatgetavalue �ϕ� � eunder 

allevaluationsofpropositionallettersinanyresiduatedlattice L ∈ K(resp. 

aresupportedby eunderall �ineveryKripkeframe L ∈ K).Thetautologiesofsubstructurallogicsthusrepresentcombinationsofresourcesthat 

arealways“forfreeorcheaper”. 

Moreimportantly,sinceallresiduatedlatticesvalidate 

e � r ⇒ s iff r � s, 

tautologiesoftheform ϕ → ψinternalizesoundrulesofresourcetransformationsthat“preserveexpenses”(inthesenseof�).Inferenceinsubstructurallogicscanthusbeunderstoodasinferencesalvisexpensis,inasimilar 

mannerasinferencesalvaveritateinclassicallogic. 5 

Classesofresiduatedlatticesadmittedaspossiblestructuresofresources 

thendetermineparticularlogicsofresourcesintheabovesense.Inparticular,bytheknowncompletenesstheorem, 

ILListhelogicofallcommutative 

residuatedlattices,andsoitisanadequatelogicifjustthegeneralstructure 

ofacommutativeresiduatedlatticeisassumedforadmissiblekindsofresources.Itsvariants 

IALL, ALL,and LLrestrictthestructureofresources 

tonarrowerclassesofcommutativeresiduatedlattices,andothersubstructurallogicscorrespondtofurtherspecificclassesofresiduatedlatticesof 

resources. 6 

InthefollowingsectionsIwillarguethatmosttypicalkindsofresources 

satisfytheso-calledprelinearitycondition,andsoareinfactgovernedby 

deductivefuzzylogicsratherthanlinearlogics. 

4 Asthisisnottheaimofthispaper,weomitthedetailsonthecorrespondencebetween 

theKripke-styleandalgebraicsemanticsofsubstructurallogics. Formoreinformation 

see(Ono&Komori,1985). 

5 Notethatthegeneralvalidityof �ϕ� � �ψ�definesthelocal consequencerelation 

(expressed,i.a.,bysequentsinSection1),whileHilbert-stylecalculiforsubstructural 

logicsusuallycapturetheglobalconsequencerelation“e � �ψ�whenever e � �ϕ�”. 

6 Forexample,classicallogiccanbeinterpretedasthelogicdistinguishingjusttwosizes 

ofresources:empty e = �1�andnon-empty f = �0� ≺ e.


7 Deductivefuzzylogics 

Deductivefuzzylogics canbedelimitedaslogicsof(classesof)linearly 

orderedresiduatedlattices(Běhounek&Cintula,2006;Běhounek,2008). 

Amongtheextensionsof ILLtheycanbecharacterizedasthosethatsatisfy 

theaxiomofprelinearity (Pre): ((A → B) ∧ 1) ∨ ((B → A) ∧ 1),orinthe 

presenceofweakening,equivalently (A → B) ∨ (B → A). 

Letuscallresiduatedlatticesforwhichasubstructurallogic Lissound, 

L-algebras. Theprelinearityaxiomensuresthatadeductivefuzzylogic L 

issoundandcomplete,notonlyw.r.t.theclassofall L-algebras(thegeneralcompletenesstheorem),butalsow.r.t.theclassofalllinear 

L-algebras 

(thelinearcompletenesstheorem). Thelinearcompletenesstheoremcharacterizesdeductivefuzzylogicsamongsubstructurallogics;thefinitaryones 

aremoreovercharacterizedbythelinearsubdirectdecompositionproperty, 

whichsaysthateach L-algebraisasubdirectproduct 7 oflinear L-algebras. 

(SeeCintula,2006fordetails.) 

Besidesthegeneralandlinearcompletenesstheorems,mostimportant 

deductivefuzzylogicsfurthermoreenjoythestandardcompletenesstheorem, 

i.e.thecompletenessw.r.t.asetof(selected) L-algebrasontheunitinterval 

[0,1]ofreals(withtheusualordering ≤),calledthestandard L-algebras. 

Since L-algebrason [0,1]arefullydeterminedbythemonoidaloperation 

∗,standard-completedeductivefuzzylogicscanbedefinedaslogicsof(sets 

of)suchmonoidaloperations ∗on [0,1].Forexample, 

•ŁukasiewiczlogicŁisthelogicoftheŁukasiewiczt-norm x ∗ y = 

(x + y − 1) ∨ 0, 

•Gödel–Dummettlogic Gisthelogicoftheminimum,i.e.of x∗y = x∧y, 

•Productfuzzylogic Πisthelogicoftheordinaryproductofreals, 

x ∗ y = x · y, 

•Hájek’sbasicfuzzylogic BListhelogicofallcontinuoust-norms, 8 

•Monoidalt-normlogic MTListhelogicofallleft-continuoust-norms, 

•Uninormlogic UListhelogicofallleft-continuousuninorms,etc. 

Formoreinformationontheselogicssee(Hájek,1998;Esteva&Godo,2001; 

Metcalfe&Montagna,2007). 

7 I.e.asubalgebraofthedirectproductwithallprojectionstotal. 

8 Acommutativeassociativemonotonebinaryoperation on [0, 1]withaneutralelement 

e ∈ [0, 1]iscalledauninorm.At-normisauninormwith e = 1.


Theweakestdeductivefuzzylogicextendingasubstructurallogic Lis 

often 9 obtainedbyaddingtheprelinearityaxiom (Pre)to L:forinstance, 

ILL + (Pre) = UL 

IALL + (Pre) = MTL 

aretheweakestdeductivefuzzylogicsextendingintuitionisticlinearlogics, 

orthelogicsoflinearcommutative(boundedintegral)residuatedlattices. 

(For LLand ILL,thedoublenegationlawistobeaddedto ULresp. 

MTL.) 

8 Fuzzylogicsaslogicsofcosts 

Sincedeductivefuzzylogicsarelogicsof(specialclassesof)residuatedlattices,theycanbeinterpretedaslogicsofresourcesinthesamewayasother 

substructurallogics. Specifically,bythelinearcompletenesstheorem(see 

Section7),deductivefuzzylogicsaresoundandcompletew.r.t.particular 

classesoflinearresiduatedlattices,andsotheyareadequateforresources 

thatarelinearlyorderedby �. Inotherwords,deductivefuzzylogicsare 

thoselogicsofresourcesinwhichwecanassumethatallresourcesarecomparable.Prototypicallinearlyorderedresourcesarecosts,thatis,resourcesconvertedtomoney.Eventhoughresourcesingeneralneednotbecomparable(cf.theexamplesinSection3),theircosts(ifspecified)canalwaysbecompared,asmoney(ofasinglecurrency)formsalinearscale. 

10 Besidesmoney, 

therearemanyotherkindsofresourcesthatarelinearlyordered,e.g.,gallonsoffuel,computationtime,operationalmemory,etc. 

Irrespectiveof 

theirnature,weshallcallalllinearlyorderedresourcescosts,todistinguishthemfromresourcesthatarenotlinearlyordered. 

Forconvenience, 

costswithvaluesintheinterval [0,+∞],e.g.,monetaryprices(where 0is 

“gratis”and +∞mayrepresentthepriceofunattainablegoods),willbe 

calledprices. 

Deductivefuzzylogicscanthusberegardedaslogicsofcosts,inthesame 

senseaslinearlogicsareregardedaslogicsofresources.Differentwaysof 

addingupcosts—givenbythefusionoperation—yielddifferentdeductive 

fuzzylogics.Themosttypicalexamplesaregivenbelow: 

•Ifpricesaresummedbyordinaryaddition,weobtaintheproduct 

logic Π,sincetheresiduatedlattice [0,+∞]withthefusion +andthe 

latticeorder ≥isisomorphic(viathefunction p ↦→ 2 −p )tothestandardproductalgebra 

[0,1]withthefusion ·andthelatticeorder ≤. 

9 Alwaysifmodusponensistheonlyderivationruleof L(Cintula,2006). 

10 ThisideaisduetoPetrCintula(pers.comm.).


Notethatinthestandardproductalgebra, 0representstheinfinite 

costand 1thenullcost. Iftheinfinitecostisnotconsidered,the 

standardproductalgebrawithout 0(calledthestandardcancellative 

hoop)anditslogic CHL(cancellativehooplogic,seeEsteva,Godo, 

Hájek,&Montagna,2003)areobtained. 11 

•Ifpricesareboundedbyavalue a ∈ (0,+∞)andsummedbybounded 

additiontruncatedat a,weobtaintheŁukasiewiczlogicŁ,sincethe 

residuatedlattice [0,a]withboundedadditionand ≥isisomorphic 

via p ↦→ (a − p)/atothestandard[0,1]algebraforŁukasiewiczlogic. 

Thebound a(correspondingto 0inthestandardalgebra)appears 

naturallyif,e.g.,afixedmaximumpriceisset,ifthereisamaximal 

possiblecostinthegivensetting,oriftheprice aisinthegiven 

contextunaffordable. 

•Ifpricesarecombinedbythemaximum,Gödellogic G(oritshoop 

variant)isobtained(bythesameisomorphism p ↦→ 2 −p asinthecase 

ofaddition).Themaximummayseemastrangeoperationforsummationofprices,butitoccursnaturallywheneverthecostscanbeshared 

bythesummands. Forexample,iftemporaryresultscanbeerased 

beforethecomputationproceeds,thememoryneededfortemporary 

resultsisonlythemaximum(ratherthansum)oftheirsizes. 

Logicsofotherparticulart-normsareobtainedbyusingvariouslydistorted‘addition’ofprices.Forinstance,thelogicofanordinalsumofthe 

threebasict-normscorrespondstousingdifferentsummationrules(ofthe 

threedescribedabove)indifferentintervalsofprices. Thelogic MTLis 

obtainedifallmonotonecommutativeassociativeleft-continuousoperations 

withthezeropriceactingastheneutralelementareadmittedas‘addition’ 

ofprices;similarlyfor BLandcontinuoussuchoperations,etc. Thelogic 

ULandotheruninormlogicsonlydifferbypermittingalsonegativeprices, 

whichexpressgainsratherthancosts. 

9 Fuzzylogicsaslogicsofresources 

Inspiteofthelinearcompletenesstheorem,whichmakesitpossibleto 

regarddeductivefuzzylogicsaslogicsoflinearlyorderedcosts,algebras 

fordeductivefuzzylogicsneednotbelinear(consider,e.g.,theirdirect 

11 Ifthecostscomeinpackages(e.g.,ifonehastobuyawholepackofcigarettesevenif 

oneneedsonlyafew),thealgebraisingeneraljustaΠMTL-chaininsteadofaproduct 

algebra,andtheresultinglogicingeneralonlyextendsthelogic ΠMTL(Esteva&Godo, 

2001)oritshoopvariant. Asimilareffectofpackaging,whichdestroysthedivisibility 

ofthealgebra(seeSection3),canbeobservedinotheralgebrasofcostsaswell. (This 

observationisbasedonremarksbyRostislavHorčíkandPetrCintula.)


products).Bythegeneralcompletenesstheorem(seeSection7),adeductive 

fuzzylogic Lisalsosoundandcompletew.r.t.theclassofall L-algebras: 

thus Lcanalsobeinterpretedasthethelogicofallkindsofresourcesthat 

formthestructureofa(possiblynon-linear) L-algebra. 

Letusrestrictourattentiontofinitarydeductivefuzzylogicsonly,as 

theyincludeallprototypicalcases;forthesakeofbrevity,letuscallthem 

justfuzzylogicsfurtheron.Bythelinearsubdirectdecompositiontheorem 

(seeSection7),any L-algebraforafuzzylogic Lcanbedecomposedinto 

asubdirectproductoflinear L-algebras.Fuzzylogicscanthusbecharacterizedaslogicsofsuchresourcesthateitherarelinearlyordered,orcan 

atleastbedecomposedintolinearlyorderedcomponents.Inotherwords, 

asoundandcompleteresource-basedsemanticsoffuzzylogicsneednotbe 

justthatofcosts,butalsothatofresourcesrepresentableastuples(possibly 

infinitary)ofcosts. 

Itcanbeobservedthatmanykindsofnon-linearresourcescanactually 

berepresentedastuplesoflinearlyorderedvalues.Forexample,ingredients 

formakingpizzaandthoseformakingspaghettiarenotsubsetsofeach 

other,thuscookingingredientsdonotformalinearlyorderedresiduated 

lattice. 12 Nevertheless,theycanbedecomposedinto(potentiallyinfinitely 

many)linearlyorderedcomponents,astheamountsofeachindividualitem 

onaningredientlistarealwayscomparable;andindeeditcanbechecked 

thattheprelinearityaxiomisvalidinthisresiduatedlattice. 13 

Infact,mosttypicalresources(includingthosementionedinSection3) 

areindeeddecomposableinthiswayintolinearcomponents. Evenmany 

resourcesforwhichsuchadecompositionisnotknown(e.g.,humanintelligence)canatleastbebelievedtobelinearlydecomposable(intosome 

unknownandveryfinelinearcomponents). Itisactuallyratherhardto 

findakindofresourcesthatdemonstrablycannotbesodecomposed. 

Thuswecanconcludethatalltypicalkindsofresourcesarelinearlydecomposable,andthereforetheysatisfytheaxiomofprelinearity,whichisnotvalidinlinearlogicnorinitsaffineorintuitionisticvariants;consequently,theyareactuallygovernedbydeductivefuzzylogicsratherthanlinearlogics.Linearlogicsarethusonlyadequateforaverygeneralstructureofresources,whichadmitseventherarekindsofresourcesthatarenot 

decomposableintolinearlyorderedcomponents.Asregardsmostusualkind 

12 Theelementsoftheresiduatedlatticeofallpossibleingredientlists(suchascanbe 

foundinrecipebooks)aretuplesofquantitiesofparticularingredienttypes(e.g.,[300g 

offlour,2tomatoes,2ltofoil],zeroamountsomitted).Thetuplesarenaturallyordered 

byinclusion(i.e.pointwisebycomponentsizes),andfusionrepresentsaddingupamounts 

ofeachingredient. 

13 Sincethefusionofamountsis(unbounded)additionineachcomponentandinfinite 

amountsdonotoccur,byextendingtheconsiderationsofSection8theresiduatedlattice 

canactuallybeidentifiedasacancellativehoop,andthelogicofcookingingredientsas 

thecancellativehooplogic CHL.


ofresources,linearlogicistooweakforthem,asitdoesnotvalidatethelaw 

ofprelinearitytheyobey. Assumingcommutativityoffusion,theweakest 

logicadequatefortypicalresourcesistheuninormlogic UL(or MTLif 

weakeningisassumed,i.e.,iftheemptyresourceisthesmallest). Specific 

structuresoftypicalresourcesaregovernedbyevenstrongerfuzzylogics 

—inparticular,productlogic Πifresourcesarecombinedbyadditionin 

eachlinearcomponent,ŁukasiewiczlogicŁiftheadditionisbounded,and 

Gödellogic Ginthecaseofsharedresources(i.e.,iftheycomponentwise 

combinebythemaximum). 

Thusitturnsoutthatdespitethecommonopinion,itisactuallyfuzzy 

logics,ratherthanlinearlogics,thatcouldbecategorizedastypicallogics 

ofresources. 14 Theinterpretationintermsofresourcesandcostsmoreover 

providesanalternativemotivationfordeductivefuzzylogicsandanexplanationofthemeaningoftheirintermediarytruth-valuesthatcaninsome 

respectsbemoreeasilyjustifiedthanthestandardaccountbasedondegrees 

ofpartialtruth. 

LiborBěhounek 

InstituteofComputerScience 

AcademyofSciencesoftheCzechRepublic 

PodVodárenskouvěˇzí2,18207Prague8,CzechRepublic 

behounek@cs.cas.cz 

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14 Thepricepaidforthemoreaccurateaccountisamorecomplexprooftheory,asprelinearitydestroysthegoodproof-theoreticalpropertiesoflinearlogics.


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Strong Paraconsistency and Exclusion Negation 

1 Trueornot? 

Francesco Berto ∗ 

Strongparaconsistency,alsocalleddialetheism,istheviewaccordingto 

whichtherearedialetheias,thatis,sentences Asuchthatboth Aand ¬A 

aretrue, 1 anditisrationaltoacceptandassertthem(aneminentcasebeing 

allegedlyprovidedbythevariousversionsoftheLiarparadox).Onecould 

thereforepicturedialetheismasdisputingtheLawofNon-Contradiction 

(LNC).Asamatteroffact,though,allthemainformulationsoftheLNC 

arenotdisputedbyatypicaldialetheist,inthesensethatsheiscommitted 

toacceptthem. Thedialetheicattitudeofthedialetheistisexpressedby 

typicallyaccepting,andasserting,boththeusualversionsoftheLNCand 

sentencesinconsistentwiththem. 

Ofcourse,thiscallsforadrasticrevisionofourstandardnotionsoftruth 

andnegation.Philosophersoftendisagreeonthecontentofbasiclogicaland 

metaphysicalconcepts(suchasidentity,existence,necessity,etc.),oronthe 

validityofsomeverybasicprinciplesofinference(suchasContraposition 

ortheDisjunctiveSyllogism).Itiswellknownthatthiskindofdiscussion 

oftenfacesimpasses,orturnsintoahardconflictofintuitions. Itisvery 

difficulttoestablishwhensomepartyorotherbeginstobegthequestion. 

Onewonderswhetheranon-standardexplanationofabasiclogicalnotion 

involvesarealdisagreementwithaclassicalaccountofthatnotion,orits 

principlessimplydescribeadifferentthingusingthesamenameorsymbol 

(thefamous“changeofsubject”Quineanmotto). 

∗ TheideasontheNOT-operatordevelopedinthispaperhavebeenhintedatin(Berto 

2007,Ch.14),andexposedextensivelyin(Berto2008)—IamgratefultotheEditorsof 

theAustralasianJournalofPhilosophyforthepermissiontoreusesomeofthatmaterial. 

ThankstoGrahamPriest,FrancescoPaoli,RossBrady,MaxCarrara,VeroTarca,Luca 

Illetterati,andDiegoMarconi,forhelpfulcomments,andtotheparticipantstoLogica 

2008forthelivelydiscussionofthetalkgiventhere. 

1 See(Berto&Priest,2008).


Thisseemstobedecidedlythecasewithstrongparaconsistency.When 

someoneclaimsthatboth Aandnot-Aaretrue,onewonderswhatismeant 

by“true”;and,ofcourse,by“not”: 

Thefactthatalogicalsystemtolerates Aand ∼Aisonlysignificant 

ifthereisreasontothinkthatthetildemeans‘not’. Don’twesay 

‘InAustralia,thewinterisinthesummer’,‘InAustralia,peoplewho 

standuprighthavetheirheadspointingdownwards’,‘InAustralia, 

mammalslayeggs’,‘InAustralia,swansareblack’?If‘InAustralia’ 

canthusbehavelike‘not’(...),perhapsthetildemeans‘InAustralia’? 

2 

IsparaconsistentnegationjustanIn-Australiaoperator?Inathoroughly 

arguedessay,CatarinaDutilhNovaeshasrecentlysuggestedthat,critics 

notwithstanding,therealphilosophicalchallengeforparaconsistentlogics 

doesnotconsistinprovidingaplausibleaccountfornegation,butforthe 

notionofcontradiction. 3 Attackstoparaconsistencydeliveredbyclaiming 

thatparaconsistentnegationisnotnegation,accordingtoDutilhNovaes, 

“canbeneutralizedifitisshownthattheconflationbetweencontradiction 

andnegationisnotlegitimate,”andthat“paraconsistentnegationisin 

principleasrealanegationasanyother;” 4 for,asthenicesurveyofthe 

historyoflogicalnegationprovidedinherpapershows,thereisnounique 

realnegationaround. 5 

Onthecontrary,itisthenotionofcontradictionwhichspellstroublefor 

(strong)paraconsistentists.Theconceptofcontradiction“canbe[defined] 

withoutusingthenegation: Aand Barecontradictorypropositionsiff 

A ∨ Bholdsand A ∧ Bdoesnothold,regardlessoftheformof Aand B.” 

Therefore“contradictionisthepropertyofapairofpropositionswhich 

cannotbothbetrueandcannotbothbefalseatthesametime;”sincetwo 

propositionsthatarecontradictoriesaccordingtoclassicallogiccanboth 

betrueaccordingtothe(strong)paraconsistentist,oneconcludesthatthe 

lattersimplyrejectstheclassicalnotionofcontradiction.So“paraconsistent 

logiciansmustgiveanaccountofwhatcontradictionamountstowithina 

paraconsistentsystem.” 6 

Astrongparaconsistentistmayobjecttothecharacterizationofthenotionofcontradictionjustgiven,forthedefinitionusesanegationinthe 

definiens:contradictories“cannotbothbetrueandcannotbothbefalse.” 

Nowisthat“not”aclassicaloraparaconsistentnegation?(Strong)paraconsistentlogicianssuchasGrahamPriestprefertoassertthatnegation 

2 (Smiley,1993,p.17). 

3 See(DutilhNovaes,2007). 

4 (DutilhNovaes,2007,pp.479and482). 

5 Onthis,seealso(Wansing,2001). 

6 (DutilhNovaes,2007,pp.479and483).


isacontradictory-formingoperator,butdefinecontradictorinesswithout 

adoptinganegationinthedefiniens: Aand Barecontradictoriesiff,if A 

istruethen Bisfalse;andif Aisfalsethen Bistrue.Manyparaconsistent 

negations,then,turnouttobecontradictory-formingoperators;forinsuch 

logicsasLP(Priest’sLogicofParadox) 7 andFDE(BelnapandDunn’sFirst 

DegreeEntailment),negationactuallyisanoperatorthattruth-functionally 

switchestruthandfalsity:if Aistrue,then ¬Aisfalse;if Aisfalse,then 

¬Aistrue;if Aisbothtrueandfalse,then ¬Ais,too(andifthesemanticsadmitstruth-valuegaps,wemayalsohavethat 

Aisneithertruenor 

false;then, ¬Aisneithertruenorfalse,too). Nowthedebatehasbeen 

movedbacktothenotionsoftruthandfalsity:dotheyoverlap?Cansome 

truth-bearerbearboth? 

Ifwewanttohaveanon-question-beggingdebateondialetheiasand 

theLNC,insteadofconcentratingontruthandfalsitywemaygobackto 

negation.Or,atleast,thisisthewaypursuedinthispaper.Musttherebe 

auniquegoodaccountofnegation? Perhaps,asDutilhNovaesforcefully 

argues,not. Wemayhavedistinctintuitionsondifferentsententialand 

predicatenegations,whichmaybecharacterizedbydifferenttheories.This 

doesnotentail,though,thatnonon-question-beggingdebateisfeasible.On 

thecontrary,Ithinkitispossibletocharacterizeanegation(Ishalllabelit 

“NOT”)withthefollowingpleasantfeatures: 

1.itsdefinitiondoesnotrefertothecontentiousconcepttruth; 

2.ithasastrongpre-theoreticalmotivation,becauseofitsindispensable 

expressivefunctioninlanguageandcommunication;and 

3.itisfullyacceptedalsobydialetheists,becauseitisbasedonadeep 

metaphysicalintuitiontheyshowtofullyshare: theintuitionofexclusion. 

IfthecharacterizationofNOTproposedinthefollowingissufficientto 

conferadeterminatemeaningtothenegationinquestion,wecanconvenientlyphraseaformulationoftheLNCviasuchanegation. 

ThisLNC 

mightbeindisputablealsofromthedialetheist’spointofview. “Indisputable”shouldbeunderstoodinthefollowingsense: 

thedialetheistis 

forcedtoacceptit,withoutalsoacceptingsomethinginconsistentwithit. 

ItmightbeaversionoftheLNConwhichboththeorthodoxfriendand 

thedialetheicfoeofconsistencycanagreeinthissense. 

7 See(Priest,1979),(Priest,1987).


2 TheExclusionProblemandPriest’sPragmaticWayOut 

Iwillstartwithaproblemfacingstrongparaconsistency,whichhasbeen 

variouslyrecognisedintheliterature. Ibelieveittobethemaintheoreticaltroublefordialetheism,andIhaveelsewhereproposedtocallitthe 

ExclusionProblem. 8 Itgoesasfollows. 

Whenyousay:“A”,andadialetheistreplies:“¬A”,shemightnothave 

managedtoruleoutwhatyouhavesaid,preciselybecauseofthefeatures 

ofherparaconsistentnegation. Inthedialetheicframework, ¬Adoesnot 

ruleout Aonlogicalgrounds: itmaybethecaseboththat Aandthat 

¬A,sothedialetheistmayacceptthemboth.Alsosaying“Aisfalse,”and 

even“Aisnottrue,”neednotruleout Aonthedialetheist’sside.Inmany 

paraconsistentlogics,beginningwithLP,givenanysetofsentences S,it 

islogicallypossiblethateverysentenceof Sistrue. Thishappensinthe 

so-calledtrivialmodelofLP:ifallatomicsentencesarebothtrueandfalse, 

thenallsentences(truth-functionally)are. Inanutshell:nothingisruled 

outonlogicalgroundsonlyinthedialetheicframework.Manyauthorshave 

inferredthatdialetheismfacestheriskofendingupinexpressible. 9 

AccordingtoPriest,though,thesetroubleswithrulingoutthingscan 

besolvedbyturningintotherealmofpragmatics. Inordertohelpthe 

dialetheistruleoutsomething,hehasprovidedaninterestingtreatmentof 

thenotionofrejection. Letuscallacceptanceandrejectiontwomental 

statesasubject xhastowards(thepropositionexpressedby)asentence. 

Acceptanceandrejectionarepolaropposites:torejectsomethingistopositivelyrefusetobelieveit. 

Assertionanddenial,ontheotherhand,are 

(typically)linguisticactsor,equivalently,illocutionaryforcesattachedto 

utterances.Roughly,assertionanddenialarethelinguisticcounterpartsof 

acceptanceandrejection.Acceptanceandassertion,and,respectively,rejectionanddenial,areoftenconflatedbyphilosophers,andanywayformost 

ofourpurposeswecanrunlinguisticactsandthecorrespondingmental 

statestogether.Let’shavetwosententialoperators,“⊢x”and“⊣x”,whose 

readingis,respectively,“rationalagent xaccepts/asserts(that)”and“rationalagent 

xrejects/denies(that).” Thestandardtreatmenthasitthat 

rejection/denialisequivalenttotheacceptance/assertionofnegation: 

⊣x A ↔⊢x ¬A. (1) 

Ifweunderstanditintermsoflinguisticacts,(1)istheclaim,famously 

heldbyFregeandPeterGeach,accordingtowhichtodenysomethingjust 

istoassertitsnegation. ButPriestsaysthataccepting ¬Aisdifferent 

fromrejecting A: adialetheistcandotheformerandnotthelatter— 

8 See(Berto,2006),(Berto,2007,Ch.14). 

9 See,e.g.,(Parsons,1990),(Batens,1990),(Shapiro,2004).


exactlywhenshethinksthat Aisparadoxical.When Aisadialetheia,the 

naturalassumption(1)breaksdown,andnegationanddenialcomeapart. 

Adenial/rejectionof Abecomesanon-derivativementalorlinguisticact,in 

thatitisdirectlyaimedat A(oratthecontentof A,orattheproposition 

expressedby A,etc). 10 

Giventhat(1)canfail,thedialetheistcanacceptboth Aand ¬A,butshe 

doesnotneedtoacceptandreject A.Actually,accordingtoPriestshecannotevendothat:Priestconsidersacceptanceandrejectionasreciprocally 

incompatible,eventhough Aand ¬Aarenot: 

Someonewhorejects Acannotsimultaneouslyacceptitanymore 

thanapersoncansimultaneouslycatchabusandmissit,orwina 

gameofchessandloseit.Ifapersonisaskedwhetherornot A,he 

canofcoursesay‘Yesandno’.Howeverthisdoesnotshowthathe 

bothacceptsandrejects A. Itmeansthatheacceptsboth Aand 

itsnegation.Moreoverapersoncanalternatebetweenacceptingand 

rejectingaclaim. Hecanalsobeundecidedastowhichtodo. But 

dobothhecannot. 11 

Andthisishowthedialetheistcanmanagetoruleoutsomething,and 

toexpressthis.Althoughtheshecannotruleout Abysimplysaying“¬A”, 

shecanreject A.Sothepragmaticincompatibilityofacceptance/assertion 

andrejection/denialplaysapivotalroleinPriest’sreplytotheExclusion 

Problem. 

3 NOT 

Thisshowsthatevendialetheistshaveanintuitionofexclusion,orincompatibilitybetweensomethingandsomethingelse.SoIproposetosearchfor 

anoperator(arguably,anegation)thatallowsustocaptureandexpressthe 

intuition.Weneedtostartfromthisverynotionpreciselybecausewewant 

toavoidexplicitlyemployingtheconceptsoftruthandfalsitytocharacterizesuchanoperator.Thedialetheistcastsdoubtsontheirbeingexclusive, 

bypointingoutthatsometruth-bearers,notably,theLiars,fallunderboth 

conceptssimultaneously.Truthtablesortruthconditionsfornegationcan 

giveusnosenseoftheconnectionbetweennegationandexclusionunless 

wealreadysharetheintuitionthattruthandfalsityruleouteachother. 

Thisbringsusbacktotheissueofthenotionofcontradiction,raisedby 

DutilhNovaes.Specifically,weshouldrefrainfromexpressingexclusionvia 

thetraditionalconceptofcontrariness: defining Aand Bascontrariesiff 

“A∧B”islogicallyfalsewon’thelpwhendiscussingwiththedialetheist.But 

10 See(Priest,2006,p.104). 

11 (Priest,1989,p.618).


onemaytrywiththeintuitivenotionofexclusionitself,takenasprimitive. 

Animalsandinfantsperceiveincompatibilitiesintheworldlongbeforethey 

havedevelopedormasteredanarticulatedlanguagetoexpressthem.One 

oftheusesofalinguisticitemthatcountsasanegationcantherefore 

beexplained,asHuwPricehasclaimed,as“initiallyameansofregistering 

(publiclyorprivately)aperceivedincompatibility.”Andif“incompatibility 

[is]averybasicfeatureofaspeaker’s(orproto-speaker’s)experienceofthe 

world,” 12 thenonecanexplainthenegationwearelookingforinterms 

ofincompatibility. Weonlyneedtoassumethatordinaryspeakersand 

rationalagentshavesomeacquaintancewithexclusions—thingsofthe 

worldrulingouteachother:theycanrecognizethemintheworld,andin 

theircommercewiththeworld. 

Ishalltalkofmaterialexclusionor,equivalently,ofmaterialincompatibility. 

Onemaycharacterizeitintermsofconcepts,properties,states 

ofaffairs,propositions,orworlds,dependingonone’smetaphysicalpreferences. 

13 Materialexclusionbearsthisnametostressthefactthatitisnot 

amerelylogical,inthesenseofformal,notion: itisbasedonthematerialcontentoftheinvolvedconcepts,orproperties,etc. 

Someexamples: 

phenomenologicalcolourincompatibilities,suchasbeing(solidly)Redand 

being(solidly)Green;conceptsthatexpressourcategorizationofphysical 

objectsinspaceandtime,suchas xbeinghererightnowand xbeingway 

overthererightnow,forasuitablysmall x. 14 Or xbeinglessthantwo 

incheslongand xbeingmorethanthreefeetlong. 15 ButalsoPriest’sabove 

x’scatchingthebusand x’smissingthebuswilldo. 

Ok,thiswastheintuition. Howdoweformalizeit? Afeasibleformal 

accountmayadapttheideadevelopedbyMichaelDunnthat“onecan 

definenegationintermsofoneprimitiverelationofincompatibility(...)in 

ametaphysicalframework.” 16 Soletustalkintermsofpropositions(that 

whichisexpressedbyasentence)andbuildasmallalgebra. Thinkofa 

structure 〈U, ⊂,V, •〉where Uisasetofpropositions; ⊂and •arebinary 

relationsdefinedon U;and Visaunaryoperationonsubsetsof U. ⊂isto 

bethoughtofasapre-order,and“p ⊂ q”canbereadas“Theproposition 

pentailstheproposition q”.Givenasetofpropositions P ⊆ U,VPisthe 

12 (Price,1990,pp.226–228). 

13 Forinstance,wemayviewitastherelationthatholdsbetweenacoupleofproperties P1 

and P2iff,byhaving P1,anobjecthasdismissedanychanceofsimultaneouslyhaving P2. 

Orwemayalsoclaimthatmaterialincompatibilityholdsbetweentwoconcepts C1and 

C2ifftheveryinstantiating C1byaputsabaronthepossibilitythataalsoinstantiates 

C2.Orwemaysaythatitholdsbetweentwostatesofaffairs S1and S2ifftheholding 

of S1precludesthepossibilitythat S2alsoholds(inworld w,attime t,etc.). 

14 See(Tennant,2004,p.362). 

15 See(Grim,2004,p.63). 

16 (Dunn,1996,p.9).


(possiblyinfinitary:moreonthissoon)disjunctionofallthepropositions 

in P.And •ispreciselyourprimitiverelationofmaterialexclusion. 

Apropositionmayhaveoneormoreincompatiblepeers: itmayrule 

outawholeassortmentofalternatives. PatrickGrim,forinstance,talks 

abouttheexclusionaryclassofagivenproperty.Theexclusionaryclassof 

aproposition p,then,istheset 

E = {x;x • p}. 

Then NOT-pisnothingbutVE. If Ehasfinitecardinality,then NOT-p 

isjustanordinarydisjunction: q1 ∨ · · · ∨ qnwhere q1,... ,qnareallthe 

membersof E. 

Supposethereareinfinitelymanypropositionsincompatiblewith p.This 

isaheavymetaphysicalassumption,butletusgrantit.Then, NOT-pturns 

outtobeaninfinitarydisjunction. Ifonehas(understandable)problems 

withinfinitarydisjunctions,wecannotavoidquantifyingonpropositions: 

NOT-p =df ∃x(x ∧ x • p). (2) 

Bothinthefinitaryandinfinitarycase,itisclearinwhichsense NOT-p 

isthelogicallyweakestamongthe nincompatibles:itisentailedbyany qi, 

1 ≤ i ≤ n,suchthat qi • p. Onemayexpressthepointviathefollowing 

equivalence: 

x ⊂ NOT-p iff x • p. (3) 

Putting NOT-pfor x,andbydetachment,weget: 

NOTp • p, (4) 

NOT-pisincompatiblewith p. Theright-to-leftdirectionof(3),then, 

tellsusthat NOT-pistheweakestincompatible,i.e.,itisentailedbyany 

incompatibleproposition. 17 

Whichlogicshouldbereadoffthealgebradependsonwhichalgebraic 

postulateswewanttoadd.Dependingonthechoiceswemake, NOTwillbecomepalatableforsomelogicians,eventhoughotherswillbedisappointed. 

Onemayassume,reasonablyenough,that •issymmetric. Butifinthe 

algebraicframework NOTisstipulatedasanoperationofperiodtwo,i.e. 

NOT-NOT-p = p, (5) 

thisislikelytoberejectedbyanintuitionist,thoughnotbymanyparaconsistentlogicians.TheintuitionistmayalsoobjecttothefactthatNOThas 

17 Variationsonthethemeofthecharacterizationofnegationviaincompatibility,and 

onnegationastheminimalincompatible,canbefoundin(Brandom,1994,pp.381ff.); 

(Harman,1986,pp.118-20);(Peacocke,1987).


beendefinedusingotheroperators,whichgoesagainsttheindependenceof 

logicalconstantsinaconstructivistframework(aremarkIowetoFrancesco 

Paoli).Or,ifwemaketheprimafacienaturalassumptionthat: 

If p ⊂ qand x • q, then x • p, (6) 

wecaneasilygetcontraposition. Butsucharesultwouldberejectedby 

thosewhowanttodismisscontrapositiononthebasisofconsiderationson 

theconditional,andalsobysomeparaconsistentists.Differentphilosophicalparties(classicists,intuitionists,paraconsistentists,etc.) 

haveopposed 

viewsonwhatnegationis,whereastheaimhereistoprovideanintuitive 

depictiononwhichallpartiescanagree;thisiswhyIfindformalization 

usefulonlytoacertainextent. 

4 MinimalLNC 

Independentlyofthepossibleadditionalcharacterization, NOThassome 

nicefeatures.First,isnotexplicitlydefinedviatheconcepttruth.AsGrahamPriesthaspointedouttome(incommunication),thismaynotprevent 

truthfromjumpinginagain.Ihavebeenforcedtoadmitthat,givensome 

(albeitdebatable)metaphysicalassumptions,wemayneedpropositional 

quantificationtospelloutthedetailsof NOT. Andsuchquantificationis 

inter-definablewithtruth. Butwhat NOTisexplicitlyreferredtoisthe 

conceptexclusion,whoseprimitivenesshasbeenarguedforabove. 

Secondly, NOThasastrongpre-theoreticalappealasanexclusion-expressingtool: 

itallowsustoruleoutthingsbyclaimingthatsomething 

incompatiblewiththemisgoingon.Thisiswhatatleastsomeoftheitems 

wequalifyasnegationshouldhelpustodo. 

Finally,dialetheistsgraspthenotionofexclusion. Theyaskustostop 

using“not”or“true”asexclusion-expressingdevices,because“not-A”is 

insufficientbyitselftoruleout A,and“Aistrue”isinsufficientbyitselfto 

ruleoutthat Aisalsofalse.ButPriest’saccountofacceptanceandrejection 

showsthatthedialetheistbelievesintheimpossibilityofsomecouplesof 

facts’,orstatesofaffairs’,simultaneouslyobtaining;or,equivalently,that 

sheassumesthatsomethingsmateriallyexcludesomeothers: x’ssimultaneouslycatchingandmissingthebus,forinstance;and,ofcourse, 

x’s 

simultaneouslyacceptingandrejectingthesame A,thisbeing,aswehave 

seen,abasicstepinPriest’sanswertotheExclusionProblem. NOTis 

supposedtoworkeveninaframeworkinwhichnothingisruledouton 

logicalgroundsalone,becauseitisnotmerelylogically,i.e.formally,but 

metaphysically(“materially”)founded. Thedialetheistmayhaveavacuousnotionoflogical,formalincompatibility.Butshedoeshaveanotionof 

materialincompatibility.


Nowforthefinalstep: expresstheLNCvia NOT. TakeAristotle’s 

traditionalformulationoftheLNC,inBook ΓoftheMetaphysics,andjust 

putinitour NOT. Theformulationcanbetakenasadefinitionof“the 

impossible”: 

ForthesamethingtoholdgoodandNOTholdgood 

simultaneouslyofthesamethingandinthesamerespect 

isimpossible. 18 

“P1does NOTholdgoodof x”shouldbeashortformfor“to xbelongs 

someproperty P2,whichismateriallyincompatiblewith P1.” Thisdoes 

notseemtobequestionablebythedialetheistanymore,providedshehas 

understood NOT—andtounderstand NOTistounderstandexclusion. 

Ifthedialetheistrefusestosubscribetothecharacterizationof NOTvia 

theintuitivenotionofexclusion,sheseemstoactuallyendupasunableto 

expresstheexclusionofanyposition(isshetryingtoexcludeexclusion?). 

AndadialetheismwithouttheLNCstatedintermsof NOTlooksvery 

muchlikeatrivialism(ItotallyagreewithDutilhNovaes,whopressesa 

pointverysimilartothisoneinheressay). 19 SuchaLNC,touseAristotle’s 

words,is“aprinciplewhicheveryonemusthavewhoknowsanythingabout 

being.” 20 

DoesthisreplytoDutilhNovaes’challengefor(strong)paraconsistency, 

namely,thatofdefining“P-contradictions,thatis,contradictionsthatare 

sothreateningtoatheorythattheyreallycompromiserationalinferencemakingwithinit”? 

21 Tosomeextent,yes—ifthestepsoftheargumentationproposedabovework. 

ButthesuccessfortheversionoftheLNC 

phrasedintermsofNOTisverylimited:forthatLNCsimplyrulesoutthe 

simultaneousobtainingofreciprocallyexclusionarystatesofaffairs. The 

questionremainsopenofwhicharetheexclusionarystatesofaffairs(or 

properties,etc.). Andnowthediscussionbetweendialetheistsandantidialetheistscandevelopwithasignificantdecreaseinissuesofquestionbeggingandclashesofintuitions.Whatisincompatiblewithwhat?Given 

twoproperties P1and P2,thequestionwhethertheyareexclusivecaninvolvebroadlyempiricalmatters,difficultanalysesofourconceptualtoolkit 

and/orofouruseofordinarylanguageexpressions.Somecasesmaybeeasy 

toresolve;butothersmayproducebattlesofintuitions:areyoungandold 

actuallyexclusive?Blueandgreen?Trueandfalse?Circularandsquare? 

Ihaveclaimedthatmaterialexclusionisbasedonthecontentoffacts,concepts,orproperties;buthowdoweknowwhatthecontentofaconcept 

18 SeeAristotleMet.1005b18–21(Aristotle,1984). 

19 See(DutilhNovaes,2007,p.487)). 

20 Arist.Met.1005b14–15(Aristotle,1984). 

21 (DutilhNovaes,2007,p.489). 

(7)


is,orwhicharetheactualfieldsofapplicationsofaproperty?Theformal 

characterizationof NOT,ofcourse,doesnotentailspecialcommitmentson 

whicharethespecificproperties,orconcepts,orstatesofaffairs,between 

whichitholds.Suchcommitmentsarefallible.Wecancometobelievethat 

someproperties,orconcepts,orstatesofaffairs,areincompatible,andthen 

findoutthattheyarenot. Wouldthisentailexplosion,thatis,anything 

beingderivable,andtrivialism? Well,not: thestandardstrategyinthis 

caseissimplytoretractourpreviousassumptionthattheywere. 

Sothedialetheistwhohasnotroubleswithourminimal(7)canstill 

objecttootherformulationsoftheLNC,e.g.,becausetheyarephrased 

intermsoftruthandfalsity: thosewhoruleoutthatanysentencecould 

bebothtrueandfalsetaketruthandfalsityasexclusionaryconcepts;the 

dialetheisthasqualmsonthis,andperhapscounterexamplestooffer(say, 

theLiarsentences). Buttheissueaddressedhereiswhetherallconcepts 

(orproperties,etc.) arelikethat;andthedialetheistagreesthatsome 

concepts(orproperties,etc.) doruleouteachother. Thisistheshared, 

basicintuition NOTappealsto. 

FrancescoBerto 

DepartmentofPhilosophyandTheoryofSciences,UniversityofVenice- 

Ca’Foscari 

Dorsoduro3484D,30123Venice,Italy 

bertofra@unive.it 

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MedievalObligationes as a Regimentation of 

‘the Game of Giving and Asking for Reasons’ 

1 Introduction 

Catarina Dutilh Novaes ∗ 

Medievalobligationesdisputationswereahighlyregimentedformoforaldisputationopposingtwoparticipants,respondentandopponent,andwhere 

inferentialrelationsbetweensentencestookprecedenceovertheirtruthor 

falsity.In(DutilhNovaes,2005),(DutilhNovaes,2006)and(DutilhNovaes, 

2007,Ch.3)Ipresentedaninterpretationofobligationesaslogicalgamesof 

consistencymaintenance;thisinterpretationhadmanyadvantages,inparticularthatofcapturingthegoal-oriented,rule-governednatureofthiskind 

ofdisputationbymeansofthegameanalogy.Italsoexplainedseveralofits 

featuresthatremainedotherwisemysteriousinalternativeinterpretations, 

suchastheroleofimpertinentsentencesandwhy,whilethereisalwaysa 

winningstrategyforrespondent,thegameremainshardtoplay.However, 

thelogicalgameinterpretationdidnotprovideafullaccountofthedeontic 

aspectofobligationes—ofwhatbeingobligedtoacertainstatementreallyconsistsin—beyondthegeneral(andsuperficial)commitmenttowards 

playing(andwinning)agame.Afterall,theverynameinvokesnormativity, 

soaninterpretationofobligationesthatdoesnotfullyaccountforthedeonticcomponentseemstobemissingacrucialaspectofthegeneralspiritof 

theenterprise.Inordertoamendthisshortcominginmypreviousanalysis 

Iherepresentanextensionofthegame-interpretationbasedonthenotion 

of‘thegameofgivingandaskingforreasons’—henceforth,GOGAR 1 — 

presentedinChapter3ofR.Brandom’sMakingitExplicit(Brandom,1994) 

asconstitutingtheultimatebasisforsociallinguisticpractices.Thebasic 

∗ ThankstoEdgarAndrade-LoteroandOleThomassenHjortlandforcommentsonan 

earlierdraftofthepaper. 

1 FollowingJ.MacFarlane’sterminology,cf.http://johnmacfarlane.net/gogar.html.

36 CatarinaDutilhNovaes 

ideaisthatobligationescanbeseenasaregimentationofsomeofthecore 

aspectsofGOGAR. 

What is to be gained from a comparison between obligationes and 

GOGAR?Fromthepointofviewofthelatter,thecomparisoncanshed 

lightonitsgenerallogicalstructure:ifobligationesreallyarearegimentationofGOGAR,thentheycancertainlycontributetomakingitsstructure 

explicit(whichisofcourseanothercrucialelementofBrandom’sgeneral 

enterprise).Indeed,anobligatioissomethingofaSprachspielforGOGAR, 

asimplifiedmodelwherebysomeofGOGAR’spropertiescanbemademanifest.Asforobligationes,whatcanbegainedfromthecomparison,besidestheemphasisonitsfundamentallydeonticnature,isabetterunderstandingofitsgeneralpurpose. 

Atfirstsight,thishighlyregimentedformof 

disputation,wheretruthdoesnotseemtohaveanymajorroletoplay,may 

seemlikesterilescholasticlogicalgymnastics. ButifitisputinthecontextofGOGAR—which(presumably)capturestheessenceofoursocial, 

linguisticandrationalbehaviors—thenitssignificancewouldappeartogo 

wellbeyondthe(mere)developmentoftheabilitytorecognizeinferential 

relationsandtomaintainconsistency. 

2 GOGAR 

AcrucialelementofthephilosophicalsystempresentedbyBrandominMakingitExplicit(andfurtherexpandedinseveralofhissubsequentwritings)is 

themodeloflanguageusethathereferstoas‘thegameofgivingandasking 

forreasons’.Brandominsiststhatlanguageuseandlanguagemeaningfulnesscanonlybeunderstoodinthecontextofsocialpracticesarticulating 

informationexchangeandactions—linguisticspeech-acts(typically,the 

makingofaclaim)aswellasnon-linguisticactions. 

Infact,GOGARshouldaccountforwhatmakesussocial,linguisticand 

rationalanimals. AsBrandomconstruesit,GOGARisfundamentallya 

normativegameinthattheproprietyofthemovestobeundertakenby 

theparticipantsisatthecentralstage.Itis,however,notatranscendental 

kindofnormativity,requiringanalmightyjudgeoutsidethegametokeep 

trackofthecorrectnessofthemovesundertaken;rather,theparticipants 

themselvesareinchargeofevaluatingwhetherthemovesundertakenare 

appropriate. Itisa“deonticscorekeepingmodelofdiscursivepractice”. 

InGOGAR,weareallplayers(speakers)andscorekeepersconcomitantly; 

weundertakemovesandkeeptrackofeverybody’smoves(includingour 

own)atthesametime. Thefocuson(givingandaskingfor)reasonsis 

animportantaspectofhowthemodelcapturestheconceptofrationality: 

weareresponsiblefortheclaimswemake,andthusmustbepreparedto

ObligationesasGOGAR 37 

providereasons 2 forthemwhenchallenged. Underlyingthisfactisthe 

ideaofalogicalarticulationofcontentssuchthatsomecontentscountas 

appropriatereasonsforothercontents. 

Inprinciple,asageneralmodeloflanguageuse,GOGARshouldencompassalldifferentkindsofspeech-acts: 

assertions,questions,butalso 

promises,orders,expressionsofdoubtetc.However,forBrandomthereis 

onefundamentalkindofspeech-actinthegame,namelythatofmaking 

anassertion. 3 Anassertionisbothsomethingthatcancountasareason 

(ajustification)foranotherassertionandsomethingthatmayconstitute 

achallenge—typically,whenaspeakerSmakesanassertionincompatiblewithsomethingpreviouslysaidby 

T—andthusprovoketheneedfor 

furtherreasons(Tmustdefendtheoriginalassertion): hence,givingand 

askingforreasons. 

Theneedtodefendone’sassertionsthreatenedbychallengesthrough 

furtherreasonsindicatesthatoneissomehowresponsibleforone’sassertions.ThisisindeedthecaseaccordingtotheGOGARmodel,andthisfact 

isaccountedforbytheabsolutelycrucialconceptofdoxasticcommitment. 

Justasapromisecreatesthecommitmenttofulfillwhathasbeenpromised, 

themakingofanassertioncreatesthecommitmenttodefendit,i.e.,tohave 

hadgoodreasonstomakeit.Thisisbecauseoneoftenreliesontheinformationconveyedbyanassertionmadebyanotherpersoninordertoassessaparticularsituationandthenactupontheassessment;butiffalseinformationistransmitted,thentheassessmentwillprobablybemistaken,and 

theactioninquestionwillprobablynothavethedesiredoutcome;itmay 

evenhavedeleteriousconsequencesfortheagent.Insuchcases,itisfairto 

saythatthepersonhavingconveyedtheincorrectinformationisresponsible 

fortheinfelicitousoutcome,justasarecklessdriverisresponsibleforthe 

accidentshe/she(directlyorindirectly)causes.Ifsomebodyshouts‘fire!’as 

aprankinacompletelyfullstadium,forexample,thiswillprobablycause 

considerablemayhem,andtheinfelicitousjokerwillbeheldaccountablefor 

allthedamagecaused.Sogiventhepotentialpracticalconsequencesofan 

assertion,itisnotsurprisingatallthatliabilityshouldbeinvolvedinthe 

makingofanassertion. 

ForBrandom,thecommitmenttothecontent 4 ofanassertioninfactgoes 

beyondtheassertionitself:oneisalsocommittedtoeverythingthatfollows 

fromtheoriginalassertion,i.e.,everythingthatcanbeinferredfromit.The 

2 Etymologically,rationalitycomesfromratio,‘reason’inLatin. 

3 “Thefundamentalsortofmoveinthegameofgivingandaskingforreasonsismaking 

aclaim—producingaperformancethatispropositionallycontentfulinthatitcanbe 

theofferingofareason,andreasonscanbedemandedforit.”(Brandom,1994,p.141). 

4 ItisnotentirelycleartomethoughwhetherBrandomseescommitmentsashaving 

contentsorsentencesorclaimsastheirobjects,butitseemstomethatcontentswould 

bethemostappropriateobjectsofcommitments.


inferentialrelationsbetweenassertionsareaprimitiveelementofBrandom’s 

system(codifiedintermsofmaterialinferences,notformalones);ashe 

sometimessays,theyare“unexplainedexplainers”(Brandom,1994,p.133). 

Materialinferencesarepainstakinglydiscussedin(Brandom,1994,Ch.2), 

butforourpurposeswhatisimportantistorealizethatcommitmenttoa 

contenttransfersovertoothercontentsbymeansofinferentialrelations. 

Butbesidesbeingcommittedtocontents,thereisanotherprimitivedeonticstatusthataspeakermayormaynotenjoywithrespecttocontents: 

entitlement.Fromthepointofviewofascorekeeper, 5 foraspeaker Stobe 

entitledtoassertingagivencontentamountsto Sbeinginthepositionto 

offergroundsthatjustifybeliefinthecontent,andthusthemakingofthe 

correspondingassertion;thisdeonticstatusisattributedwhenthespeaker 

hasgood(enough)reasonstobelievethecontenttobethecase.Brandom 

remarksthat“commitmentandentitlementcorrespondtothetraditional 

deonticprimitivesofobligationandpermission”(Brandom,1994,p.160); 

herejectsthisterminologybecausehewishestoavoidthestigmataofnorms 

associatedwithhierarchyandcommands(asnotedabove,thescorekeeping 

isdonehorizontallybyallparticipants). Butultimately,acommitmentis 

indeedanobligation,andanentitlementisindeedapermission,andthus 

beingcommittedtoacontentamountstobeingobligedtoitinexactlythe 

samesenseofbeingobligedduringanobligatiodisputation(asweshall 

see):onehasadutytowardsacertaincontent,whichtransfersovertoall 

thecontentsthatfollowfromit. 

Fromthetwoprimitiveconceptsofcommitmentandentitlement,Brandomderivestheequallyimportantconceptofincompatibility: 

content p 

beingincompatiblewithcontent qamountstocommitmentto pprecluding 

entitlementto q.Itisnotsomuchthatitisfactuallyimpossibleforoneto 

becommittedto pwhilebelievingoneselftobeentitledto q;thiscanoccur, 

justasonecanmakeconflictingpromisesandholdinconsistentbeliefs.But 

again,thisisamatterofdeonticscorekeeping: fromthepointofviewof 

thescorekeepers,ifaspeakeriscommittedto pthereisawholeseriesof 

contents q, t,etc.towhichthespeakerinquestionissimplynotentitledas 

longashemaintainshiscommitmentto p.Butifheneverthelessinsistsin 

beingcommittedto pandentitledto qatthesametime,thenheissimply 

makingabadmovewithinGOGAR. 

Brandomcorrectlynoticesthatincompatibility,asmuchasentailment, 

isessentiallyarelationbetweensetsofcontents,notbetweencontentsthem- 

5 Thedeonticstatusesofcommitmentandentitlementarealwaysperspectival,i.e.definedbythedeonticattitudesof(self-)attributingcommitmentsandentitlementsofeach 

scorekeeper.“Suchstatusesarecreaturesofthepracticalattitudesofthemembersofa 

linguisticcommunity—theyareinstitutedbypracticesgoverningthetakingandtreating 

ofindividualsascommitted.”(Brandom,1994,p.142).


selves. 6 Takeasetofthreecontents,e.g.,thoseexpressedbythesentences 

‘Everymanisrunning’,‘Socratesisaman’and‘Socratesisnotrunning’. 

Commitmenttoeitheroneofthetwofirstcontentsalonedoesnorprecludeentitlementtothethirdcontent,butcommitmenttobothofthem 

doesprecludeentitlementtothethird,justascommitmenttothefirsttwo 

contentssimultaneouslyentailscommitmenttothecontent‘Socratesisrunning’. 

Thisaspectwillbesignificantforthecomparisonwithobligationes 

lateron,asithintsatthefundamentallydynamicnatureoftheGOGAR 

model:everynewassertionmaderequirestherecalibrationofeverybody’s 

deonticstatusesbythescorekeepers—oftheasserter,inparticular,butin 

factofeverybodyelseaswell,asGOGARalsoaccountsforinter-personal 

transmissionofentitlementbytestimony.Inotherwords,aspeaker’sdeonticstatus—hercommitmentsandentitlements—ismodifiedeverytimean 

assertionismade,moresalientlybutnotexclusivelybythespeakerherself. 

Indeed,thereseemtobefourmainsourcesofentitlementaccordingto 

theGOGARmodel. 

1.Interpersonal,intracontentdeferentialentitlement: Speaker 1isentitledto(asserting)content 

pbecausespeaker 2,areliablesource, 

asserted p. 

2.Intrapersonal,intercontentinferentialentitlement: Speaker 1isentitledto(asserting) 

qbecausesheisentitledto(asserting) pand p 

entails q. 

3.Perception:Speaker 1isentitledto(asserting) pbecauseshehashad 

a(reliable)perceptualexperiencecorrespondingto p. 

4.Defaultentitlement: ‘freemoves’,thecontentsentitlementtowhich 

issharedbyallspeakersinsofarasthesecontentsconstitutecommon 

knowledge—everybodyknowsit,andeverybodyknowsthateverybodyknowsit. 

AfinalpointIwishtoaddressinmybriefpresentationofGOGAR 

isthenotionofinference,morespecificallymaterialinference. Brandom 

criticizestheformalistviewofinference,accordingtowhicheveryvalid 

inferenceisaninstanceofaformallyvalidschema;rather,theinferential 

relationsthataretheprimitiveelementsofhisinferentialsemanticsareofa 

conceptualnature,whilealsofirmlyembeddedinpractices:“Inferringisa 

kindofdoing.”(Brandom,1994,p.91)Thefocusonthenotionofmaterial 

inferencealsoechoesimportantfeaturesofobligationes,asinthelatter 

6 (Brandom,2008,Lect.5). Thecasesofrelationsinvolvingsinglecontentscanbeseen 

aslimit-cases,relatingsingletonsets.


frameworktherelationof‘following’(sequitur)inquestionisnotrestricted 

toformallyvalidschemata. 7 

3 Medievalobligationes 

Anobligatiodisputationhastwoparticipants,OpponentandRespondent. 

Inthecaseofpositio,themostcommonandwidelydiscussedformofobligationes,thegamestartswithOpponentputtingforwardasentence,usuallycalledthepositum,whichRespondentmustacceptforthesakeofthedisputation,unlessitiscontradictoryinitself. 

Opponentthenputsforward 

othersentences(theproposita), oneatatime,whichRespondentmust 

eithergrant,denyordoubtonthebasisofinferentialrelationswiththe 

previouslyacceptedordeniedsentences—or,incasetherearenone(and 

thesearecalledimpertinent 8 sentences)onthebasisofthecommonknowledgesharedbythosewhoarepresent.Inotherwords,ifRespondentfails 

torecognizeinferentialrelationsorifhedoesnotrespondtoanimpertinent 

sentenceaccordingtoitstruth-valuewithincommonknowledge,thenhe 

respondsbadly.Respondent‘losesthegame’ifheconcedesacontradictory 

setofpropositions.ThedisputationendsifandwhenRespondentgrantsa 

contradiction,orelsewhenOpponentsays‘cedattempus’,‘timeisup’.Opponentandpossiblyalargerpanelofmasterspresentatthedisputationare 

inchargeofkeepingtrackofRespondent’srepliesandofevaluatingthem 

oncethedisputationisover. 

Anobligatiodisputationcanberepresentedbythefollowingtuple: 

Ob = 〈KC,Φ,Γ,R(φn)〉 

KCisthestateofcommonknowledgeofthosepresentatthedisputation. 

Φisanorderedsetofsentences,namelytheverysentencesputforward 

duringthedisputation. Γisanorderedsetofsetsofsentences,whichare 

formedbyRespondent’sresponsestothevarious φn. Finally, R(φn)isa 

functionfromsentencestothevalues 1, 0,and ?,correspondingtotherules 

Respondentmustapplytoreplytoeach φn. 

Therulesforthepositumare 

• R(φ0) = 0iff φ0 � ⊥, 

• R(φ0) = 1iff φ0 � ⊥. 

7 Indeed,theterminologyofformalvs.materialconsequences,fromwhichtheterminology 

usedbyBrandom(directlyborrowedfromSellars)ultimatelyderives,wasconsolidated 

inthe 14 th century;see(DutilhNovaes,2008). 

8 Throughoutthetext,Iwillusetheterms‘pertinent’and‘impertinent’,theliteraltranslationsoftheLatinterms‘pertinens’and‘impertinens’. 

Butnoticethattheyareoften 

translatedas‘relevant’and‘irrelevant’,forexampleinthetranslationofBurley’streatise 

(Burley,1988).


Therulesforthepropositaare 

•Pertinentpropositions: Γn−1 � φnor Γn−1 � ¬φn; 

–If Γn−1 � φnthen R(φn) = 1; 

–If Γn−1 � ¬φnthen R(φn) = 0; 

•Impertinentpropositions: Γn−1 � φnand Γn−1 � ¬φn; 

–If KC � φnthen R(φn) = 1; 

–If KC � ¬φnthen R(φn) = 0; 

–If KC � φnand KC � ¬φnthen R(φn) =?. 

Asthedisputationprogresses,differentsetsofsentencesareformedat 

eachround,namelythesetsformedbythesentencesthatRespondenthas 

grantedandthecontradictoriesofthesentenceshehasdenied.Thesesets 

Γncanbeseenasmodelsofthesuccessivestagesofdeonticstatusesof 

Respondentwithrespecttothecommitmentsundertakenbyhimateach 

reply.Thesets Γnaredefinedasfollows: 

•If R(φn) = 1then Γn = Γn−1 ∪ {φn}; 

•If R(φn) = 0then Γn = Γn−1 ∪ {¬φn}; 

•If R(φn) =?then Γn = Γn−1. 

Forreasonsofspace,Ishallkeepmypresentationofobligationesvery 

brief. Theinterestedreaderisurgedtoconsultthevastprimaryandsecondaryliteratureonthetopic, 

9 butfurtheraspectsoftheframeworkwill 

bediscussedinthenextcomparativesectionsaswell. 

4 Comparison 

InthissectionIundertakeasystematiccomparisonofthetwoframeworks. 

Theemphasiswillbelaidonsimilarities,butIwillalsomentionsomeimportantdissimilarities.Essentially,whatisatstakeduringanobligatiodisputationistheabilitytoappreciatethe(logicalandpractical)consequences 

ofthecommitmentsundertakenbyRespondent. Respondediscommitted 

(i.e.obligated)tothesentenceshegrantsaswellastothecontradictories 

ofthesentenceshedenies. Thedeonticstatusofentitlementplaysaless 

prominentrolewithinobligationes,asthepointreallyistoexplorewhat 

oneisobligatedtoonceoneobligatesoneselftothepositum. Besidesthis 

9 Myownpreviouswork(DutilhNovaes,2005),(DutilhNovaes,2006),(DutilhNovaes, 

2007)canserveasastartingpoint.


generalandfundamentalpointofsimilarity,thereareseveralspecificsimilaritiesbetweenGOGAR’sandobligationalconcepts: 

10 

Thekeyroleofinferentialrelations Inbothmodels,(intra-personal, 

inter-content)transferofcommitmenttakesplacethroughinferentialrelations,notrestrictedtoformallyvalidinferences.Bymeansofthetransferof 

commitment,Respondentisobligatedtoeverythingthatfollowsfromwhat 

hehasgranted/deniedsofar,aswellastothecontradictoriesofwhatis 

incompatible(repugnans)withwhathehasgranted/deniedsofar.Indeed, 

thenotionof‘repugnant’sentencescorrespondspreciselytoBrandom’snotionofincompatibility. 

Therelationofinferencerelatessetsofsentences/contents Both 

frameworkscorrectlytreattherelationofinference(andthecorresponding 

transferofcommitment)asrelatingsetsofsentencestosetsofsentences(althoughusuallytheconsequentsetisasingleton). 

Indeed,withinrational 

discursivepractices,whatcountsarenotsomuchtheinferentialrelations 

betweenindividualsentences/contents; asamatteroffact,weareusuallycommittedtoawiderangeofsentences/contents.Itistheinteractionbetweenthesedifferentcommitmentsthatcountstodefineourfurthercommitments:whatveryoftenhappensisthatcommitmentto 

paloneorto q 

alonedoesnotcommitthespeakerto t,butjointcommitmentto pandto 

qdoes.Intheobligationalframework,everypropositumthatisgrantedor 

deniedmodifiesRespondent’scommitments. 

Thedynamicnatureofbothmodels Acorollaryofthepreviouspoint 

isthatbothmodelsaredynamic,i.e.,temporalityisanimportantfactor. 

In(DutilhNovaes,2005),Ihaveexploredindetailthedynamicnatureof 

obligationes,andGOGARisdynamicinverymuchthesameway. Both 

modelsdealwithphenomenathattakeplaceinsuccessivesteps,andeach 

stepistosomeextentdeterminedbytheprevioussteps(afeaturethat 

isaccuratelycapturedbythegamemetaphor). Inbothcases,theorder 

ofoccurrenceofthesestepsiscrucial. Forexample,ifthepositumofan 

obligatiois‘Everymanisrunning’,andthenextstepis‘Youarerunning’, 

thispropositum mustbedeniedasimpertinentandfalse(sincenothing 

hasbeensaidaboutRespondentbeingamansofar).However,ifafterthe 

samepositum‘Youareaman’isproposedandaccepted(asimpertinentand 

true),andafterward‘Youarerunning’isproposed,thenthelattershould 

10 Forreasonsofspace,Iheretreatonlythemostsalientpointsofsimilarity. Notice 

thoughthatthereareothers,forexampletheroleplayedbypragmaticelementsinboth 

cases.


beacceptedasfollowingfromwhathasbeengrantedsofar,contrarytothe 

firstscenario. 

Impertinentpropositionsanddefaultentitlement Eventhoughobligationesdealessentiallywithcommitmentsandlesssowithentitlements, 

onespecifickindofentitlementisneverthelesspresentintheframework. 

WhileRespondent’srepliestopertinentsentencesarefullydeterminedby 

hispreviouscommitments,therearenocommitmentsconcerningimpertinentsentences(asthisisexactlywhattheyare:thusfaruncommitted-to 

contents). Whatmustdeterminehisrepliestoimpertinentsentencesare 

exactlytheuncontroversialentitlementssharedbyallthosewhoarepresent 

atthedisputation.Theseincludecircumstantialinformation(suchasbeing 

inParisorbeinginRome),aswellasverygeneralcommonknowledge,for 

examplethatthePopeisaman.Inotherwords,Respondentisentitledto 

accepting,denyingordoubtingasentenceonthebasisofhisfactualknowledgeconcerningthem;theseareBrandom’s‘freemoves’,withthesame 

socialdimensioninsofarasitconcernscommonknowledge. 

Scorekeeping WithinGOGAR,scorekeepingissomethingofametaphor 

ratherthanareality—nobodyexplicitlywritesdownthecommitments 

andentitlementsofotherspeakers.Scorekeepingisrathersomethingdone 

tacitly,andusuallyoneisnotevenreallyawareofdoingit. Butwithin 

obligationes,scorekeepingisforreal. ThisisexactlywhatImeanwhenI 

saythatthelatterisaregimentedmodeloftheformer:someimplicit,tacit 

elementsofGOGARaremadeexplicitandtangiblewithinobligationes. 

Indeed,thosepresentatthedisputation(inparticularOpponent)explicitlykeepscoreofRespondent’ssuccessivedeonticstatusesofcommitments 

duringadisputation;whenhethenfailstorecognizeapreviouslytaken 

commitment,herespondsbadlyandlosesthegame. Moreover,oncethe 

disputationisover,Respondent’sperformanceisexplicitlyevaluatedbya 

panelofMasterspresentattheoccasion. 

Caveats Whiletheresemblancebetweenthetwoframeworksisoverwhelming,thereareofcourseimportantpointsofdissimilarity.Morespecifically,andasnotedbefore,obligationesisalessencompassingmodel,treatingonlyasubclassofthephenomenacapturedbyGOGAR. 

•Obligationesonlyaccountforthecommitmentsandentitlementsof 

onespeaker,namelyRespondent. 

•Askingforreasonsisnotpartofanobligatio:OpponentcannotchallengeRespondent, 

exceptbysaying‘cedat tempus’ifRespondent 

grantsacontradiction.


•Obligationesoffernoextensivetreatmentofthedifferentkindsofentitlementsandofthemechanismsoftransferofentitlement. 

•GOGARismeanttobeamodeloftheverymeaningfulnessoflanguage—i.e.,therelationsofcommitment-preservingentailmentand 

entitlement-preservingentailmentdefinethemeaningofutterances— 

whereasobligationesoperatewithalanguagethatismeaningfulfrom 

thestart. 

Forthesereasons,anobligatioisbestseenasasimplifiedmodelofhow 

aspeakermustbehavetowardsassertions.Thissimplificationmayonthe 

onehandentaillossofgenerality,butontheotherhanditmayoffera 

viewpointfromwhichsomepropertiesofoursocialdiscursivepracticesare 

mademanifestandcanthusmoreeasilybestudied. 

5 Whatisgainedthroughthecomparison? 

Forobligationes 

Thedeonticnatureofobligationes Eversincescholarsofmedieval 

philosophybecameinterestedinobligationeshalfwaythe 20 th century,the 

verynameofthisformofdisputationwasasourceofpuzzlement.Inwhat 

senseexactlydidsuchadisputationconsistinanobligation? Whowas 

obliged,andwhatwasheobligedto?Althoughsomemodernanalysesdid 

emphasizeitsdeonticnature(see(Knuuttila&Yrjonsuuri,1988)),itisfair 

tosaythatthedeonticcomponentwasessentiallyoverlookedinmostof 

them(includingmyowngameinterpretation). Onapersonalnote,Ican 

saythatIonlyfullyunderstoodhowthoroughlydeontictheobligationes 

frameworkreallywasagainstthebackgroundofGOGAR,andinparticular 

bymeansoftheconceptofcommitment. 

RecallthatIhaveaccountedforthenotionofcommitmenttoastatement/contentintermsofthepracticalconsequencesthattherelianceonitstruthcanhaveforotherpeople’slives,insofarastheyassumethestatementtobetrueunlesstheyhavegoodreasonsnotto(Brandom’s‘default 

entitlement’andLewis’‘conventionoftruthfulnessandtrust’)andinsofar 

astheymakepracticaldecisionsonthebasisontheirrelianceonitstruth. 

Ofcourse,giventhesomewhat‘artificial’settingofanobligatiodisputation,nopracticalconsequencesaretobeexpected.Nevertheless,thebasic 

ideaseemstobethatcommitment—obligation—transfersoverbymeans 

ofinferentialrelations: ifrespondentiscommittedto φnand φnimplies 

φm,thenrespondentisalsocommittedto φm. Now,sincerespondentis 

alwayscommittedtoatleastonestatement,thepositum,thisfirstcommitmentsetsthewholewheelofcommitmentsinmotion.Soanobligatioisnot 

onlyaboutlogicalrelationsbetweensentencesandconsistencymaintenance;


moreimportantly,itisaboutthedeonticstatusesofcommitmentsandentitlementsandthe(intrapersonal,inter-content)mechanismsofinheritance 

ofthesestatuses. 

Thegeneralpurposeofobligationes Morepervasiveandsignificant 

thanthepuzzlementcausedbythetermobligationesitselfisthestillwidespreadperplexityofscholarsconcerningtheverypurposeofsuchdisputations:afterall,what’sthepoint?Whatareobligationesabout?Theyare 

notabouttruth,asmorestraightforwardformsofdisputationare,giventhat 

thepositumisgenerally,andconspicuously,apossiblebutfalsesentence. 

Manyofthemoderninterpretationshavesoughttoestablisharationalefor 

obligationes—alogicofconditionals,aframeworkforbeliefrevision—, 

buttheshortcomingsofeachoftheseinterpretationsonlycontributedto 

thegrowingfrustrationrelatedtotheapparentelusivenessofthe‘point’ 

ofobligationes. Itcouldn’tpossiblybeamereformoftestingastudent’s 

skills,i.e.“schoolboy’sexercise”,assuggestedintheearlysecondaryliteratureofthe1960’s. 

Ifthereisnorealpurposetoitbeyondtheintricate 

logicalstructureoftheframework,thenitmightbemerelysterilescholastic 

logicalgymnasticsafterall,justasmostofthetechniquesofscholasticism 

accordingtothestandardpost-scholastic(i.e.Renaissance)criticism. 

ButwhenputinthecontextofGOGAR,obligationesseemtoprovidea 

modelofwhatitmeanstoactandtalkrationally,i.e.totakepartin(mainly, 

butnotexclusively)discursivesocialpractices. Thusviewed,obligationes 

couldalsomostcertainlyfulfillanimportantpedagogicaltask,namelythat 

ofteachingastudenthowtoarguerationally—i.e.,howtoarguemindfulofone’sentitlementsandcommitments,ofthereasons(grounds)forendorsingorrejectingstatements,andoftheneedtodefendone’sowncommitments—butitsimportanceclearlygoesbeyondmerelypedagogical 

purposes. Interestingly,throughoutthelaterMiddleAgestheformatof 

obligationeswasextensivelyadoptedforscientificinvestigations,precisely 

becauseitprovidesagoodmodelforrationalargumentation. Inawide 

varietyofcontexts(rangingfromlogictotheology,fromethicstophysics), 

oneencountersextensiveuseoftheobligationesvocabularyandconceptsin 

thepresentationofarguments.Thusseen,theframeworkisfarfrombeing 

afutilelogicalexercise:rather,itpresentsaregimentationofsomecrucial 

aspectsofwhatitistoargueandactrationally,ofwhichGOGARisalsoa 

(moreencompassing)model. 

Forthegameofgivingandaskingforreasons 

Underlyinglogicalstructure Whileintermsofthe‘biggerpicture’,it 

ismostlyobligationesthatcanbenefitfromthecomparison,onthelevel 

of(logical)detailGOGARhasmuchtolearnfromobligationes.Eversince


thepublicationof(Brandom,1994),Brandomhasbeenrefiningthelogical 

structureunderlyingGOGARinparticularandhisinferentialistsemantics 

ingeneral,especiallythroughthedevelopmentofwhathecalls‘incompatibilitysemantics’.Nevertheless,anddespitethepowerfulnessofthemodern 

logicaltechniquesoftenemployedbyBrandomandhiscollaborators,these 

fairlyrecentdevelopmentsarestillsomewhatovershadowedbythecenturies 

ofresearch(involvingaverylargenumberoflogicians)onthelogicofobligationes. 

11 Indeed,the(primaryandsecondary)literatureonthetopic 

containssophisticatedanalysesofthelogicalandpragmaticpropertiesof 

theframework,whichare(presumably)applicabletoGOGARsothatthe 

comparisoncancontributetomakingGOGAR’slogicalstructureexplicit. 

Forexample,Ihaveprovedelsewhere(DutilhNovaes,2005)thattheclass 

ofmodelssatisfying Γnbecomessmallerinthenextstepofthegameonlyif 

φn+1isimpertinent;if φn+1ispertinent,thentheclassofmodelssatisfying 

Γnisthesameastheclassofmodelssatisfying Γn+1,eventhough Γnand 

Γn+1arenotthesame. 12 ThisresultcanbeinterpretedintermsofGOGAR 

inthefollowingmanner:whenaspeakermakesanassertion pwhichactually 

followsfromanysentenceorsetofsentencespreviouslyassertedbyhim, 

thenhissetofcommitmentsistherebynotaugmented.Inotherwords,his 

deonticstatusremainsthesame,ashewasdefactoalreadycommittedto p. 

Mixingthetwovocabularies,onecansaythataspeaker’sdeonticstatusis 

modifiedonlyifheassertsanimpertinentsentence;assertionsofpertinent 

sentenceshavenoeffectwhatsoeverinthissense. Now,thisisjustone 

exampleofhow,giventhattheobligationesframeworkisamoreregimented 

formoftherational,discursivepracticesalsomodeledbyGOGAR,such 

logicalpropertiesaremoreeasilyinvestigatedagainstthebackgroundof 

theformerratherthanthelatter. 

Strategicperspective Whenspeakingof‘thegameofgivingandasking 

forreasons’,Brandomseemstobetakingseriouslytheanalogybetweenthe 

rationaldiscursivepracticespresumablycapturedbyGOGARandactual 

games.ItisundoubtedlyalsoareferencetoWittgenstein’slanguage-games, 

butthequestionimmediatelyarises:howmuchofagameisGOGAR,actually?Tothebestofmyknowledge,Brandomdoesnotfurtherexplorethe 

comparisontogames,justashedoesnotdiscussspecificgame-theoretic 

propertiesofGOGAR;thisseemstome,however,tobeapromisingline 

ofinvestigation. Twoimportantgame-theoreticalpropertiesthatcometo 

mindarethegoal(s)tobeattainedwithinacertaingame,i.e.theexpected 

11 ObligationeswereoneofthemaintopicsofinvestigationinthelatemedievalLatin 

tradition,asattestedbytheverylargenumberofsurvivingtextsrangingfromthe 12 th 

tothe 15 th Century. 

12 Assuming,ofcourse,thatRespondenthasrepliedaccordingtotherules.


outcome,andthepossiblestrategiestoplaythegame(usually,oneisinterestedinmaximizingthepayoff,i.e.intheratioofbestpossibleoutcome 

vs.themosteconomicalstrategy).Basedonthesetwoconcepts,itwould 

seemthatGOGARisinfactafamilyofgames,notasinglegame,aseach 

particulargameoftheGOGARfamilyhasitsowngoals.Mostofthemare 

cooperativegames,whereparticipantshaveacommongoalratherthanthat 

ofbeatingtheopponent,e.g.,dialogueswherepeopleexchangeinformation 

andcoordinatetheiractions. Nevertheless,thereareofcoursenumerous 

situationsofdiscursivepracticeswherethepointreallyistobeattheopponent,suchas,e.g.,inacourtoflaw. 

Ineachcase,differentstrategies 

mustbeemployed:inthecaseofcooperativegames,Griceanmaximsmay 

beseenasagoodaccountofstrategiestomaximizeunderstandingbetween 

theparties;inthecaseofcompetitivegames,however,acompletelydifferentstrategymustbeused,onewheredeceit,forinstance,mayevenhave 

someroletoplay. 

Obligationesisobviouslyacompetitivegame: ifRespondentgrantsa 

contradiction,helosesthegame;butifheisabletomaintainconsistency, 

hebeatsOpponent.Andeventhoughthemedievalauthorsthemselvesdid 

notaccountforobligationesintermsofgames(nordidtheyhaveknowledge 

ofthespecificgame-theoreticalconceptsjustdiscussed),medievaltreatises 

onobligationesarefilledwithstrategicadviceforRespondentonhowtoperformwellduringanobligatio.Thesetreatisespresentnotonlyrulesdefiningthelegitimatemoveswithinthedisputationbutalsopractical,strategicadvice. 

13 SomeofthestrategicrulespresentedinBurley’streatiseare:“One 

mustpayparticularattentiontotheorder[ofthepropositions]”(Burley, 

1988,p.385);“Whenapossiblepropositionhasbeenposited,itisnotabsurdtograntsomethingimpossibleperaccidens”(Burley,1988,p.389); 

“Whenafalsecontingentpropositionisposited,onecanproveanyfalse 

propositionthatiscompossiblewithit”(Burley,1988,p.391). 

ThepointhereisthatthestrategicperspectivepresentintheseobligationestreatisescanverylikelybetransposedtotheGOGARframework 

toproduceinterestingresults. InthecaseofGOGARgameswherethe 

differentspeakersaretrulyopposedtooneanotherandthepointisreally 

tobeattheopponent,thenthestrategictipsfromtheobligationestreatisescanbeusedstraightforwardly. 

Buteveninthecaseofcooperative 

gamesofGOGAR,theobligationalstrategiesmaystillbequitehelpful,as 

theyessentiallydescribeproceduresthatmayenableonetomaintainconsistency—certainlyadesirableoutcomeinthecontextofrationaldiscursive 

practices. TheheartofthematteristhatGOGARdoesnotemphasize 

theplayer-perspective:rather,Brandom’sdescriptionofGOGARisthatof 

13 “Itisimportanttoknowthattherearesomerulesthatconstitutethepracticeofthis 

artandothersthatpertaintoitsbeingpracticedwell.”(Burley,1988,p.379)


thetheoriststandingoutsidethegameandofferingamodeltoexplainthe 

use(s)andmeaningfulnessoflanguage.Inthissense,theplayer-perspective 

offeredbyobligationesmaycomeasaninterestingcomplement. 

Theroleofdoubting BrandompresentsGOGARashavingonlyone 

quintessentialkindofmove,i.e.makingaclaim. Wehaveseenthatchallengingisalsoanimportantmove,butachallengeismadebymeansof 

theassertionofanincompatiblecontent.Incontrast,obligationesfeature 

threemainkindsofmovesforRespondent:granting,denyinganddoubting. 

Grantingobviouslycorrespondstoasserting,andinasense,denyingisalso 

akindofassertionwithintheobligationesframework,namelytheassertion 

ofthecontradictoryofthedeniedsentence. Ihavealsopointedoutthat 

challengingisnotalegitimatemoveforeitherRespondentorOpponent,a 

factthatisrelatedtotheregimentedandsimplifiednatureofobligationes 

asamodelofrationaldiscursivepractices.ButGOGARclaimstobemuch 

moreencompassingthanobligationesdoes,sowhileitseemsreasonablefor 

obligationestoleavesomeimportantelementsout,thesamedoesnothold 

ofGOGAR.Now,GOGARhasnoresourcestodealwiththephenomenon 

ofnotbeingsure,ofrecognizingthatonedoesnotdisposeofsufficient 

groundstoassertacontentoritscontradictory(knowingthatyoudon’t 

know),whereasthisseemstobeaveryimportantelementofourrational 

discursivepractices. Incontrast,byhavingdoubtingasoneofthelegitimatemovesforRespondent,theobligationalframeworkfaresbatterinthis 

respect. 

ItmightbearguedthatdoubtingisnotrelevantforGOGARinsofarasit 

hasnoimpactonaspeaker’sdeonticstatus,asitissimplythelackofcommitmentorentitlement;notso.AparticularrulepresentedinKilvington’s 

treatmentofobligationesinhisSophismata(Kilvington,1990,sophism48) 

showsthatdoubtingcanindeedalteraspeaker’sdeonticstatus. Therule 

isthefollowing:if‘pimplies q’isagoodconsequence,andifRespondent 

hasdoubted p,thenhemustnotdeny q,i.e.,heisnotentitledto ¬q. 

Thisisbecause,inavalidconsequence,iftheconsequentis(knowntobe) 

false,thentheantecedentwillalsobe(knowntobe)false,soifRespondent 

hasdoubtedtheantecedent,hemusteitherdoubtorgranttheconsequent. 

Thisisjustanexampleoftheintricaciesofthelogicofdoubtingandof 

howdoubtingcanindeedhaveanimpactonone’sdeonticstatus.Theobligationalliteratureisfilledwithmanymoreofsuchexamples,inparticular 

inthetreatmentsofdubitatio, 14 oneoftheformsofanobligationaldis- 

14 Inadubitatio,thefirstsentence(theobligatum)isnotapositum,itisadubium,a 

sentencewhichRespondentmustdoubtforthesakeofthedisputationjustasheaccepts 

theposituminapositio;hemustthenseewhatfollows(intermsofhiscommitmentsand 

entitlements)fromhavingdoubtedthefirstsentence.


putationalongwithpositio(whichisinsomesensethe‘standard’formof 

obligationes,andtheonediscussedinthistextsofar). Thus,Isuggest 

thatGOGARshouldpaymoreattentiontospeech-actsotherthanassertionsasalsohavinganimpactonaspeaker’sdeonticstatus—doubtingin 

particular,asshownwithintheobligationalframework. 

6 Conclusion 

InthisbriefcomparativeanalysisofGOGARandmedievalobligationesI 

hopetohaveindicatedhowfruitfulasystematiccomparisonbetweenthe 

twoframeworkscanbe. ForreasonsofspaceIhaveheremerelysketched 

suchacomparison,andamorethoroughanalysisshallremainatopicfor 

futurework. 

CatarinaDutilhNovaes 

DepartmentofPhilosophyandILLC,UniversityofAmsterdam 

NieuweDoelenstraat15,1012CPAmsterdam,TheNetherlands 

c.dutilhnovaes@uva.nl 

http://staff.science.uva.nl/∼dutilh/ 

References 

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Press. 

Brandom,R. (2008). Betweensayinganddoing. Oxford: OxfordUniversity 

Press. 

Burley,W.(1988).Obligations(selection).InN.Kretzmann&E.Stump(Eds.), 

TheCambridgeTranslationsofMedievalPhilosophicalTexts:LogicandthePhilosophyofLanguage(pp.369–412).Cambridge:CambridgeUniversityPress. 

DutilhNovaes,C.(2005).Medievalobligationesaslogicalgamesofconsistency 

maintenance.Synthese,145(3),371–395. 

DutilhNovaes,C. (2006). RogerSwyneshed’sobligationes: Alogicalgameof 

inferencerecognition?Synthese,151(1),125–153. 

DutilhNovaes,C.(2007).Formalizingmedievallogicaltheories.Berlin:Springer. 

DutilhNovaes,C.(2008).Logicinthe 14 th CenturyafterOckham.InD.Gabbay 

&J.Woods(Eds.),Handbookofthehistoryoflogic(Vol.2: Mediaevaland 

RenaissanceLogic,pp.433–504).Amsterdam:Elsevier. 

Kilvington,R. (1990). Sophismata. Cambridge: CambridgeUniversityPress. 

(Englishtranslation,historicalintroductionandphilosophicalcommentarybyN. 

KretzmannandB.E.Kretzmann.)


Knuuttila,S.,&Yrjonsuuri,M. (1988). Normsandactionsinobligationaldisputations. 

InO.Pluta(Ed.),DiePhilosophieim14.und15.Jahrhundert(pp. 

191–202).Amsterdam:B.R.Güner.

Truth Value Intervals, Bets, and Dialogue Games 

Christian G. Fermüller ∗ 

FuzzylogicsinZadeh’s‘narrowsense’(Zadeh,1988),i.e.,truthfunctional 

logicsreferringtotherealclosedunitinterval [0,1]assetoftruthvalues,areoftenmotivatedaslogicsfor‘reasoningwithimprecisenotionsand 

propositions’(see,e.g.,(Hájek,1998)).Howevertherelationbetweenthese 

logicsandtheoriesofvagueness,asdiscussedinaprolificdiscourseinanalyticphilosophy(Keefe&Rosanna,2000),(Keefe&Smith,1987),(Shapiro, 

2006)ishighlycontentious.Wewillnotdirectlyengageinthisdebatehere 

butratherpickoutso-calledintervalbasedfuzzylogicsasaninstructive 

exampletostudy 

1.howsuchlogicsareusuallymotivatedinformally, 

2.whatproblemsmayarisefromthesemotivations,and 

3.howbettinganddialoguegamesmaybeusedtoanalyzetheselogics 

withrespecttomoregeneralprinciplesandmodelsofreasoning. 

Themaintechnicalresult 1 ofthisworkconsistsinacharacterizationofan 

importantintervalbasedlogic,considered,e.g.,in(Esteva,Garcia-Calvés, 

&Godo,1994),intermsofadialoguecumbettinggame,thatgeneralizes 

RobinGiles’sgamebasedcharacterizationofŁukasiewiczlogic(Giles,1974), 

(Giles,1977). However,ouraimistoaddressfoundationalproblemswith 

certainmodelsofreasoningwithimpreciseinformation.Wehopetoshow 

thatthetraditionalparadigmofdialoguegamesasapossiblefoundationof 

logic(goingback,atleast,to(Lorenzen,1960))combinedwithbetsas‘test 

cases’forrationalityinthefaceofuncertaintymighthelptosortoutsome 

oftherelevantconceptualissues.Thisisintendedtohighlightaparticular 

meetingplaceoflogic,games,anddecisiontheoryatthefoundationofa 

fieldoftencalled‘approximatereasoning’(see,e.g.,(Zadeh,1975)). 

∗ ThisworkissupportedbyFWFprojectI143–G15. 

1 Duetolimitedspace,westatepropositionswithoutproofs.

52 ChristianG.Fermüller 

1 T-normbasedfuzzylogicsandbilattices 

PetrHájek,intheprefaceofhisinfluentialmonographonmathematical 

fuzzylogic(Hájek,1998)asserts: 

Theaimistoshowthatfuzzylogicasalogicofimprecise(vague) 

propositionsdoeshavewelldevelopedformalfoundationsandthat 

mostthingsusuallynamed‘fuzzyinference’canbenaturallyunderstoodaslogicaldeduction.(Hájek,1998,p.viii) 

Asthequalification‘vague’,addedinparenthesisto‘imprecise’,betrays, 

someterminologicaland,arguably,alsoconceptionalproblemsmaybelocatedalreadyinthisintroductorystatement. 

Theseproblemsrelateto 

thefactthatfuzzylogicisoftensubsumedunderthegeneralheadingsof 

‘uncertainty’and‘approximatereasoning’. Inanycase,Hajekgoesonto 

introduceafamilyofformallogics,basedonthefollowingdesignchoices 

(comparealso(Hájek,2002)): 

1.Thesetofdegreesoftruth(truthvalues)isrepresentedbythereal 

unitinterval [0,1].Theusualorderrelation ≤modelscomparisonof 

truthdegrees; 0representsabsolutefalsity,and 1representsabsolute 

truth. 

2.Thetruthvalueofacompoundstatementshallonlydependonthe 

truthvaluesofitssubformulas. Inotherwords:thelogicsaretruth 

functional. 

3.Thetruthfunctionfor(strong)conjunction(&)shouldbeacontinuous,commutative,associative,andmonotonicallyincreasingfunction 

∗ : [0,1] 2 → [0,1],where 0 ∗ x = 0and 1 ∗ x = x.Inotherwords: ∗ 

isacontinuous t-norm. 

4.Theresiduum ⇒∗ofthe t-norm ∗—i.e.,theuniquefunction ⇒∗: 

[0,1] 2 → [0,1]satisfying x ⇒∗ y = sup{z | x ∗ z ≤ y}—servesasthe 

truthfunctionforimplication. 

5.Thetruthfunctionfornegationis λx[x ⇒∗ 0]. 

ProbablythebestknownlogicarisinginthiswayisŁukasiewiczlogicL 

(Łukasiewicz,1920),wherethe t-norm ∗Lthatservesastruthfunctionfor 

&isdefinedas x ∗L y = max(0,x + y − 1). Itsresiduum ⇒Lisgivenby 

x ⇒L y = min(1,1 − x + y).Apopularalternativechoiceforconjunction 

takestheminimumasitstruthfunction.Besides‘strongconjunction’(&), 

alsothislatter‘weak(min)conjunction’(∧)canbedefinedinall t-norm 

basedlogicsby A ∧ B def 

= A&(A → B). Maximumastruthfunctionfor 

disjunction(∨)isalwaysdefinablefrom ∗and ⇒∗,too.

TruthValueIntervals 53 

Otherimportantlogics,likeGödellogicG,andProductlogicP,can 

beobtainedinthesameway,butwewillconfineattentiontoL,here.At 

thispointweliketomentionthatarich,deep,andstillgrowingsubfield 

ofmathematicallogic,documentedinhundredsofpapersandanumberof 

books(beyond(Hájek,1998))istriggeredbythisapproach.Consequently 

itbecameevidentthatdegreebasedfuzzylogicsareneithera‘poorman’s 

substituteforprobabilisticreasoning’noratrivialgeneralizationoffinitevaluedlogics. 

Anumberofresearchershavepointedoutthat,whilemodellingdegrees 

oftruthbyvaluesin [0,1]mightbeajustifiablechoiceinprinciple,itis 

hardlyrealistictoassumethatthereareproceduresthatallowustoassign 

concretevaluestoconcrete(interpreted)atomicpropositionsinacoherent 

andprincipledmanner. Whilethisproblemmightbeignoredaslongas 

weareonlyinterestedinanabstractcharacterizationoflogicalconsequence 

incontextsofgradedtruth,itisdeemeddesirabletorefinethemodelby 

incorporating‘imprecisionduetopossibleincompletenessoftheavailable 

information’(Estevaetal.,1994)abouttruthvalues. Accordingly,itis 

suggestedtoreplacesinglevalues x ∈ [0,1]bywholeintervals [a,b] ⊆ [0,1] 

oftruthvaluesasthebasicsemanticunitassignedtopropositions. The 

‘naturaltruthordering’ ≤canbegeneralizedtointervalsindifferentways. 

Following(Estevaetal.,1994)wearriveatthesedefinitions: 

Weaktruthordering: [a1,b1] ≤ ∗ [a2,b2]iff a1 ≤ a2and b1 ≤ b2 

Strongtruthordering: [a1,b1] ≺ [a2,b2]iff b1 ≤ a2or [a1,b1] = [a2,b2] 

Ontheotherhand,setinclusion(⊆)iscalledimprecisionorderinginthis 

context.Thesetofclosedsubintervals Int [0,1]of [0,1]isaugmentedbythe 

emptyinterval ∅toyieldso-calledenrichedbilatticestructures 〈Int [0,1], ≤ ∗ , 

0,1, ∅,L,N ∗ 〉aswellas 〈Int [0,1], ≺,0,1, ∅,L,N ∗ 〉,where Listhestandard 

latticeon [0,1],withminimumandmaximumasoperators,and N ∗ isthe 

extensionofthenegationoperator Ntointervals;inourparticularcase 

N ∗ ([a,b]) = [1 − b,1 − a]and N ∗ (∅) = ∅. 

Quiteanumberofpapershavebeendevotedtothestudyoflogics 

basedonsuchintervalgeneratedbilattices. Letusjustmentionthatthe 

GhentschoolofKerre,Deschrijver,Cornelis,andcolleagueshasproduced 

animpressiveamountofworkonintervalbilatticebasedlogics(see,e.g., 

(Cornelis,Deschrijver,&Kerre,2006)). 

Whileitisstraightforwardtogeneralizebothtypesofconjunction(tnormandminimum)aswellasdisjunction(maximum)from 

[0,1]to Int [0,1] 

byapplyingtheoperatorspoint-wise,itseemslessclearhowthe‘right’ 

generalizationofthetruthfunctionforimplicationshouldlooklike. In 

(Cornelis,Arieli,Deschrijver,&Kerre,2007), (Cornelis,Deschrijver,& 

Kerre,2004) [a,b] ⇒∗ C [c,d]def = [min(a ⇒ c,b ⇒ d),b ⇒ d]isstudied,butin


(Estevaetal.,1994)theauthorssuggest [a,b] ⇒∗ E [c,d]def = [b ⇒ c,a ⇒ d]. 

Ashasbeenpointedoutin(Hájek,n.d.)thereseemstobeakindoftradeoff 

involvedhere.While ⇒∗ Cpreservesalotofalgebraicstructure—inpartic ularityieldsaresiduatedlatticewhichcontainstheunderlyinglatticeover 

[0,1]asasubstructure—thefunction ⇒∗ Eisnotaresiduum,butleads tothefollowingdesirablepreservationpropertythatismissingfor ⇒∗ C .If 

M2isaprecisiationof M1(meaning:foreachpropositionalvariable p, M2 

assignsasubintervaloftheintervalassignedto pby M1),thananyformula 

satisfiedby M1isalsosatisfiedby M2. 2Below,wewilltrytoshowthata gamebasedapproachmightjustifythepreferenceof ⇒∗ Eover ⇒∗C against 

abackgroundthattakesthechallengeofderivingformalsemanticsfrom 

firstprinciplesaboutlogicalreasoningmoreseriouslythanthementioned 

literatureon‘intervallogics’. 

2 Worriesabouttruthfunctionality 

Itisinterestingtonotethatboth,(Estevaetal.,1994)and(Cornelisetal., 

2007),(Cornelisetal.,2004),refertoGinsberg(Ginsberg,1988),whoexplicitlyintroducedbilatticesfollowingideasof(Belnap,1977).MostprominentlyGinsbergconsiders 

B = 〈{0, ⊤, ⊥,1}, ≤t, ≤k, ¬〉asendowedwiththe 

followingintendedmeaning: 

• 0and 1represent(classical)falsityandtruth,respectively, ⊤represents‘inconsistentinformation’and 

⊥represents‘noinformation’. 

Theideahereisthattruthvaluesareassignedafterreceivingrelevant 

informationfromdifferentsources. Accordingly ⊤isidentifiedwith 

theinformationset {0,1}, ⊥with ∅andtheclassicaltruthvalueswith 

theirsingletonsets. 

• ≤t,definedby 0 ≤t ⊤/⊥ ≤ 1,isthe‘truthordering’. 

• ≤k,definedby ⊥ ≤t 0/1 ≤ 1,isthe‘knowledgeordering’. 

•Negationisdefinedby ¬(0) = 1, ¬(1) = 0, ¬(⊤) = ⊤, ¬(⊥) = ⊥. 

Whilethefour‘truthvalues’of Bmayjustifiablybeunderstoodtorepresent 

differentstatesofknowledgeaboutpropositions,itisveryquestionableto 

trytodefinecorresponding‘truthfunctions’forconnectivesotherthannegation. 

Indeed,itissurprisingtoseehowmanyauthors 3 followed(Belnap, 

1977)indefendingafourvalued,truthfunctionallogicbasedon B.Itshould 

beclearthat,intheunderlyingclassicalsettingthatistakenforgranted 

byBelnap,theformula A ∧ ¬Acanonlybefalse (0),independentlyofthe 

2 Here,aformulaisdefinedtobesatisfiedifitevaluatestothedegenerateinterval [1, 1]. 

3 DozensofpapershavebeenwrittenaboutBelnap’s4-valuedlogic.


kindofinformation,ifany,wehaveaboutthetruthof A. Ontheother 

hand,ifweneitherhaveinformationabout Anorabout B,then B ∧ ¬A 

couldbetrueaswellasfalse,andtherefore ⊥shouldbeassignednotonly 

to A, B,and ¬A,butalsoto B ∧ ¬A(incontrastto A ∧ ¬A).Thissimple 

argumentillustratesthatknowledgedoesnotpropagatecompositionally— 

awellknownfactthat,however,hasbeenignoredrepeatedlyintheliterature.(Forarecent,forcefulreminderontheincoherencyoftheintended 

semanticsforBelnap’slogicwereferto(Dubois,n.d.).) 

Inourcontextthiswarningaboutthelimitsoftruthfunctionalityis 

relevantattwoseparatelevels. First,itimpliesthat‘degreesoftruth’for 

compoundstatementscannotbeinterpretedepistemicallywhileupholding 

truthfunctionality. Indeed,mostfuzzylogicianscorrectlyemphasizethat 

theconceptofdegreesoftruthisorthogonaltotheconceptofdegreesof 

belief. Whiletruthfunctionsfordegreesoftruthcanbemotivatedand 

justifiedinvariousways—belowwewillreviewagamebasedapproach 

—degreesofbeliefsimplydon’tpropagatecompositionallyandcallfor 

othertypesoflogicalmodels(e.g.,‘possibleworlds’). Second,concerning 

theconceptofintervalsofdegreesoftruth,oneshouldrecognizethatitis 

incoherenttoinsistonbothatthesametime: 

1.truthfunctionsforallconnectives,liftedfrom [0,1]to Int [0,1],and 

2.theinterpretationofaninterval [a,b] ⊆ [0,1]assignedtoa(compound) 

proposition F asrepresentingasituationwhereourbestknowledge 

aboutthe(definite)degreeoftruth d ∈ [0,1]of Fisthat a ≤ d ≤ b. 

Giventhemathematicaleleganceof1,thatresults,amongotherdesirable 

properties,inalowcomputationalcomplexityoftheinvolvedlogics, 4 one 

shouldlookforalternativesto2.GodoandEsteva 5 havepointedoutthat, 

ifweinsiston2justforatomicpropositions,thenatleastwecanassert 

thatthecorresponding‘real’,butunknowntruthdegreeofanycomposite 

proposition F cannotlieoutsidetheintervalassignedto F accordingto 

thetruthfunctionsconsideredin(Estevaetal.,1994)(describedabove). 

However,theseboundsarenotoptimal,ingeneral. AswewillseeinSection5,takingcluesfromGiles’sgamebasedsemanticforL(Giles,1974), 

(Giles,1977),atightercharacterizationemergesifwedismisstheideathat 

intervalsrepresentsetsofunknown,butdefinitetruthdegrees. 

4 ItiseasytoseethatcoNP-completenessoftestingvalidityforL(andmanyother t-norm 

basedlogics)carriesovertotheintervalbasedlogicsdescribedabove. 

5 Privatecommunication.


3 RevisitingGiles’sgameforL 

Giles’sanalysis(Giles,1974),(Giles,1977)ofapproximatereasoningoriginallyreferredtothephenomenonof‘dispersion’inthecontextofphysicaltheories.Later(Giles,1976)explicitlyappliedthesameconcepttotheproblemofproviding‘tangiblemeanings’to(composite)fuzzypropositions. 

6 For 

thispurposeheintroducesagamethatconsistsoftwoindependentcomponents: 

Bettingforpositiveresultsofexperiments. 

Twoplayers—say:meandyou—agreetopay1€totheopponentplayer 

foreveryfalsestatementtheyassert.By [p1,... ,pm�q1,...,qn]wedenote 

anelementarystateofthegame,whereIasserteachofthe qiinthemultiset 

{q1,... ,qn}ofatomicstatements(representedbypropositionalvariables), 

andyouasserteachatomicstatement pi ∈ {p1,... ,pm}. 

Eachpropositionalvariable qreferstoanexperiment Eqwithbinary 

(yes/no)result.Thestatement qcanbereadas‘Eqyieldsapositiveresult’. 

Thingsgetinterestingastheexperimentsmayshowdispersion;i.e.,thesame 

experimentmayyielddifferentresultswhenrepeated.However,theresults 

arenotcompletelyarbitrary:foreveryrunofthegame,afixedriskvalue 

〈q〉 r ∈ [0,1]isassociatedwith q,denotingtheprobabilitythat Eqyieldsa 

negativeresult.Forthespecialatomicformula ⊥(falsum)wedefine 〈⊥〉 r = 

1. Theriskassociatedwithamultiset {p1,... ,pm}ofatomicformulas 

isdefinedas 〈p1,...,pm〉 r = m� 

〈pi〉 r .Itspecifiestheexpectedamountof 

i=1 

money(in €)thathastobepaidaccordingtotheaboveagreement. The 

risk 〈〉 r associatedwiththeemptymultisetis 0. Theriskassociatedwith 

anelementarystate [p1,... ,pm�q1,...,qn]iscalculatedfrommypointof 

view. Thereforethecondition 〈p1,...,pm〉 r ≥ 〈q1,... ,qn〉 r expressesthat 

Idonotexpect(intheprobabilitytheoreticsense)anyloss(butpossibly 

somegain)whenwebetonthetruthoftheinvolvedatomicstatementsas 

stipulatedabove. 

6 E.g.,Gilessuggeststospecifythesemanticsofthefuzzypredicate’breakable’byassigninganexperimentlike’droppingtherelevantobjectfromacertainheighttoseeifit 

breaks’. Theexpecteddispersivenessofsuchanexperimentrepresentthe’fuzziness’of 

thecorrespondingpredicate.Anarguablyevenbetterexampleofadispersiveexperiment 

intheintendedcontextmightconsistinaskinganarbitrarilychosencompetentspeaker 

forayes/noanswertoquestionslike’IsChristall?’ or‘IsShakirafamous?’ forwhich 

truthmaycogentlybetakenasamatterofdegree.


Adialoguegameforthereductionofcompositeformulas. 

GilesfollowsideasofPaulLorenzenthatdatebackalreadytothe1950s(see, 

e.g.,(Lorenzen,1960))andconstrainsthemeaningoflogicalconnectives 

byreferencetorulesofadialoguegamethatproceedsbysystematically 

reducingargumentsaboutcompositeformulastoargumentsabouttheir 

subformulas. 

Thedialogueruleforimplicationcanbestatedasfollows: 

R → IfIassert A → Bthen,wheneveryouchoosetoattackthisstatement 

byasserting A,Ihavetoassertalso B.(Andviceversa,i.e.,forthe 

rolesofmeandyouswitched.) 

Thisrulereflectstheideathatthemeaningofimplicationisspecifiedby 

theprinciplethatanassertionof‘if A,then B’(A → B)obligesoneto 

assert B,if Aisgranted. 7 

Inthefollowingweonlystatetherulesfor‘me’;therulesfor‘you’are 

perfectlysymmetric.Fordisjunctionwestipulate: 

R∨IfIassert A1 ∨ A2thenIhavetoassertalso Aiforsome i ∈ {1,2} 

thatImyselfmaychoose. 

Therulefor(weak)conjunctionisdual: 

R∧IfIassert A1 ∧ A2thenIhavetoassertalso Aiforany i ∈ {1,2}that 

youmaychoose. 

Onemightaskwhetherassertingaconjunctionshouldn’tobligeonetoassert 

bothdisjuncts.Indeed,forstrongconjunction 8 wehave 

R&IfIassert A1&A2thenIhavetoasserteitherboth A1and A2,or 

just ⊥. 

Thepossibilityofasserting ⊥insteadoftheattackedconjunctionreflects 

Giles’s‘principleofhedgedloss’: oneneverhastoriskmorethan1€for 

eachassertion.Asserting ⊥isequivalentto(certainly)paying1€. 

Incontrasttodialoguegamesforintuitionisticlogic(Lorenzen,1960), 

(Felscher,1985)orfragmentsoflinearlogic,nospecialregulationonthe 

successionofmovesinadialogueisrequiredhere. Moreover,weassume 

thateachassertionisattackedatmostonce:thisisreflectedbytheremoval 

oftheoccurrenceoftheformula Ffromthemultisetofformulasasserted 

byaplayer,assoonasithasbeenattacked,orwhenevertheotherplayer 

hasindicatedthatshewillnotattackthisoccurrenceof Fduringthewhole 

7 Notethat,since ¬Fisdefinedas F → ⊥,accordingto R →andtheabovedefinitionof 

risk,theriskinvolvedinasserting ¬pis 1 − 〈p〉 r . 

8 Gilesdidnotconsiderstrongconjunction.Theruleisfrom(Fermüller&Kosik,2006).


runofthedialoguegame. Everyrunthusendsinfinitelymanystepsin 

anelementarystate [p1,...,pm�q1,...,qn]. Givenanassignment 〈·〉 r of 

riskvaluestoall piand qiwesaythatIwinthecorrespondingrunofthe 

gameifIdonothavetoexpectanylossinaverage,i.e.,if 〈p1,...,pm〉 r ≥ 

〈q1,... ,qn〉 r . 

AsanalmosttrivialexampleconsiderthegamewhereIinitiallyassert 

p → qforsomeatomicformulas pand q;i.e.,theinitialstateis [�p → q].In 

response,youcaneitherassert pinordertoforcemetoassert q,orexplicitly 

refusetoattack p → q.Inthefirstcase,thegameendsintheelementary 

state [p�q];inthesecondcaseitendsinstate [�].Ifanassignment 〈·〉 r of 

riskvaluesgives 〈p〉 r ≥ 〈q〉 r ,thenIwin,whatevermoveyouchoosetomake. 

Inotherwords:Ihaveawinningstrategyfor p → qinallassignmentsof 

riskvalueswhere 〈p〉 r ≥ 〈q〉 r . 

Notethatwinning,asdefinedhere,doesnotguaranteethatIdonot 

loosemoney.Ihaveawinningstrategyfor p → p,resultingeitherinstate 

[�]orinstate [p�p]dependingonyour(theopponents)choice.Inthesecond 

case,althoughthewinningconditionisclearlysatisfied,Iwillactuallyloose 

1€,iftheexecutionoftheexperiment Epassociatedwithyourassertion 

of phappenstoyieldapositiveresult,buttheexecutionofthesameexperimentassociatedwithmyassertionof 

pyieldsanegativeresult. Itis 

onlyguaranteedthatmyexpectedlossisnon-positive.(‘Expectation’,here, 

referstostandardprobabilitytheory. Underafrequentistinterpretation 

ofprobabilitywemaythinkofitasaverageloss,resultingfromunlimited 

repetitionsofthecorrespondingexperiments.) 

TostateGiles’smainresult,recallthatavaluation vforŁukasiewicz 

logicLisafunctionassigningvalues ∈ [0,1]tothepropositionalvariables 

and 0to ⊥,extendedtocompositeformulasusingthetruthfunctions ∗L, 

max, min,and ⇒L,forstrongandweakconjunction,disjunctionandimplication,respectively. 

Theorem1((Giles,1974),(Fermüller&Kosik,2006)).Eachassignment 

〈·〉 r ofriskvaluestoatomicformulasoccurringinaformula Finducesa 

valuation vforLsuchthat v(F) = xifandonlyifmyoptimalstrategy 

for Fresultsinanexpectedlossof (1 − x)€. 

Corollary1. FisvalidinLifandonlyifforallassignmentsofriskvalues 

toatomicformulasoccurringin FIhaveawinningstrategyfor F. 

4 Playingunderpartialknowledge 

ItisimportanttorealizethatGiles’sgamemodelforreasoningaboutvague 

(i.e.,here,unstable)propositionsimpliesthateachoccurrenceofthesame 

atomicpropositioninacompositestatementmaybeevaluateddifferently


atthelevelofresultsofassociatedexecutionsofbinaryexperiments.This 

featureinducestruthfunctionality:thevaluefor p∨¬pisnottheprobability 

thatexperiment Epeitheryieldsapositiveoranegativeresult,whichis 1 

bydefinition;itratheris 1−x,where x = min(〈p〉 r ,1−〈p〉 r )ismyexpected 

loss(in €)afterhavingdecidedtobeteitherforapositiveorforanegative 

resultofanexecutionof Ep(whatevercarrieslessriskforme). 

Theplayersonlyknowtheindividualsuccessprobabilities 9oftherelevant experiments.Alternatively,onemaydisregardindividualresultsofbinary 

experimentsaltogetherandsimplyidentifytheassignedprobabilitieswith 

‘degreesoftruth’. Inthisvariantthe‘pay-off’justcorrespondstothese 

truthvalues,andGiles’sgameturnsintoakindofHintikkastyleevaluation 

gameforL. 

Howdoesallthisbearonthementionedproblemsofinterpretationfor 

intervalbasedfuzzylogics?Rememberthatboth,(Estevaetal.,1994)and 

(Cornelisetal.,2007,2006,2004)seemtosuggestthatanintervaloftruth 

values [a,b]represents‘impreciseknowledge’aboutthe‘realtruthvalue’ c, 

inthesensethatonly c ∈ [a,b]isknown. Forthebettinganddialogue 

gamesemanticthissuggeststhattheplayers(oratleastplayer‘I’)now 

havetochoosetheirmovesinlightofcorresponding‘imprecise’(partial) 

knowledgeaboutthesuccessprobabilitiesoftheassociatedexperiments. 

However,whilethismayresultinaninterestingvariantoftheGiles’sgame, 

itsrelationtothetruthfunctionalsemanticssuggestedforlogicsbasedon 

Int [0,1]andL-connectivesisdubious. 

Thefollowingsimpleexampleillustratesthisissue.Supposetheinterval 

v ∗ (p) = [v ∗ 1 (p),v∗ 2 

(p)]assignedtothepropositionalvariable pis [0,1],re- 

flectingthatwehavenoknowledgeatallaboutthe‘realtruthvalue’ofthe 

propositionrepresentedby p. Accordingtothetruthfunctionspresented 

inSection1,theformula p ∨ ¬pevaluatesalsoto [0,1],since v∗ (¬p) = [1 − 

v∗ 2 (p),1−v∗ 1 (p)] = [0,1]andhence v(p∨¬p) = [max(0,0),max(1,1)] = [0,1]. 

Stickingwiththe‘impreciseknowledge’interpretation,theresultinginterval 

shouldreflectmyknowledgeaboutmyexpectedlossifIplayaccordingto 

anoptimalstrategy.However,while 1 − v∗ 2 (p ∨ ¬p) = 0isthecorrectlower 

boundonmyexpectedlossafterperformingtherelevantinstanceof Ep,to 

requirethat 1 − v∗ 1 (p ∨ ¬p) = 1isthebestupperboundforthelossthat 

Ihavetoexpectwhenplayingthegameisproblematic. Whenplayinga 

mixedstrategythatresultsinmyassertionofeither porof ¬pwithequal 

probability,thenmyresultingexpectedlossis0.5€,not1€. 

Weintroducesomenotationtoassistprecisestatementsabouttherelationbetweentheintervalbasedsemanticsof(Estevaetal.,1994)andGiles’s 

9 Thesemightwellbepurelysubjectiveprobabilitiesthatmaydifferforthetwoplayers. 

ToproveTheorem1oneonlyhastoassumethatIcanactonassignedprobabilitiesthat 

determine‘myexpectation’ofloss.


game.Let v ∗ beanintervalassignment,i.e.anassignmentofclosedintervals 

⊆ [0,1]tothepropositionalvariablesPV.Then v ∗ L denotestheextensionof 

v ∗ fromPVtoarbitraryformulasviathetruthfunctions ⇒ ∗ E forimplicationandthepoint-wisegeneralizationsof 

max, min,and ∗Lfordisjunction, 

weakconjunction,andstrongconjunction,respectively. Callanyassignment 

vofreals ∈ [0,1]compatiblewith v ∗ if v(p) ∈ v ∗ (p)forall p ∈PV. 

Thecorrespondingriskvalueassignment 〈·〉 r v,definedby 〈p〉 r v = 1 − v(p),is 

alsocalledcompatiblewith v ∗ . 

Proposition1.If,givenanintervalassignment v∗ ,theformula Fevaluatesto 

v∗ L (F) = [a,b]thenthefollowingholds: 

∗ForthegameinSection3,playedon F:All(pure)strategiesforme 

thatareoptimalwithrespecttosomefixedriskvalueassignment 〈·〉 r v 

compatiblewith v ∗ resultinanexpectedlossofatmost (1 − a)€,but 

atleast (1 − b)€. 

Notethatintheabovestatementmyexpectedlossreferstoariskvalue 

assignment 〈·〉 r vthatisfixedbeforethedialoguegamebegins. Iwillplay 

optimallywithrespecttothisassignment.Sincethecorrespondingexpected 

pay-offisallthatmattershere,wetechnicallystillhaveagameofperfect 

informationandthereforenogeneralityislostbyrestrictingattentionto 

purestrategies. Theboundsgivenby v∗ Lformyexpectedlossarenot optimalingeneral. Inotherwords,theinversedirectionofProposition1 

doesnothold.Toseethis,consideragaintheintervalassignment v∗ (p) = 

[0,1]resultingin v∗ L (p ∨ ¬p) = [0,1]. Obviously,Icannotloosemorethan 

1€,evenifIplaybadly,butmyexpectedlossunderanyfixedriskvalue 

isnevergreaterthan 0.5€ifIplayoptimallywithrespect 

assignment 〈·〉 r v 

to 〈·〉 r v . 

Ontheotherhand,stickingwithourexample‘p ∨ ¬p’,onecanobserve 

thatthebestupperboundformylossisindeed1€ifIdonotknowtherelevantriskvaluesandIstillhavetostickwithsomepurestrategy.Thisisbecausethechosenstrategymightsuggesttoassert 

pevenif,unknowntome, 

theexperiment Epalwayshasanegativeresult.Inotherwords,thebounds 1 

and 0areoptimalnowandcoincidewiththelimitsof v∗ L (p ∨ ¬p).However, 

ingeneral,thisscenario—playingapurestrategyreferringtoriskvalues 

thatneednotcoincidewiththeriskvaluesusedtocalculatetheexpected 

pay-off—mayleadtoanexpectedlossoutsidetheintervalcorresponding 

to v∗ L .Forasimpleexampleconsider p ∨q,where v∗ (p) = [0.4,0.4],i.e.,the 

playersknowthattheexpectedlossassociatedwithanassertionof pis 0.6€, 

and v∗ (q) = [0,1],i.e.,theriskassociatedwithasserting qcanbeanyvalue 

between 1and 0.Wehave v∗ L (p ∨ q) = [max(0,0.4),max(0.4,1)] = [0.4,1]. 

Undertheassumptionthat 〈q〉 r v = 0,whichiscompatiblewith v ∗ (q),my 

beststrategycallsforasserting qinconsequenceofasserting p ∨ q. But


ifthestate [�q]isevaluatedusingtheriskvalue 〈q〉 r v = 1,whichisalso 

compatiblewith v∗ (q),thenIhavetoexpectasurelossof 1€,although 

1 − 1 = 0isoutside [0.4,1]. 

5 Cautiousandboldbettingonunstablepropositions 

Wesuggestthatamoreconvincingjustificationoftheformalsemantics 

of(Estevaetal.,1994)arisesfromthefollowingalternativegamebased 

modelofreasoningunderimpreciseknowledge.Likeabove,let v∗beanas signmentofintervals ⊆ [0,1]tothepropositionalvariables.Again,weleave 

thedialoguepartofGiles’sgameunchanged.Butinreferencetothepartial 

informationrepresentedby v∗ ,weassigntwodifferentsuccessprobabilities 

toeachexperiment Eqcorrespondingtoapropositionalvariable q,reflecting 

whether qisassertedbymeorbyyouandconsiderbestcaseandworstcase 

scenarios(frommypointofview)concerningtheresultingexpectedpay-off. 

Moreprecisely,myexpectedlossforthefinalstate [p1,... ,pm�q1,... ,qn] 

whenevaluated v∗-cautiouslyisgivenby n� 

〈qi〉 

i=1 

r m� 

h − 〈pi〉 

i=1 

r l ,butwhenevaluated 

v∗-boldlyitisgivenby n� m� 

− 

〈qi〉 

i=1 

r l 〈pi〉 

i=1 

r h ,wheretheriskvalues 〈q〉r h 

and 〈q〉 r laredeterminedbythelimitsoftheinterval v∗ (q) = [a,b]asfollows: 

〈q〉 r h = 1 − aand 〈q〉r l = 1 − b. 

Proposition2.Givenanintervalassignment v ∗ ,thefollowingstatements 

areequivalent: 

(i)Formula Fevaluatesto v∗ L (F) = [a,b]. 

(ii)ForthedialoguegameinSection3,playedof F:ifelementarystates 

areevaluated v ∗ -cautiouslythentheminimalexpectedlossIcanachievebyanoptimalstrategyis 

(1−b)€;ifelementarystatesareevaluated 

v ∗ -boldlythenmyoptimalexpectedlossis (1 − a)€. 

6 Conclusion 

Wehavebeenmotivatedbyvariousproblemsthatarisefrominsistingon 

truthfunctionalityforaparticulartypeoffuzzylogicintendedtocapture 

reasoningunder‘impreciseknowledge’. Mostimportantlyforthecurrent 

purpose,wehaveemployedadialoguecumbettinggameapproachtomodel 

logicalinferenceinacontextof‘dispersiveexperiments’fortestingthetruth 

ofatomicassertions.Thisanalysisnotonlyleadstodifferentcharacterizationsofanimportantintervalbasedfuzzylogic,butrelatesconcernsabout 

propertiesoffuzzylogicstoreflectionsonrationalityquaplayingoptimally 

inadequategamesfor‘approximatereasoning’.


ChristianG.Fermüller 

InstitutfürComputersprachen,TUWien 

Favoritenstraße9–11,A–1040Vienna,Austria 

chrisf@logic.at 

http://www.logic.at/staff/chrisf/ 

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Procedural Semantics for Mathematical Constants 


Bjørn Jespersen Marie Duˇzí ∗ 

Considernumericalconstantslike‘1’and‘π’.Whatistheirsemantics?We 

aregoingtoargueinfavourofarealistproceduralsemantics,accordingto 

whichsenseanddenotationarecorrelatedasprocedureandproduct.Soit 

isobviousthatourproceduralsemanticsbearssimilaritiestoMoschovakis’s 

asbasedonalgorithmandvalue. WeareinoppositiontoKripke’sunrealisticrealistcontentionthatthesemanticsof‘π’consistsinnothingother 

than‘π’rigidlydenoting π.Yes,‘π’doesdenote π—indeed,‘π’qualifies 

asastronglyrigiddesignatorof π,cf.(Kripke,1980,p.48)—butthere 

issubstantiallymoretothesemanticsof‘π’thanmerelythedenotation 

relation.Inthispaperwefocuson‘π’,sinceourgeneraltop-downstrategy 

istodevelopasemanticsforthehardest(oraveryhard)caseandthen 

generalisedownwardstoincreasinglylesshardcasesfromthere. 

Inoutline,ourproceduralsemanticssaysthat‘π’expressesasitssensea 

procedurewhoseproductis π.Theprocedureis,asamatterofmathematicalconvention,adefinitionofπandtheproductis,asamatterofmathematicalfact,the(transcendental)numbersodefined.Forcomparison,‘1’ 

expressesasitssensetheprocedureconsistinginapplyingthesuccessor 

functiontozeroonceanddenoteswhatever(natural)numberemergesas 

theproductofthisprocedure. 

Theupsideofaproceduralsemanticsfor‘π’isthattounderstand,asa 

readerorhearer,andexerciselinguisticcompetence,asawriterorspeaker, 

onemustmerelyunderstandaparticularnumericaldefinitionandneednot 

knowwhichnumberitdefines.Proceduralsemantics,whetherrealistoridealist,construessenseasanitinerariomentisabstractingfromtheitinerary’s 

destination. Makingthedenotationofanumericalconstantirrelevantto 

∗ ThisworkissupportedbyGrantNo.401/07/0451,SemantisationofPragmatics,ofthe 

GrantAgencyoftheCzechRepublic.

66 BjørnJespersen&MarieDuˇzí 

understandingandlinguisticcompetenceisnotpressinginthecaseof‘1’, 

butitissointhecaseof‘π’.Thedownside,however,isthatatleasttwo 

equivalent,butobviouslydistinct,definitionsof πarevyingfortheroleas 

thesenseof‘π’.Oneistheratioofacircle’sareaanditsradiussquared;the 

otheristheratioofacircle’scircumferencetoitsdiameter.Theyareequivalent,becausethesamenumberisharpoonedbybothdefinitions.Butthe 

proceduresareconceptuallydifferent,sotheyshouldnotbothbeassigned 

to‘π’asitssenseonpainofinstallinghomonymy.Thiskindofpredicament 

hasbecomehistoricallyfamous.SaysFrege, 

SolangenurdieBedeutungdieselbebleibt,lassensichdieseSchwankungendesSinnesertragen,wiewohlauchsieindemLehrgebäudeeinerbeweisendenWissenschaftzuvermeidensindundineinervollkommenenSprachenichtvorkommendürften. 

(Frege,1892,n.2, 

p.42) 

Weshallsuggestasolutiontothispredicament.Thecrustofthesolution 

istorelegateeachdefinitionof πtoindividualconceptualsystems. Since 

aninterpretedsignsuchas‘π’isapairwhoseelementsareacharacter(in 

thiscasetheGreekletter‘π’)andasense,therewillbeasmanysuchpairs 

asthereareconceptualsystemsdefining π.Disambiguationof‘π’-involving 

discoursewillconsistinmakingexplicitwhichparticular π-definingsystem 

shouldsupplythesenseofatokenofthecharacter‘π’. 

Arelatedpredicament,whichweshallalsoaddress,iswhether‘π’isbest 

construedasanamefor πorasashorthandforadefinitedescription.Ifa 

name,thesenseof‘π’will,inoursemantics,beaprimitiveprocedureconsistingintheinstructiontoobtain,oraccess, 

πinonestep.Theprocedure 

willnottellushowtoobtain π,butonlythat πistobeobtained. This 

doesnotsitwellwith πbeingsomethingascomplicatedasatranscendental 

number. Butitdoessitwellwith‘π’beingitselfaprimitive,orsimple, 

characternotdisclosinganyinformationaboutitsdenotation. Soatleast 

onaliteralanalysis,accordingtowhichsyntacticandsemanticstructures 

arebyandlargeisomorphic,‘π’shouldbepairedoffwithanon-compound 

sense.If‘π’isadefinitedescription(indisguise),thesenseof‘π’will,inour 

semantics,beacompoundprocedureconsistingintheinstructiontomanipulatevariousmathematicaloperationsandconceptsinordertodefinea 

number. Onlytheproblem,aswejustpointedout,is,whichprocedure? 

Isittheinstructiontocalculatetheratioofacircle’sareaanditsradius 

squared,orisittheinstructiontocalculatetheratioofacircle’scircumferenceanditsdiameter,orisitsomeyetotherinstruction?Whicheverit 

maybe,though,thegrammaticalconstant‘π’willbesynonymouswiththe 

definitedescription‘theratio...’chosen.Theproblemofhomonymydoes 

notrearitsheadincasethesenseof‘π’isaprimitiveprocedure,forthen


‘π’isonlyequivalent(co-denoting)withaparticulardefinition. Infact, 

sinceallthevariantsofdefinitionsco-denotethesamenumber,‘π’willbe 

equivalentwithallsuchdescriptions. 

Ourunderlyingsemanticschemaisdepictedinthefollowingfigure. 

procedure 

(sense) 

expresses 

constant 

produces 

Semanticschema 

denotes(names) 

denotedentity 

(ifany) 

Therelationaprioriofexpressingasobtainingbetweenconstantandsense 

exhauststhepuresemanticsoftheconstant. Assoonasaprocedureis 

explicitlygiven,itsproduct(ifany)isimplicitlygiven,fortherelationfrom 

proceduretoproductisaninternalone: aprocedurecanhaveatmost 

oneproduct,andthatproductisinvariant.Theprocedurewillproduceits 

productindependentlyofanyalgorithm;thisiswhytherelationbetween 

procedureandproductisaninternalone.Butforepistemologicalreasons 

wewillneedsomewayorotherofcalculatingitsproducttolearnwhatit 

is,soweneedaπ-calculatingalgorithmtoshowuswhatnumbersatisfies 

whatever π-definingcondition.Suchanalgorithmwill,ipsofacto,revealto 

uswhatthedenotationof‘π’is.Thenumber 3.14159...whichis πisitself 

noplayerinthepuresemanticsof‘π’. πisjustwhatevernumberrollsout 

asthevalueofthegivenprocedure.Thenumber 3.14159...isitselfoflittle 

mathematicalinterestandofnosemanticimport.Thepropertiesof π,by 

contrast,areofgreatinterest;e.g.,whether πisnormalinsomebase;and 

establishingthat πistranscendental(andnotjustrational)wasamajor 

mathematicalachievement. 

Analgorithmmayappearinoneoftwocapacities. Eitheritisanintermediarybetweenthedefinitionandthenumbersodefined: 

thenthe 

algorithm(whicheveritis)isnoplayerinthepuresemanticsof‘π’. Or 

analgorithmistheverysenseof‘π’:thenthealgorithmisaplayerinthe 

puresemanticsof‘π’.Ourproceduralsemanticsallowsthataπ-calculating 

algorithmmayitselfbeelevatedtoplayingtheroleofsenseof‘π’.Insucha 

case‘π’willhaveasitssenseoneparticularwayofcalculating π.Analgorithmisaparticularkindofprocedureandcanassuchfigureasalinguistic 

senserelativetoaproceduralsemantics. 

Intheformercase,ifthedefinitionisaconditionthenthealgorithm 

willcalculatethesatisfierofthecondition.Fullcompetencewithrespectto


thedefinitiontheratio...willyieldknowledgeofaconditiontobesatisfied 

byarealnumber,butwillnotyieldknowledgeofwhichnumbersatisfies 

it. Sothedefinitionis,strictlyspeaking,adefinitionofsomethingfora 

numbertobe;namely,theratiooftwogeometricproportions.Ifthesense 

defines πastheratiobetweentheareaofacircleanditsradiussquared, 

amatchingalgorithmmustcalculatethisratio.Fulllinguisticcompetence 

withrespectto‘π’neitherpresupposes,norneedinvolve, knowledgeof 

howtocalculate π.Whatcompetenceconsistsindependsonwhetherthe 

senseof‘π’isaprimitiveorcompoundprocedure.Ifprimitive,competence 

requiresknowingwhichtranscendentalreal 3.14159... is π.Ifcompound, 

competencerequiresunderstandingtheconcepttheratioof,aswellaseither 

theconceptstheareaof,theradiusof,thesquareof,ortheconceptsthe 

circumferenceof andthediameterof,togetherwithknowledgeofhowto 

mathematicallymanipulatethem. Aschoolchildwillunderstandsucha 

complexprocedure;ittakesaprofessionalmathematiciantodevelopand 

comprehendaπ-calculatingalgorithm.Thetaskfacingthemathematician 

istocomeupwithanalgorithmequivalentwiththedefinitiondefiningthe 

givenratio. 

Inthelattercase,whereanalgorithmisthesenseof‘π’,fulllinguisticcompetencewithrespectto‘π’istounderstandadefinitionof 

πand, 

again,notofthenumbersodefined.Butsincethealgorithmisnownotan 

intermediarybetweendefinitionandnumber,linguisticcompetencewillbe 

hardertocomeby,sincethesenseof‘π’isnowlikelytoinvolvemuchmore 

complicatedmathematicalnotionsthanjust,say,thoseofratio,area,and 

circumference,suchasthelimitofaninfiniteseries. 

2 BeyondBenacerraf 

Assumethatthetruth-conditionof“...π...”requires πtoexistasanindependent,abstractentity.Assume,further,thatwecanhavenoepistemic 

accesstoentitiesthatwecanhavenocausalinteractionwith. Thennext 

stopisBenacerraf’sdilemmaasformulatedfor π: wedonotknowwhat 

numberis π;yetwewanttodub π‘π’inordertotalkabout πin“...π...”. 

Sohowis‘π’tobeintroducedintomathematese?Moreover,nowthat‘π’ 

hasactuallybeenintroducedintostandardmathematicalvocabularyand 

beeninuseforthreehundredyears,whatwouldarealist(asopposedto 

constructivistorotherwiseidealist)construalofitssemanticslooklike? 

WeproposeplacingourproceduralsemanticswithinthegeneralFregean 

programmeofexplicatingsense(Sinn)asthemodeofpresentation(Artdes 

Gegebenseins)oftheentity(Bedeutung)thatasensedetermines.Muskens 

correctlypointsoutthat“TheideawasprovidedwithextensivephilosophicaljustificationinTichý[(1988)]”andthat“[Tichý’s]notionofsensesas


constructions essentiallycapturesthesameidea.” (Tichý,2004,p.474) 

GoingwiththisFregeanprogramme,however,raisesabatchofquestions 

deservinganddemandingtobeanswered.Justhowfinelyaresensessliced? 

Whatistheontologicalstatusofasense?Whatdoesasense‘looklike’;in 

particular,whatisitsstructure? Andhowdoesasensedeterminesomething? 

WeagreewithMoschovakis’conceptionofsense(‘referentialintension’, 

inhisvernacular)as‘an(abstract,idealized,notnecessarilyimplementable) 

algorithmwhichcomputesthedenotationof[aterm]’(Moschovakis,2006, 

p.27);seealso(Moschovakis,1994). 1 

Moschovakisoutlineshisconceptionthus: 

Thestartingpoint... [is]theinsightthatacorrectunderstanding 

ofprogramminglanguagesshouldexplaintherelationbetweenaprogramandthealgorithmitexpresses,sothatthebasicinterpretation 

schemeforaprogramminglanguageisoftheform 

program P →algorithm(P) →den(P). 

Itisnothardtoworkoutthemathematicaltheoryofasuitably 

abstractnotionofalgorithmwhichmakesthiswork;andoncethis 

isdone,thenitishardtomissthesimilaritywiththebasicFregean 

schemefortheinterpretationofanaturallanguage, 

term A →meaning(A) →den(A). 

Thissuggestedatleastaformalanalogybetweenalgorithmsand 

meaningswhichseemedworthinvestigating,andprovedaftersome 

worktobemorethanformal:whenweviewnaturallanguagewitha 

programmer’seye,itseemsalmostobviousthatwecanrepresentthe 

meaningofaterm AbythealgorithmwhichisexpressedbyAand 

whichcomputesitsdenotation.(Moschovakis,2006,p.42) 

Inmodernjargon,TILbelongstotheparadigmofstructuredmeaning. 

However,Tichýdoesnotreducestructuretoset-theoreticsequences,as 

1 Moschovakis’notionofalgorithmbordersonbeingtoopermissive,sincealgorithmsare 

normallyunderstoodtobeeffective.(See(Cleland,2002)fordiscussion.)Tichýseparates 

algorithmsfromconstructions:“Thenotionofconstructionis... correlativenotwiththe 

notionofalgorithmitselfbutwithwhatisknownasaparticularalgorithmiccomputation, 

thesequenceofstepsprescribedbythealgorithmwhenitisappliedtoaparticularinput. 

Butnoteveryconstructionisanalgorithmiccomputation.Analgorithmiccomputation 

isasequenceofeffectivesteps,stepswhichconsistinsubjectingamanageableobject... 

toafeasibleoperation. Aconstruction,ontheotherhand,mayinvolvestepswhichare 

notofthissort.”(Moschovakis,1994,p.526),(Moschovakis,2006,p.613)


doKaplanandCresswell. 2 NordoesTichýfailtoexplainhowthesense 

ofamoleculartermisdeterminedbythesensesofitsatomsandtheir 

syntacticarrangement(asMoschovakisobjectsto‘structural’approaches 

in(Moschovakis,2006,p.27)). 

Ingeneral,aprocedureisastructureencompassingoneormoresteps 

thatindividuallydetailhowtodetermineaproductandjointlydetailhow 

todeterminetheproductoftheprocedurethattheyaresub-proceduresof. 

(Thisholdsevenforone-stepprocedures.)Structuresareneededasmolecularunitsinwhichtoorganiseatomicsub-proceduresinaparticularorder.Acompoundstructureconstitutesahierarchyofsub-procedures.Thephilosophicalideainformingourproceduralsemanticsisthatsincesensesareprocedures,anytwosensesareidenticaljustwhentheyare,roughlyspeaking,procedurallyindistinguishable. 

(Weshallindividuatesensesinterms 

ofproceduralisomorphism;seebelow.)Intuitively,anytwoproceduresare 

identicaljustwhentheyareinstructionstodothesametothesamethings 

inthesameorder. 

3 Logicalfoundations 

TILconstructionsareprocedures. Constructionsdivideintoatomicand 

compound,accordingastheyencompassoneormoresteps. Theatomic 

onesareVariableandTrivialization;thecompoundones,Compositionand 

Closure. 3 Avariable xconstructsanobjectrelativetoavaluationfunction 

pairingvariablesandentitiesoff,suchthat xconstructsthevalueassigned 

toit.TheTrivialization 0 Xconstructstheentity X(whichmaybewhatever 

sortofentityfoundintheontologyofTIL).ACompositionistheprocedure 

ofapplyingafunctionatoneofitsargumentstoobtainthevalue(ifany) 

atthatargument;thefunctionalvalueistheproductofthatprocedure.A 

Closureistheprocedureofarrangingobjects x1,...,xnand yasfunctional 

argumentsandvalues,respectively;theresultingfunctionistheproductof 

thatprocedure. Ifthesenseof‘π’issimple,itssenseistheTrivialization 

of π: 0 π.Ifcomplex,itisaComposition.Ineithercasetheproductofthe 

respectiveprocedureisthesametranscendentalnumber. 

2 Kaplanmaywellhavebeentheonetoreintroducethenotionofstructuredmeaning 

intomainstreamanalyticphilosophyoflanguage. See(Kaplan,1978),writtenin1970; 

butseealso(Lewis,1972).(Cresswell,1985)hasbecomethestandardpointofreference. 

Allthreeagreethatstructure,especiallyastructuredproposition,is(orcanbemodelled 

as)anordered n-tuple.Thiswon’tdo,though,sincesequencesunderdeterminestructure 

andsocannotsolveRussell’soldproblemofpropositionalunity. 

3 Andfourothers—Execution,DoubleExecution,Tuple,Projection—thatwedonot 

needhere.


Herefollowsanoutlineofthelogicalbackboneofourproceduralsemanticsfor‘π’. 

TILconstructions,aswellastheentitiestheyconstruct,all 

receivealogical(asopposedtolinguistic)type. 

Definition1(Typeoforder1). 

Let Bbeabase,whereabaseisacollectionofpair-wisedisjoint,non-empty 

sets.Then: 

(i)Everymemberof Bisanelementarytypeoforder 1over B. 

(ii)Let α, β1,...,βm(m > 0)betypesoforder 1over B. Thenthe 

collection (αβ1 ...βm)ofall m-arypartialmappingsfrom β1×· · ·×βm 

into αisafunctionaltypeoforder 1over B. 

(iii)Nothingisatypeoforder 1over Bunlessitsofollowsfrom1and2. 

Definition2(Construction). 

(i)TheVariable xisaconstructionthatconstructsanobject Oofthe 

respectivetypedependentlyonavaluation v;it v-constructs O. 

(ii)Trivialization: Where Xisanobjectwhatsoever(anextension,an 

intensionoraconstruction), 0 XistheconstructionTrivialization.It 

constructs Xwithoutanychange. 

(iii)TheComposition [XY1 ...Ym]isthefollowingconstruction. 

If X v-constructsafunction fofatype (αβ1 ...βm),and Y1 ... Ym 

v-constructentities B1,...,Bmoftypes β1,... ,βm,respectively,then 

theComposition [XY1 ...Ym] v-constructsthevalue(anentity,ifany, 

oftype α)of fonthetuple-argument 〈B1,... ,Bm〉. Otherwisethe 

Composition [XY1 ...Ym]doesnot v-constructanythingandsois vimproper. 

(iv)TheClosure [λx1 ...xmY ]isthefollowingconstruction. 

Let x1,x2,... ,xmbepairwisedistinctvariables v-constructingentitiesoftypes 

β1,...,βmand Y aconstruction v-constructingan αentity.Then 

[λx1 ...xmY ]istheconstruction λ-Closure(orClosure). 

It v-constructs thefollowingfunction f oftype (αβ1 ... βm). Let 

v(B1/x1,...,Bm/xm)beavaluationidenticalwith vatleastupto 

assigningobjects B1,... ,Bmoftypes β1,... ,βm,respectively,tovariables 

x1,... ,xm. If Y is v(B1/x1,... ,Bm/xm)-improper(see(iii)), 

then fisundefinedon 〈B1,... ,Bm〉. Otherwisethevalueof fon 

〈B1,...,Bm〉istheentityoftype α v(B1/x1,... ,Bm/xm)-constructed 

by Y. 

(v)Nothingisaconstruction,unlessitsofollowsfrom(i)through(iv).


Definition3(Ramifiedhierarchyoftypes). 

Let Bbeabase.Then: 

T1(typesoforder 1):definedbyDefinition1. 

Cn(constructionsoforder n) 

(i)Let xbeavariablerangingoveratypeoforder n. Then xisa 

constructionoforder nover B. 

(ii)Let X beamemberofatypeoforder n. Then 0 X, 1 X, 2 X are 

constructionsoforder nover B. 

(iii)Let X,X1,...,Xm(m > 0)beconstructionsoforder nover B.Then 

[XX1 ... Xm]isaconstructionoforder nover B. 

(iv)Let x1,...,xm,X(m > 0)beconstructionsoforder nover B.Then 

[λx1 ...xmX]isaconstructionoforder nover B. 

(v)Nothingisaconstructionoforder nover Bunlessitsofollowsfrom 

Cn(i)–(iv). 

Tn+1(typesoforder n + 1) 

Let ∗nbethecollectionofallconstructionsoforder nover B. 

(i) ∗nandeverytypeoforder naretypesoforder n + 1. 

(ii)If 0 < mand α,β1,... ,βmaretypesoforder n + 1over B,then 

(αβ1 ...βm)(seeT12)isatypeoforder n + 1over B. 

(iii)Nothingisatypeoforder n + 1over Bunlessitsofollowsfrom(i) 

and(ii). 

Theontologicalstatusofaconstructionisanobjective,abstract,structuredprocedureresidinginaPlatonicrealm.Constructionsarenotinherentlylinguisticsenses,fortheyexistpriortoandindependentlyoflanguage. 

Buttheymaybemade,vialinguisticconvention,toserveaslinguistic 

senses. Thatis,intruerealistfashion,TILconsiderslanguageacode. 4 

Programmaticallystated,oursemanticsfor‘π’complementstheontology 

for πputforwardin(Brown,1990). 

Aconstructiondetermineswhatitconstructsbyconstructingit.Sothe 

logicofdeterminationconsistsintheconstructionaldescentfromaproceduretoitsproduct,asspecifiedforeachparticularkindofconstruction 

inDefinition2. Constructionsaretoofinelyindividuatedtofigureaslinguisticsenses,sincesomeoftheproceduraldifferencestheyembodyare 

logicallyinsignificantandarenotencodedlinguistically. Mostobviously, 

4 See(Tichý,1988,pp.228ff.).


two α-equivalentconstructionslike λx[ 0 > x 0 0]and λy[ 0 > y 0 0]arejustthat 

—twoconstructionsoftheclassofpositivenumbersandnotone;yetthe 

differencebetweenthe λ-boundvariables xand yisprocedurallyirrelevant. 

Thesolutiontothegranularityproblemconsistsinformingequivalence 

classesofprocedurallyisomorphicconstructionsandprivilegingamember 

ofeachsuchclassastheproceduralsenseofagivenunambiguoustermor 

expression.Technically,thequestisforasuitabledegreeofextensionality 

inthe λ-calculus. Needlesstosay,itremainsanopenresearchquestion 

exactlywhatthedesirablecalibrationoflinguisticsensesshouldbe,butour 

currentthesisisthatprocedures,andhencesenses,shouldbeidentifiedup 

to α-and η-equivalence. 5 

4 Kripke’s‘π’andours 

CentraltoKripke’sdenotationalsemanticsisthedistinctionbetweenfixing 

thereferenceandgivingthemeaning/asynonym. 6 OneofKripke’sillustrationsisthis: 

[‘π’]isnotbeingusedasshortforthephrase‘theratioofthecircumferenceofacircletoitsdiameter’[...]Itisusedasanameforareal 

number,whichinthiscaseisnecessarilytheratioofthecircumference 

ofacircletoitsdiameter.(Kripke,1980,p.60) 

Kripke’ssemanticsfor‘π’issimple(simplistic,asitturnsout): 

‘π’ 

rigidlydesignates 

Thedescription‘theratio...’ servestosingleouttheuniqueratioshared 

byallcircles,afterwhichthatnumberisbaptised‘π’. Thedescriptionis 

subsequentlykickedoffandsodoesnotformpartofthesemanticsproper 

ofKripke’s‘π’.Thisisproblematic.Nobodyknowsofsomeoneparticular 

realthatitis π.Sonobodyknowsofsomeoneparticularrealthatitisthe 

referenceof‘π’.Soitisobscurewhatlinguisticcompetencewithrespectto 

‘π’wouldconsistin.Notethatitisnotanoptiontosaythat‘π’designates 

whateverrealistheratioofacircle’scircumferencetoitsdiameter,forthis 

uniquenessconditionformsnopartofKripke’ssemanticsfor‘π’. 7 Kripke’s 

introductionof‘π’isimpeccable,andhis‘π’doesdenote π.Butwecannot 

5 See(Duˇzí,Jespersen,&Materna,ms.1, §2.2)or(Jespersen,ms). Fordiscussionof 

Frege’squestfortherightcalibrationofSinn,see(Sundholm,1994)and(Penco,2003). 

6 —adistinctionanticipatedatleastby(Geach,1969). 

7 TheKripkeancanhaverecoursetosomecausaltheoryofreferenceinthecaseofwords 

forempiricalentitiesliketigers,lemonsandgold. ButBenacerraf’sfirsthornblocks 

thisavenue.WesurmisethatKripkeanrigiddesignationcannotpossiblybeextendedto 

numericalconstantsandothertermsdenotingabstractentities. 

π


usehis‘π’todenote π,norcanweunderstandanyoneelse’suseof‘π’,since 

wecannotknowwhichparticulartranscendentalnumberis π. Inshort, 

Kripke’s‘π’hasbeenseveredradicallyfromanyhumanlypossiblelinguistic 

practice,soitisinoperative. 

Intheidiomofproceduralsemantics,Kripkefocusesentirelyonthe 

productattheexpenseoftheprocedure.Asamatterofmathematicalfact, 

3.14159...is π,butwhyintroduceanon-descriptivenamewhenthatname 

seversthelinkbetweencondition/procedureandsatisfier/product?Itseems 

thatonKripke’ssemanticsitwillbeadiscovery,andnotaconvention,that 

πistheratioofacircle’scircumferencetoitsdiameter.Ifso,italsoseems 

thatKripke’s‘π’misconstruesmathematicalpractice. 

Some π-producingproceduremustfigureinthesemanticsof‘π’;only 

how?TILfacesadilemmaofitsown,aswesawabove.Ontheonehand,a 

literalanalysisof‘π’woulddictatethatthesenseof‘π’be 0 π,yieldingthe 

schema 

‘π’ 0 π π 

expresses constructs 

Theadvantageofthisconstrualisthatwhatlookslikeaconstantisa 

constant(andnotadefinitedescriptionmasqueradingasone). However, 

thisistooclosetoKripke’s‘π’forcomfort. Wewouldbereinstatingthe 

problemthatthesemanticsof‘π’pairsnomathematicalconditionoffwith 

‘π’.Tomaster‘π’, 0 πwouldsuffice.TheTrivializationmerelyinstructsus 

toconstruct πandnotalsohowtoconstructit. 

Ontheotherhand,notleastepistemicconcernsdictatethatthesense 

of‘π’oughttobeanontologicaldefinitionof π,yieldingtheschema 

‘π’ [ιx[∀y[x = [ 0 Ratio [... y ...][... y ...]]]]] π 

expresses constructs 

(By‘ontologicaldefinition,wemeanacompoundconstruction(here,aComposition)that,inthiscase,constructsthenumber 

π,therebylayingdown 

what πis. Anontologicaldefinitioncontrastswithalinguisticdefinition, 

whichintroducesanewtermassynonymouswithanexistingterm.)This 

makes‘π’ashorthandtermsynonymouswith‘theratio...’,anditssense 

isanontologicaldefinitionof π.Theadvantageofthisconstrualisthatit 

pairsamathematicalconditionoffwith‘π’;butagain,which?Thereisno 

criteriontohelpdecidewhichofthepossibleontologicaldefinitionsshould 

bethesenseof‘π’.Itwouldbearbitrarytoselectoneandassignitassense; 

butassigningthemallintroduceshomonymy. 

Itwouldseemevidentthatalanguage-userneedstoknowatleastone 

definitionof πinordertouseandunderstand‘π’. Ifwegowiththe 

Trivialization-basedanalysisof‘π’,thefirststeptowardenhancingitis


tomakethelogico-semanticfactthat 0 πisequivalent with [ιx[∀y[x = 

[ 0 Ratio [...y ... ][...y ...]]]]]partofthesemanticsof‘π’. 0 πisindifferent 

tohow πisconstructedbythisorthatcompoundconstruction,soasfaras 

theequivalencerelationgoes,anycompound π-constructionisasgoodas 

anyother.‘π’maybeintroducedasequivalentwith 

or 

[ιx∀y[x = [ 0 Ratio [ 0 Area y][ 0 Square[ 0 Radiusy]]]]] (1) 

[ιx∀y[x = [ 0 Ratio [ 0 Circumferencey][ 0 Diameter y]]]], (2) 

oranyothercompound π-constructingconstruction.Understandingisanothermatter.Onethingistounderstand(1);anotherthingistounderstand 

(2).Onemaywellknowthat‘π’isequivalenttothisCompositionwithout 

knowing,ipsofacto,thatitisequivalenttothatComposition. 

5 Realisticrealism? 

Bothcausaltheoryofreferenceanddenotationalsemanticsareneitherhere 

northereasatheoryoftermsforabstractentitiessuchasnumbers. We 

areputtingforwardaproceduralsemanticsasarivaltheoryinordernot 

togetgoredbyBenacerraf’shornsorturninglinguisticcompetencewith 

mathematicalconstantsintoanenigma. Wesuggest,inthefinalanalysis, 

thatthesemanticsof‘π’oughttobethatitisshorthandfor,andthereforesynonymouswith,adefinitedescriptionexpressingadefinitionof 

π 

anddenotingthenumbersodefined. Butforeachdefinitionnof πthere 

isgoingtobeapair 〈‘π’,definitionn(π)〉.Sohowdowehandletheresultinghomonymy?SchwankungendesSinnesareneitherherenorthereina 

regimentedlanguagesuchasmathematese. Oursolutionrevolvesaround 

conceptualsystems. 

By‘conceptualsystem’wemeanasetofconstructionsthatisfullydeterminedbythechosensetofsimpleconcepts.SimpleconceptsareTrivialisationsofnonconstructionalentitiesoforder 

1.Thecompoundconceptsofa 

conceptualsystemarethenallthecompoundconstructionsthatareformed 

accordingtotherulesofDefinition2(plusperhapsinvolvingadditional 

constructions)usingsimpleconceptsandvariables.Theexactdefinitionof 

conceptualsystemcanbefoundin(Materna,2004). 

Relativetoaparticularconceptualsystem,apair 〈‘π’,definitionn(π)〉is 

anunambiguousassignmentofexactlyonedefinitionof πto‘π’,provided 

theconceptualsystemisindependent,i.e.,itssetofsimpleconceptsis 

minimal.Consequently,‘π’isnotambiguous,forthischaractermustalways 

begiventogetherwithaparticulardefinitionof πculledfromaparticular 

conceptualsystem. Theappearanceofambiguityarisesonlywhentwoor


moreconceptualsystemsareinvokedinthecourseofadiscourseinwhich 

tokensof‘π’occur. 

Theupshotofoursolutionisthatthereareseveral π-denotingconstants 

sharingthesamefirstelement,‘π’.Sowhentwomathematiciansareboth 

deployingtokensof‘π’,thereisariskofthemtalkingatcrosspurposes, 

untilandunlesstheycomparenotesand,incaseofinvokingdifferentconceptualsystems,cometoagreeonthesamedefinitionof 

πintheinterestof 

synonymy.Yetthemathematicalresultstheymayhave πindividuallyobtainedwithrespectto 

πareboundtobeequivalent,foranytwodefinitions 

of πareboundtoconvergeinthesamenumber.Afterall,theproblemwas 

alwaystodowithSchwankungendesSinnesandneverSchwankungender 

Bedeutung. 8 

BjørnJespersen 

SectionofPhilosophy,DelftUniversityofTechnology 

TheNetherlands 

b.t.f.jespersen@tudelft.nl 

MarieDuˇzí 

DepartmentofComputerScience,VSB–TechnicalUniversityOstrava 

17.listopadu15,70833Ostrava,CzechRepublic 

marie.duzi@vsb.cz 

http://www.cs.vsb.cz/duzi/ 

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Neighborhood Incompatibility Semantics for 

Modal Logic 

Kohei Kishida ∗ 

Thispaperintroducesneighborhoodsemanticsforpropositionalmodallogic 

intotheframeworkofBrandom’s(2008)incompatibilitysemantics.Neighborhoodsemanticsformodallogic,asitisconventionallystudied,canbe 

consideredtobeakindofpossible-worldsemantics,inthesensethata 

systemofneighborhoodscodifiesageneralizedaccessibilityrelationamong 

pointsofthespace,orworlds,atwhichthetruthvaluesofpropositionsare 

evaluated.Suchasemanticsfeaturestherepresentationalnotionsoftruth 

andpossibleworldasitsbasicprimitiveconstituents. Brandom’sincompatibilitysemantics,incontrast,isfoundedupontheinferentialnotionsof 

incoherenceandincompatibilityofsentences.Thechiefgoalofthispaperis 

toshowthatthisinferentialistframeworkofincompatibilitysemanticscan 

alsoadoptthenotionofneighborhoodtointerpretmodality,asthecoreidea 

ofneighborhoodsemanticsworksindependentlyoftherepresentationalnotions. 

1 IncompatibilitySemantics:AQuickReview 

Herewequicklyreviewthebasicdefinitionsandfactsinincompatibility 

semanticsthatarerelevanttothispaper;see(Brandom,2008)forafull 

expositionofthesemantics. 

Wewrite Lbothforagivensententiallanguageandforthesetofits 

sentences. Let Incbeanysubsetof PL,thepowersetof L,thatisclosed 

upwardintermsof ⊆,i.e.,if X ∈ Incand X ⊆ Y then Y ∈ Inc. Wesay 

Xisincoherentif X ∈ Inc;then Incbeing ⊆-upwardclosedmeansthat 

addingmoresentencestoanincoherentset Xofsentencesnevercuresthe 

incoherence. Wealsosay Y isincompatiblewith Xif X ∪ Y ∈ Inc,and 

∗ TheauthorwouldliketothankAlpAker,RobertBrandom,JaroslavPeregrin,José 

MartínezFernández,andespeciallyNuelBelnap,forinsightfulcommentsandhelpful 

suggestions.

80 KoheiKishida 

write I(X)forthecollectionof Y ⊆ PLincompatiblewith X,i.e. 

I(X) = {Y ⊆ L | X ∪ Y ∈ Inc}. 

When p ∈ L,wewrite I(p)for I({p}). 

Theentailmentrelation, p � q,isdefinedby I(q) ⊆ I(p),i.e.,thatif X 

isincompatiblewith qthenitisincompatiblewith p. Ingeneral, X � Y, 

i.e.,theconjunctionof Xentailingthedisjunctionof Y,isdefinedby 

� 

X � Y ⇐⇒ I(p) ⊆ I(X), 

thatis,anythingthatrulesoutall p ∈ Yrulesout X.Applyingthistothe 

case Y = ∅inparticular,with � 

I(p) = PL,wehavethefollowing,which 

p∈∅ 

p∈Y 

agreeswithwhatisusuallymeantby X � ∅: 

X � ∅ ⇐⇒ PL ⊆ I(X) ⇐⇒ X ∈ Inc. 

When Lhasanegationoperator ¬,weassumethat Isatisfies 

X ∈ I(¬p) ⇐⇒ X � p forevery p ∈ L; 

i.e., ¬pisincompatiblewithallandonly Xthatentail p. Thenwehave 

I(¬¬p) = I(p),andhence X ∈ I(p) ⇐⇒ X � ¬p. Also,when Lhasa 

disjunctionoperator ∨, Iisassumedtosatisfy I(p ∨ q) = I(p) ∩ I(q);i.e., 

Ximpliesneither pnor qisthecaseifandonlyifitdeniesboth pand q. 1 

Apair (L, I)ofsuch Land Iiscalledanincompatibilityframe. 

2 NeighborhoodSemanticsinthePossible-WorldFramework 

Tointroduceneighborhoodincompatibilitysemantics,itishelpfultofirst 

reviewtheneighborhoodsemanticsasconventionallystudiedinthepossibleworldframeworkandtothendrawaformalanalogy. 

Letusrecallthatpossible-worldsemanticsinterpretsamodallanguage L 

withasetÏofpossibleworldsbyassigningtoeachsentence p ∈ Lasubset 

�p�ofÏ,sometimescalledaproposition.Thenthataworld w ∈Ïliesin 

theinterpretation �p�of pmeans pistrueat w,andhence psemantically 

entails qifandonlyif �p� ⊆ �q�.So,forexample,theTaxiom ✷p ⊢ p ⊢ ✸p 

ofmodallogiccorrespondsto �✷p� ⊆ �p� ⊆ �✸p�.Thissuggests,fromthe 

pointofviewoftopology,thatthe ✷operatorcorrespondstoageneralized 

interioroperation,while ✸correspondstoageneralizedclosure,definedon 

asystemofneighborhoodsonÏasfollows. 

1 Theofficialdefinitionin(Brandom,2008)firstdefines I(p ∧ q) = I({p, q})andthen 

defines p ∨ qas ¬(¬p ∧ ¬q),whichstillentails I(p ∨ q) = I(p) ∩ I(q).

NeighborhoodIncompatibilitySemantics 81 

Eachworld w,orpointofthespaceÏ,isassignedacollectionofsubsets 

ofthespace,calledtheneighborhoodsof w;wewrite Nwforthiscollection. 

Then,given A ⊆ X,apoint wisintheinterior(inthegeneralizedsense) 

of Aifandonlyifithasaneighborhood U containedin Atowitness 

that wis“wellinside” A. And wisintheclosureof Aifandonlyifall 

ofitsneighborhoodsintersect A,orinotherwords,ifandonlyif whas 

noneighborhood Udisjointfrom Atowitnessthat wis“welloutside” A. 

Now,givenaninterpretation �p� ⊆ Xof p,itsinteriorandclosureinterpret 

✷pand ✸p,respectively. Moreformally,wehavethefollowing(✷pw)and 

(✸pw)(thesubscript“pw”isjusttoconnote“thepossible-worldcase”): 2 

w ∈ �✷p� ⇐⇒ ∃U ∈ Nw · U ⊆ �p�, (✷pw) 

w ∈ �✸p� ⇐⇒ ∀U ∈ Nw · U ∩ �p� �= ∅. (✸pw) 

Letusnotetwothingshere.First,thisneighborhoodsettingsubsumes 

Kripkesemantics.ThatisbecauseKripkesemanticsisobtainedfromthis 

neighborhoodsemanticsbyfurtherassumingthateachworld whasexactly 

oneneighborhood Rw,calledtheworlds“accessiblefrom w”.Then(✷pw) 

and(✸pw)boildowntothefollowingconditions,whichareclearlyequivalent 

totheusualtruthconditionsfor ✷pand ✸p: 

w ∈ �✷p� ⇐⇒ Rw ⊆ �p�, 

w ∈ �✸p� ⇐⇒ Rw ∩ �p� �= ∅. 

Second,thisneighborhoodsemanticshas ✷and ✸dualtoeachother,i.e., 

✸isjust ¬✷¬,while ✷isjust ¬✸¬,becausethecondition �¬p� =Ï\�p� 

(i.e.,thatnegationisinterpretedbycomplementinÏ)implies 

w ∈ �✸p� ⇐⇒ ∀U ∈ Nw · U ∩ �p� �= ∅ by(✸pw) 

⇐⇒ ¬∃U ∈ Nw · U ⊆Ï\�p� = �¬p� 

⇐⇒ w /∈ �✷¬p� by(✷pw) 

⇐⇒ w ∈Ï\�✷¬p� = �¬✷¬p�. 

2 Thisdefinitionofinteriorandclosureislessgeneralthanthestandardversioninneighborhoodsemantics(see,e.g.,(Chellas,1980)). 

Indeed,theformerisequivalenttothe 

latterwiththeassumptionthatallfamiliesofneighborhoodsare ⊆-upwardclosed.The 

reasonIadoptthelessgeneraldefinitioninthispaperisaphilosophicalonethat,when 

importedtotheincompatibilityframework,itrendersavailabletousthecounterfactualrobustnessinterpretationofneighborhoods,whichwewillseeinSection3. 

Thereisno 

technicalreasonwecouldnotadoptandimportthestandard,fullygeneralformulation 

totheincompatibilityframework.


3 NeighborhoodSemanticsintheIncompatibilityFramework 

Toimporttheideafromthepossible-worldframeworktotheincompatibility 

framework,letuscomparethetwoframeworksattheground(i.e.nonmodal)level. 

First,whilethepossible-worldsemanticsinterpretssentences pwithsubsets 

�p�ofÏcontainingworlds w,theincompatibilitysemanticsinterprets 

themwithsubsets I(p)of PLcontainingsets Xofsentences. Thiscomparisonsuggeststhatweconsider 

PLratherthanÏtobethespace,and 

X ∈ PLratherthan w ∈Ïtobepointsinthisspace. 

Thenrecallthat,becausethatapoint Xofthisspace PLliesinasemanticinterpretant 

I(p)meansincompatibility,entailment p � qcorrespondsto 

thereverseinclusion I(p) ⊇ I(q)intheincompatibilityframework,rather 

thantheinclusion �p� ⊆ �q�inthepossible-worldcase.Forexample,theT 

axiom ✷p ⊢ p ⊢ ✸pcorrespondsto I(✸p) ⊆ I(p) ⊆ I(✷p).Thissuggests 

that,intheincompatibilityframework, ✸shouldbeinterpretedbyinterior 

ratherthanclosure,and ✷byclosureratherthaninterior. 

Therefore,theneighborhoodincompatibilitysemanticsshouldsimplyreplaceÏwith 

PL,andswitch ✸and ✷intheinterpretation.Thisideacan 

beputasfollows.Aneighborhoodincompatibilityframeisatriple (L, I, N) 

consistingof: 

•A(sentential)language Lwithmodaloperators; 

•Amap Isuchthat (L, I)isanincompatibilityframeon L,treating 

thenon-modaloperatorsof Lproperly(inthemannerreviewedin 

Section1); 

•Aneighborhoodfunction N : PL → PPPLwhoseinteriorandclosure 

operationsinterpret ✸and ✷,i.e.,thatsatisfiesthefollowing: 

X ∈ I(✸p) ⇐⇒ ∃U ∈ NX · U ⊆ I(p), (✸) 

X ∈ I(✷p) ⇐⇒ ∀U ∈ NX · U ∩ I(p) �= ∅. (✷) 

Wedefine 

� 

�asbefore: (L, I, N)has X � Y ifandonlyif (L, I)has 

I(p) ⊆ I(X). 

p∈Y 

Asthisdefinitionispurelyformal,weneedtoexplainwhat,conceptually, 

isgoingonhere.First,fixapoint X ∈ PL.Then Xisasetofsentences, 

e.g., p /∈ X, q ∈ X,... When U ⊆ PLisaneighborhoodof X,namely 

U ∈ NX,itcontainsotherpoints Y, Z,... eachofwhichisasetofsentences,e.g., 

p ∈ Y, q /∈ Y,... Then Ymightbeobtainedbyadding pto X, 

dropping qfrom X,andsoon,andsimilarlyfor Z.Hencewecanconsider 

Y or Ztobemodifying Xwithcounterfactualhypotheses;forexample,


whenweareattheinformationstate X, Y isjust Xwiththecounterfactualsupposition“If 

pwereknowntrue,but qwereunknown,andsoon”. 

Theneachneighborhood U ⊆ PLof Xisan“admissible”wayofgrouping 

togethersuchcounterfactualhypotheseson X,wherethe“admissibility”is 

formallyexpressedby Ulyingin NX. 

Now,underthisinterpretationofneighborhoods U ∈ NX, U ⊆ I(p)(i.e., 

∀Y ∈ U[Y ∈ I(p)])means“Whatevercounterfactualhypothesis Y (within 

therangeof U)wemaymakeon X,itwouldstillbeincompatiblewith 

p”. Inshort, U ∈ NXsuchthat U ⊆ I(p)witnessestheincompatibility 

of X with piscounterfactuallyrobust. Accordingly,(✸)statesthat X 

isincompatiblewithpossibly-pifandonlyifsomeneighborhood Uof X 

witnessesthecounterfactualrobustnessof Xbeingincompatiblewith p.On 

theotherhand, U ∈ NXsuchthat U ∩ I(p) = ∅(i.e., ∀Y ∈ U[Y /∈ I(p)]) 

witnessesthatthecompatibilityof Xwith piscounterfactuallyrobust,and 

hence(✷)statesthat Xisincompatiblewithnecessarily-pifandonlyifthe 

counterfactualrobustnessof Xbeingcompatiblewith pisneverwitnessed. 

Or,rewriting(✷)intermsofentailmentwith X � ¬p ⇐⇒ X ∈ I(p),we 

have 

X � ¬✷p ⇐⇒ X ∈ I(✷p) ⇐⇒ ∀U ∈ NX · U ∩ I(p) �= ∅ 

⇐⇒ ∀U ∈ NX∃Y ∈ U · Y ∈ I(p) 

⇐⇒ ∀U ∈ NX∃Y ∈ U · Y � ¬p; 

thatis, Xentailsnot-necessarily-pifandonlyifeveryneighborhood Uof 

Xcontainsacounterfactualhypothesis Ythatentailsnot-p. 

4 LogicforNeighborhoodIncompatibilitySemantics 

Thissectionreviewswhatrulesandaxiomsarevalidorinvalidinneighborhoodincompatibilitysemantics.Bysayingthatanaxiom(scheme) 

X ⊢ Y 

isvalidinaneighborhoodincompatibilityframe (L, I, N),wemeanthat 

(L, I, N)has X � Y,i.e. � 

I(p) ⊆ I(X)(forallinstancesofthescheme). 3 

p∈Y 

Also,bysayingthatarule(scheme) 

X0 ⊢ Y0 

X1 ⊢ Y1 

isvalidin (L, I, N),wemeanthatif (L, I, N)has X0 � Y0thenithas 

X1 � Y1(forallinstancesofthescheme). 

3 Weuse �and ⊢differentlyasfollows: X � Yisastatementthat Xentails Y(inagiven 

frame);incontrast, X ⊢ Y isasequentratherthanastatement.


First,the ✸operatorpreservestheorderofentailment �;thatis,the 

ruleMbelowisvalidinallneighborhoodincompatibilityframes. 4 

p ⊢ q 

✸p ⊢ ✸q 

Misvalidbecause p � q(i.e., I(q) ⊆ I(p))implies ✸p � ✸q(i.e., I(✸q) ⊆ 

I(✸p))inanyframe. Toshowthis,assume I(q) ⊆ I(p)and X ∈ I(✸q). 

Then,by(✸), Xhassome U ∈ NXsuchthat U ⊆ I(q) ⊆ I(p),which 

means,againby(✸),that X ∈ I(✸p).Asimilarargumentshowsthat ✷also 

preservesentailment,becauseanyneighborhoodintersecting I(q) ⊆ I(p) 

intersects I(p)aswell. 

Apartfromthesepreservationrules,neighborhoodsemanticsissogeneral 

astoprovidecounterexamplestomanyrulesandaxiomsthatarevalidin 

other(mostnotablyKripke’srelational)semantics.Forexample, ✸maynot 

preserveincoherence;thatis,evenwhen pisincoherent(viz., I(p) = PL), 

✸pmaybecoherent(viz., I(✸p) �= PL),therebyfailingtherule 

M 

p ⊢ 

. N✸ 

✸p ⊢ 

Thisfailsinapathologicalframeofneighborhoodswheresome X ∈ PL 

has NX = ∅(andhence,by(✸), X /∈ I(✸q)forany q).Also,therule 

p ⊢ 

✷p ⊢ 

canfailinanotherkindofpathologicalframewheresome X ∈ PLhas 

∅ ∈ NX(andhence,by(✷), X /∈ I(✷q)forany q).Infact,N✸andN✷are 

validintheframeswithoutthesepathologies. 

Thefactsdescribedsofarapplytoneighborhoodsemanticsingeneral, 

notonlyintheincompatibilityframeworkbutalsointheconventional 

possible-worldframework. Onemajordivergenceoftheformerfromthe 

latteristhefollowingpointregardingcompleteness.Notethateveryframe 

satisfiesoneofthefollowing(1)–(3): 

N✷ 

N∅ �= ∅but ∅ /∈ N∅, (1) 

∅ ∈ N∅, (2) 

N∅ = ∅. (3) 

Themodallogic MNthatisobtainedbyaddingM,N✸,N✷toclassical 

logicissoundandcompletewithrespecttotheframessatisfying(1),inthe 

4 Thisrulecanbeavoidedifweadoptthestandard,moregeneraldefinitionofinterior; 

seeFootnote2.TherulethatisvalidinsteadofMinthemoregeneralformulationisto 

infer ✸p ⊢ ✸qfromboth p ⊢ qand q ⊢ p.


sensethatasequentoraninferencefromsequentstoanotherisvalidinall 

thoseframesif(soundness)andonlyif(completeness)itisatheoremora 

derivableruleof MN. Also,let MI✸and MI✷bethelogicsobtainedby 

addingMaswellasthefollowingaxiomsI✸andI✷,respectively,toclassical 

logic: 

✸p ⊢, I✸ 

✷p ⊢ . I✷ 

Then MI✸and MI✷aresoundandcompletewithrespecttotheframes 

satisfying(2)and(3),respectively. Moreover,weneedtoformulatethe 

completenessofanaxiomorruleinamannerthatdependsonthesethree 

classesofframes,asfollows. WesayanaxiomorruleAiscompletewith 

respecttoasemanticconditionCifthelogicsobtainedbyaddingAto MN, 

MI✸, MI✷,respectively,arecompletewithrespecttotheclassesofframes 

satisfyingCaswellas(1),(2),(3),respectively.Ontheotherhand,wecan 

definethesoundnessofAwithrespecttoCintheusualmanner. 

Themostnotabledivergenceoftheneighborhoodincompatibilitysemanticsfromtheconventionalneighborhoodsemanticsisthatthedualityof 

✸ 

and ✷mayfailintheformer,eventhoughthenon-modalbaseofthelogic 

isclassical. Recallthattheproof(attheendofSection2)oftheduality 

inthepossible-worldframeworkwasbasedontheinterpretationofnegation, 

¬,intermsofcomplementinthespaceÏ,sothat �p� ∩ �¬p� = ∅ 

and �p� ∪ �¬p� =Ï. Theincompatibilityframeworkinterprets ¬differently,whichiswhythedualityfailsintheframework.Eventhoughafull 

constructionofcounterexamplesrequirestoomanydetailstocoverhere, 

aroughbutheuristicdescriptioncanbegivenasfollows. Intheincompatibilityframework, 

I(p)and I(¬p)normallyintersectwitheachother; 

indeed, I(p) ∩ I(¬p)equalstheset Incofincoherentsetsofsentences.So,a 

coherentpoint Xmayhaveanonempty U ⊆ Inc = I(p) ∩ I(¬p)asitsonly 

neighborhood.Then U ∩ I(p) �= ∅andhence X ∈ I(✷p)by(✷)(because 

NX = {U}),butatthesametime U ⊆ I(¬p);thismeans X ∈ I(✸¬p) 

by(✸),whichthenimplies X /∈ I(¬✸¬p)since Xiscoherent,thatis, 

X /∈ Inc = I(✸¬p) ∩ I(¬✸¬p).Sothis Xwitnesses I(✷p) �⊆ I(¬✸¬p),i.e., 

¬✸¬p � ✷p.Theupshotisthatthisentailmentfailswhenneighborhoods 

aretoostrong(i.e.,whenpointsinthemlieinboth I(p)and I(¬p)),which 

cannothappenintheclassicalpossible-worldframework(i.e.,nopointever 

liesinboth �p�and �¬p�).Quiteexpectably,theotherdirection ✷p � ¬✸¬p 

ofthedualityfailswhenneighborhoodsaretooweak(i.e.,whenpointsin 

themlieinneither I(p)nor I(¬p)),whichcannothappenintheclassical 

possible-worldframework(i.e.,everypointhastolieineither �p�or �¬p�). 

Whiletherearecertainsemanticconditionstotheeffectthatneighborhoodsarenottoostrong,ornottooweak,withrespecttowhicheitherof


¬✸¬p ⊢ ✷pand ✷p ⊢ ¬✸¬pissoundandcomplete,themoreimportant 

pointisthatneighborhoodsemanticsismoreexpressiveintheincompatibilityframeworkthanintheclassicalpossible-worldframework.Thisisnot 

merelyinthesensethatthedualityaxiomsareinvalid,butthesemantics 

separatesthe ✸and ✷versionsofmanyaxioms;forexample, 

p ⊢ ✸p, T✸ 

✷p ⊢ p. T✷ 

Eventhoughtheseaxiomsaretreatedjustasequivalentinpossible-world 

semantics,theycorrespondtotwodifferentsemanticconditionsonneighborhoodincompatibilityframes.Tolayouttheseconditions,weneedtointroduceapreorder(i.e.areflexiveandtransitiverelation) 

�on PLdefinedasfollows: 

X � Y ⇐⇒ I(X) ⊆ I(Y ). 

So, X � Y roughlymeans Xisweakerthan Y,orthat Y conjunctively 

entails Xconjunctively. Then �generalizes ⊆inthesensethat X ⊆ Y 

entails X � Y, 5 andmoreovereverysemanticinterpretant I(p)isclosed 

upwardintermsof �.Then,givenanyset U ⊆ PLofpoints,wewrite ↑U 

and ↓Uforthe �-upwardand �-downwardclosuresof U ⊆ PL,respectively, 

i.e., 

↑U = {Y ∈ PL | X � Yforsome X ∈ U}, 

↓U = {X ∈ PL | X � Yforsome Y ∈ U}. 

Inthepossible-worldframework,theTaxiom p ⊢ ✸p(or ✷p ⊢ p)correspondstothesemanticconditionthat 

w ∈ Uforall U ∈ Nw.Incontrast, 

intheincompatibilityframework,usingthenotionsdefinedabovewecan 

modifythisconditiontoobtainthefollowingtwoversions: 

X ∈ ↑U forall U ∈ NX, (4) 

X ∈ ↓U forall U ∈ NX. (5) 

Then(4)and(5)havetheaxiomsT✸andT✷,respectively,soundand 

complete. 

Hereweonlyshowthesoundness. ThatT✸issoundwithrespectto 

(4)meansthat(4)entails I(✸p) ⊆ I(p). Toshowthis,letusassume 

X ∈ I(✸p). Itmeanssome U ∈ NXisincludedin I(p). Then,because 

I(p)is �-upwardclosed, ↑Uisstillincludedin I(p). Hence(4)implies 

X ∈ ↑U ⊆ I(p),therebyestablishingthesoundness. Wecansimilarly 

5 Therearemoresensesinwhichwecansay �generalizes ⊆.See(Kishida,n.d.-b).


showthesoundnessofT✷withrespectto(5),i.e.,(5)entailing I(p) ⊆ 

I(✷p),byassuming X /∈ I(✷p).Thismeanssome U ∈ NXisdisjointfrom 

I(p).Then,againbecause I(p)is �-upwardclosed, ↓Uisstilldisjointfrom 

I(p).Hence(5)implies X ∈ ↓Uandtherefore X /∈ I(p),establishingthe 

soundness. 

Upwardanddownwardclosuresdifferentiatethe ✸and ✷versionsinthe 

caseoftheTaxioms,andtheydosointhefollowingcaseaswell.Consider 

thecondition: 

If U ∈ NX,thereis V ∈ NXsuchthat 

every Y ∈ Vhassome UY ∈ NYwith UY ⊆ U. (6) 

Thismeansthateveryneighborhood Uof Xhasanother Vof Xsuchthat 

Uis(asupersetof)aneighborhoodofeverypointin V;inshort,(6)says 

thateachneighborhoodisaneighborhoodofaneighborhood.Nowreplace 

thelast Uin(6)with ↑U,toobtain: 

If U ∈ NX,thereis V ∈ NXsuchthat 

every Y ∈ Vhassome UY ∈ NYwith UY ⊆ ↑U. (7) 

Thenthefollowing ✸versionoftheS4axiomissoundandcompletewith 

respectto(7): 

✸✸p ⊢ ✸p. S4✸ 

Toshowthesoundness,i.e.,that(7)entails I(✸p) ⊆ I(✸✸p),assume 

X ∈ I(✸p). Thismeans I(p)includessome U ∈ NXand ↑U. Then(7) 

yields V ∈ NXsuchthatevery Y ∈ V hassome UY ⊆ ↑U ⊆ I(p),i.e. Y ∈ 

I(✸p),whichmeans V ⊆ I(✸p).Therefore Vwitnesses X ∈ I(✸✸p). 

Replacing ↑Uwith ↓Uin(7),wecanshowbyessentiallythesameidea 

thatS4✷issound: 

✷p ⊢ ✷✷p. S4✷ 

Hereis,however,someasymmetrybetween ✸and ✷.EventhoughS4✷is 

soundwithrespecttothecondition(7)with ↓Uinplaceof ↑U,itisnot 

complete.Toachievethecompleteness,weneedtoweakentheconditiona 

littlebit,toobtain: 

If N∅ �= ∅,thenthefollowingholdsforevery X ∈ PL: 

if U ∈ NX,thereis V ∈ NXsuchthat 

every Y ∈ Vhassome UY ∈ NYwith UY ⊆ ↓U.


Inthefollowingcasetheasymmetrybetween ✸and ✷isevenbigger. 

Considerthefollowingaxioms(whicharedualtoeachotherintheconventionalframework): 

C✸issoundandcompletewithrespectto: 

✸(p ∨ q) ⊢ ✸p ∨ ✸q, C✸ 

✷p ∧ ✷q ⊢ ✷(p ∧ q). C✷ 

U0,U1 ∈ NX =⇒thereis U2 ∈ NXsuchthat U2 ⊆ ↑U0and U2 ⊆ ↑U1. 

(8) 

Forthesoundness(i.e.,(8)entailing I(✸p∨✸q) ⊆ I(✸(p∨q))),assume X ∈ 

I(✸p ∨✸q) = I(✸p) ∩ I(✸q).Thismeans Xhassomeneighborhoods U0 ⊆ 

I(p)and U1 ⊆ I(q),which,asalways,entails ↑U0 ⊆ I(p)and ↑U1 ⊆ I(q). 

Now,(8)yields U2 ∈ NXsuchthat U2 ⊆ ↑U0 ⊆ I(p)and U2 ⊆ ↑U1 ⊆ I(q), 

i.e. U2 ⊆ I(p)∩I(q) = I(p∨q).Hence U2witnesses X ∈ I(✸(p∨q)).Inthis 

way,C✸issoundwithrespectto(8),andinfactcomplete. Nevertheless, 

theannouncedasymmetrybetween ✸and ✷isthatitisanopenproblem 

evenwhatconditionhasC✷soundandcomplete,becauseitdoesnotseem 

toworktoreplace ↑Uwith ↓U. 6 Itisalsointerestingtoseehow ✸and ✷ 

interactwitheachother.Theaxiom 

✷p ⊢ ✸p D 

issoundandcompletewithrespecttotheconditionthat U ∩ V �= ∅forall 

U,V ∈ NX.ItiseasytoshowDtobesoundbecauseif X ∈ I(✸p),i.e.,if 

a U ∈ NXhas U ⊆ I(p),thentheconditionsaysevery V ∈ NXintersects 

U,therebyintersecting I(p),i.e. X ∈ I(✷p). 

Itisanopenproblemwithrespecttowhatconditionthefollowingaxiom 

Biscomplete: 

p ⊢ ✷✸p; B 

butthereisaconditionwithrespecttowhichBissound: 

Givenacollection {Xi ∈ PL | i ∈ I }ofanysizeand Y ∈ PL, 

ifeach Xihassome Ui ∈ NXiwith Y /∈ Ui, 

thenaV ∈ NYhas Xi /∈ Vforall Xi. (9) 

Toshowthesoundness(i.e.,(9)entailing I(✷✸p) ⊆ I(p)),suppose Y /∈ 

I(p). Write I(✸p) = {Xi ∈ PL | i ∈ I};theneach Xiliesin I(✸p),i.e., 

Xihassome Ui ∈ NXi suchthat Ui ⊆ I(p)andhence Y /∈ Ui. Then(9) 

yields V ∈ NY suchthat Xi /∈ V forall Xi,i.e. V ∩ I(✸p) = ∅;therefore 

Y /∈ I(✷✸p). 

6 Thedifficultyarises partlyfromthelackofunderstandingofwhat I(✷p ∧ ✷q) = 

I({✷p, ✷q})lookslike,incontrastto I(✸p ∨ ✸q)understoodsimplyas I(✸p) ∩ I(✸q).


5 Conclusion 

Wehaveshownthattheideaofneighborhoodsemanticstointerpretmodal 

operatorswithinteriorandclosureoperationscanbestraightforwardlyimported—quiteindependentlyofthenotionsoftruthandpossibleworlds—totheframeworkofincompatibilitysemantics.Thephilosophicaladvantageofputtingthenotionofneighborhoodinthisframeworkisthattheconnectionbetweenneighborhoodsandmodalitycanbedirectlyandnaturallyinterpretedintermsoftheideaofcounterfactualrobustness.Wehave 

alsoshownthetechnicalmeritofthesemanticsthatwecanseparatethe 

behaviorsof ✸and ✷whilekeepingthenon-modalbaseofthelogicclassical,which,combinedwiththecounterfactual-robustnessinterpretation,will 

enableustoapplymodallogictoanevenwiderrangeofcases. 

KoheiKishida 

DepartmentofPhilosophy,UniversityofPittsburgh 

1001CathedralorLearning,Pittsburgh,PA15260U.S.A. 

kok6@pitt.edu 

References 

Brandom,R. (2008). Incompatibility,modalsemantics,andintrinsiclogic. In 

Betweensayinganddoing:Towardsananalyticpragmatism(chap.5). Oxford: 

OxfordUniversityPress. 

Chellas,B.(1980).Modallogic:Anintroduction.Cambridge-NewYork:CambridgeUniversityPress. 

Kishida,K. (n.d.-a). Neighborhoodincompatibilitysemanticsandcompleteness. 

(Draft.) 

Kishida,K.(n.d.-b).Possible–worldrepresentationofincompatibility.(Draft.)

What do Gödel Theorems Tell us about 

Hilbert’s Solvability Thesis? 

Vojtěch Kolman ∗ 

Whendealingwiththefoundationalquestionsofelementaryarithmetic, 

wefindourselvesstandingintheshadowofGödel,justasourpredecessors 

stoodintheshadowofKant,totheextentthatwetendtoseeGödel’s 

famousincompletenesstheoremsasanewCritiqueofPureReason.Inits 

mostexuberantform(commonparticularlyamongtheso-calledworking 

mathematicians)thisamountstoclaimingthat 

humanreasonhasencountereditslimitsbyprovingthatthereare 

truthswhicharehumanlyunprovable(“inaccessible”)andthatitis 

impossibleforourmindtoproveitsownconsistency. 1 

Thisattitudeisnotonlyatvariancewiththe(Kantian)doubtsaboutthe 

possibilityofprovingtheunprovabilityinanabsolutesense,but,more 

specificallyandfamously,withtheso-calledHilbertprogramofsolvingeverymathematicalproblembyaxiomaticmeans. 

InhisParisianaddress, 2 

Hilbertnotonlyphrasedtheconjecturethatallquestionswhichhuman 

mindasksmustbeanswerable(theso-calledaxiomofsolvability) 3 but 

supplementedit,asakindofchallenge, withalistoftenandlaterof 

twenty-threeproblemsofprimeinterest,includingtheSecondProblemof 

theconsistency(andcompleteness)ofarithmeticalaxioms. 

InHilbert’slaterwritings,particularlyinhisKönigsbergaddress, 4 the 

solvabilityargumenttakesamoresubtleform.Introducingthefinitemode 

∗ WorkonthispaperhasbeensupportedinpartbygrantNo.401/06/0387oftheGrant 

AgencyoftheCzechRepublicandinpartbytheresearchprojectMSM0021620839of 

theMinistryofEducationoftheCzechRepublic. 

1 (Gödel,1995,p.310)himselfphraseditlikethis: “thereexistabsolutelyunsolvable 

diophantineproblems[...],wheretheepithet‘absolutely’meansthattheywouldbe 

undecidable,notjustwithinsomeparticularaxiomaticsystem,butbyanymathematical 

proofthehumanmindcanconceive.” 

2 See(Hilbert,1900). 

3 See(Hilbert,1900,p.297). 

4 See(Hilbert,1930).

92 VojtěchKolman 

ofthought(finiteEinstellung) 5 asanewkindofKantianintuition,Hilbert 

arguesthattheharmonybetweennature(experience)andthought(theory) 

mustlieexactlyinthetranscendentalfacttheyarebothfinite. 6 Asa 

consequence,theseeminginfinityofhumanknowledge(particularlyinthe 

realmofmathematics)musthavefiniterootswhicharetobeidentified 

withafinite(orfinitelydescribable)systemofrulesandaxioms,andfinite 

deductionsfromthem. 7 Hence,“wemustknow,weshallknow.” 8 Obviously, 

thisisatranscendentaldeductionofitsownkind,namelyofinferentialismor 

broaderaxiomatismfromfinitism,startingwiththewords:inthebeginning 

wasasign. 9 

Gödel(1931),soweareusuallytold,putanendtoHilbert’soptimism 

byprovingthattheSecondProblemisessentiallyunsolvable.Thisverdict 

issometimessupportedbytheseeminglyanalogouscaseofHilbert’sFirst 

Problem,theContinuumHypothesis,which,partiallyalsoduetoGödel, 

wasprovedtobeundecidableonthebasisofcurrentlyacceptedaxioms.In 

thispaperIwouldliketopresentGödel’stheoremsnotasadirectrefutation 

ofHilbert’saxiombutonlyasanimpulsetophraseitwithmorecaution, 

insuchawaythattheContinuumHypothesisisnolongerregardedasa 

realproblem. Iwilldrawontworatherdifferentsources,both,however, 

connectedtoHilbert’sphilosophy,namely 

•thelatemetamathematicalviewsofZermeloand 

•Lorenzen’spost-Hilbertianprogramofoperativemathematics. 

Thiswillleadmetoacloseranalysisofthedistinctionbetweenproofand 

truthwhichdoesnotendorseoneofthemattheexpenseoftheother,as 

Lorenzen,theconstructivist,andZermelo,thePlatonist,stilltendtodo. 

1 

First,letusdiscussthepossibilityofprovingtheunsolvabilityofsomething. 

Thereisageneralpattern:ifsomeonecomesalongwithapositivesolution 

toagivenproblem,onecanchecktoseethatitdoestherequiredwork. 

Butifitistobeshownthattheproblemisunsolvable,onehastogivea 

precisedelimitationofthemethodsthatcanbeemployed.Thisbringsusto 

thedifferencebetweenmethodinthebroader(general)andinthenarrower 

(limited)sense. 

5 See(Hilbert,1930,p.385)andalso(Hilbert,1926,p.161). 

6 See(Hilbert,1930,pp.380–381). 

7 See(Hilbert,1930,p.379)andalso(Hilbert,1918). 

8 (Hilbert,1930,p.387). 

9 See(Hilbert,1922,p.163).

WhatdoGödelTheoremsTellusaboutHilbert’sSolvabilityThesis? 93 

Toillustratethepointletustakesomefamousgeometricalproblemslike 

thetrisectionofanangleorthequadratureofthecircle.Duetothemethods 

ofmodernalgebrawepositivelyknowthattheseproblemsareunsolvable 

bystraightedgeandcompass. However,wealsoknowthattheancient 

mathematicians(Hippias,Archimedes)alreadysolvedthembyextended 

—so-calledmechanical—means(quadratix,spiral) 10 wheretheepithet 

“mechanical”meansmainlythattheyweredevisedadhoc. Similarly,ifI 

giveyou—meaningsomebodysufficientlyeducatedinpredicatelogic—a 

formula,Iamquitesureyouwillbeabletodecide,inafinitenumberof 

steps,whetheritisatautologyornot.Whatyoumightnotbeabletodo, 

however,istosolvetheproblemwithpre-chosenschematicmethodssuch 

aswithoneparticularTuringmachine. 

Now,asmaybeexpected,asimilarobservationappliestoGödel’stheorems,onlythistimeitistheprovabilityitselfthelimitsofwhichbegthe 

question.Gödelshowedthatforanyaxiomaticsystemofarithmeticthere 

willalwaysbeanindividualsentencethatisundecidablebyit. Thegist 

ofhisargumentliesinthefactthatthisunprovablesentenceofarithmetic 

(informallysaying“Iamunprovable”)isunprovablebecauseitistrue(it 

isunprovable),itstruthbeingprovenasapartoftheargument. So,the 

wholeargumentworksonlybecauseitemploystwodifferentconceptsof 

proof,thefirstbeingthatofPrincipiaMathematica(orPeanoarithmetic) 

andthesecondbeingthebroaderoneinwhichtheargumentisclinched. 

Zermelo,inhisunjustlyinfamouscorrespondencewithGödel,wasprobablythefirstpersontomakethisobservation.Settinghimselfthenatural 

question,“Whatdoesoneunderstandbyaproof?”,hisanswerwentlike 

this: 

Ingeneral,aproofisunderstoodasasystemofpropositionsthat, 

whenacceptingthepremises,yieldsthevalidityoftheassertionas 

beingreasonable. Andthereremainsonlythequestionofwhatmay 

be“reasonable”. Inanycase—asyouareshowingyourself—not 

onlythepropositionsofsomefinitaryschemethat,alsoinyourcase, 

mayalwaysbeextended. So,inthisrespect,weareofthesame 

opinion,however,Iaprioriacceptamoregeneralschemethatdoes 

notneedtobeextended.Andinthissystem,reallyallpropositions 

aredecidable. 11 

Whatneedstobeexplainednowisthenatureofthedifferencebetween 

proofinthenarrowerandbroadersense,orbetweentheproof andtruth, 

andthesenseinwhichthesecondoneis“decidable”,orbetter:complete 

andunextendable,asZermeloclaims. 

10 See,e.g.,(Heath,1931). 

11 See(Gödel,2003,p.431).


2 

Theanalogousdifferencesbetweenthegeneralandnarrowerconstruability 

ordecidabilityislessproblematicsincetheadhocconstructiveordecision 

methods(likequadratixorspiral)arestillboundtosomehumanlyfeasible 

means,andsoquitenaturallycountedasconstructionsandalgorithms.The 

traditionalproblemofarithmeticisitsveryrelationshiptotheempirical 

world,as(alreadybeforeKant)expressedintheclaimitisascienceof 

analyticalnature. Hence,thewholeissueofthedifferencebetweenthe 

truthandproofcanbeboileddowntoasinglequestion: 

whatisarithmeticaltruthoutsideofaspecificaxiomaticsystem? 

Itisexactlythelackofanyexplicitanswertothisquestionthatleadsto 

thePlatonistaccountofarithmeticaltruth. Theusualmodel-theoretical 

expositionoperatingwithanunexplainedconceptofstandardmodel(“2 + 

2 = 4”istrueifandonlyif 2 + 2 = 4)confirmsthisimage,particularly 

whenitstartstoinvokeour“intuitions”. 

However, tounderstandsentenceslike“2 + 2 = 4”and“23 + 4 < 

(6 × 3) + 2”youneednomoremathematicsthanthatprovidedbyagood 

secondaryeducation.Thisistosaythattheyarenottrueorfalse,atleast 

notinthefirstplace,becausetheyarededucibleinPeanoarithmetic,or 

happentoinexplicablyholdinthestandardmodel,butbecausetheyare 

transformableintothesimplerformsof“4 = 4”and“27 < 20”whereonly 

knowledgeofthesequence 1,2,3,4,... andtheabilitytocomparesymbolsisneeded.Thisisthebasisoftheoperativistaccountofarithmetical 

truthasdevelopedbyLorenzeninhisEinführungindieoperativeLogikund 

Arithmetik(Lorenzen,1955),inoppositiontotheusualstandardsofFrege 

thatconsidersuchjustificationsprescientific.AccordingtoLorenzen, 12 the 

ultimatefoundationofarithmetic(includinghigheranalysis)liesexactlyin 

theseprescientificpracticesofcountingandoperatingwithsymbols.They 

canbemadeexplicitinsynthetic(recursive)definitionslike 

⇒ |, ⇒ x + | = x| 

x ⇒ x|, x + y = z ⇒ x + y| = z| 

⇒ | × x = x ⇒ | < x| 

x × y = p, p + y = q ⇒ x| × y = q x < y ⇒ x < y| 

introducing(inunaryform)thenumberseries,theoperations +, ×andthe 

relation


theconsequencesofthesedefinitions,prospectivelywithinthebroaderframe 

ofgame-orproof-theoreticalsemantics(Lorenzen’sdialogicalgames). 13 

AsforGödel’sresults,Lorenzen 14 claimsthatinsteadofbeingabout 

arithmetic,ascompletelygivenbyitsoperativedefinition,theymerelytell 

ussomethingaboutPeano’sformalisminitsparticularshapeofafirst-order 

schemewithinthelanguagecontaining 0, s, +and ×. So,comingfrom 

theotherside,LorenzenarrivedatthesamebasicdifferenceasZermelo. 

ItisalsoinaccordbothwithLorenzen’slaterviews,asdevelopedinhis 

Metamathematik(1962),andwithZermelo’slateprojectofinfinitistlogic, 15 

torephrasethisdifferenceininferentialisttermsasthedistinctionbetween 

twodifferentkindsofconsequence:stronglyeffectiveorfull-formal ⊢and 

themoreliberalorsemi-formal |=. 16 Now,simplifyingheavily: 

Full-formalarithmetic,likethearithmeticofPeano,isarithmeticinthe 

narrowersense,anddealswithschematicallyormechanicallygivenand 

controllableaxiomsandrules. Semi-formalarithmeticorthearithmetic 

properemploys—inaccordwiththeinfinitenatureofthenumbersequence 

1,2,3,...—ruleswithinfinitelymanypremises,particularlythe (ω)-rule 

A(1),A(2),A(3),etc. ⇒ (∀x)(Ax). (ω) 

Asanarithmeticalruleitistransparentandsoundenough(or“reasonable”, 

asZermelowouldsay),aslongasoneinterpretsthe“etc.” correctly. In 

fact,Tarski’sideaofsemantics 17 employsthiskindofrulessystematically, 

withthe (ω)-ruleasaspecialcaseofthemoregeneral 

A(N)forallsubstituents N ⇒ (∀x)A(x). (∀) 

Thisruleisthennothingelsethanthewell-knownpartoftheso-called 

semanticdefinitionoftruth.Hence,thesignificanceofsemi-formalismisto 

makeusthinkofsemanticdefinitionsasspecial(moregenerouslyconceived) 

systemsofrules(proofsystems)which—startingwithsomeelementary 

sentences—evaluatethecomplexonesbyexactlyoneoftwotruthvalues. 

Themostimportantpointtonoticeisthatthesemi-formalrulesarecalled 

semanticnotbecausetheyareinfinitebutbecausethey,unlikePeano’s 

formalism,workwithauniquelydeterminedrangeofquantification. 

Asaconsequence,arithmeticaltruthneednotbeguaranteedbyGod 

orbyintuition,but,as(Zermelo,1932,p.87)putit,simplybythefact 

thatthebroaderconceptof“mathematicalproofisnothingotherthana 

systemofpropositionswhichiswell-foundedbyquantification.”Zermelo’s 

13 See(Lorenzen&Lorenz,1978). 

14 See(Lorenzen,1974,p.21–22). 

15 See(Zermelo,1932). 

16 Bothdistinctionsaredueto(Schütte,1960). 

17 Seeespecially(Tarski,1936).


claimthatallthesentencesaredecidedbyhis“moregeneralscheme”,i.e., 

completelyandcorrectlyevaluatedbyarithmeticalsemi-formalism,canbe 

“proved”byaneasymeta-inductionlikethis: 

1.Elementaryarithmeticalsentences(M = P, M < N)areevaluated 

unambiguouslyastrueorfalseonlyonthebasisofcalculationswith 

numerals. 

2.Tarski’sdefinitionprovidesfortheevaluationofmorecomplexsentences,particularlybecause:eitherforeveryterm 

Nfrom 1,2,3,... , 

thesentence A(N)istrueandhence (∀x)A(x)istrue,orthereis N 

from 1,2,3,...suchthat A(N)isfalse,and (∀x)A(x)isfalse,tertium 

nondatur. 

Itisaknownfactthattheintuitionistsandsomeconstructivists(including 

Lorenzen, 18 butnot,e.g.,Weyl 19 )questionthecompletenessofthisevaluation,arguingthattheexistenceofconcretestrategiesforprovingorrefuting 

every A(N)doesn’tentailtheexistenceofageneralstrategyfor A(x).To 

giveafamiliarexample:thereisnoproblemindemonstratingwhether,for 

anygivenevennumber M,itisthesumoftwoprimes.However,thetruth 

valueofthegeneraljudgmentthateveryevennumberisthesumoftwo 

primes(GoldbachConjecture)isstillunknown, 250yearsaftertheproblem 

wasfirstposed.Hence,itispossiblethatwehaveproofsforallthesentences 

A(N)withoutknowingit,i.e.,withouthavingthegeneralstrategyofhow 

toproveapropositionconcerningthemall. 

Consequently,adecisionmustbemadewhethertheinfinitevehiclesof 

truthandjudgmentsuchas(∀)or(ω)shouldbereferredtoasrules 

3 

1.onlyinthecasewhenwepositivelyknowthatalltheirpremisesare 

true,i.e.,whenwehaveatourdisposalsomegeneralstrategyfor 

provingallofthematonce,or 

2.moreliberally,ifweknowsomehowthatalltheirpremisesarepositivelytrueorfalse.Thegeneraldistinctionbetweentheconstructive 

andclassicalmethodsinarithmeticisbasedonthis. 

Now,ifoneleaves,like,e.g.,LorenzenandBishop,theconceptofeffective 

procedureorprooftoalargeextentopenanddoesnottieit,like,e.g., 

GoodsteinandMarkov,totheconceptoftheTuringmachine, 20 thereisstill 

18 See,e.g.,(Lorenzen,1968,p.83). 

19 See(Weyl,1921,p.156). 

20 Forfurtherdiscussionofthesedifferencessee,e.g.,(Bridges&Richman,1987).


roomforaneffective,yetliberalenoughsemantics(semi-formalsystem) 

andastronglyeffectiveor‘mechanical’syntaxoraxiomatics(full-formal 

system). Hence,theconstructivistreadingdoesnotnecessarilywipeout 

thedifferencesbetweentheproofandtruth,as,e.g.,Brouwer’smentalism 

orWittgensteins’sverificationismseemto. Asaresult,onecanofficially 

differentiatenotonlybetweenfull-formal ⊢andsemi-formal |=consequence, 

butalsobetweensemi-formalconsequenceinastricter(constructive)sense 

andinthemoreliberal(classical)sense. Allthesedifferencesstemfrom 

(Gödel,1931)forthefollowingreason: 

Gödel’stheoremaffectsonlythefull-formalsystems,becausetheirschematicnaturemakesitpossibletodeviseageneralmeta-strategyforconstructingtruearithmeticalsentencesnotprovableinthem.Theunprovable 

sentenceofGödelisoftheso-calledGoldbachtype,i.e.,itisoftheform 

(∀x)A(x)where A(x)isadecidablepropertyofnumbers.Now,Gödel’sargumentshowsthatthisdecisionisdonealreadybyPeanoaxiomsinthesense 

thatalltheinstances A(N)arededucibleand,hence,setastrue.So,with 

Gödel’sproofwehaveageneralstrategyforprovingallthepremises A(N) 

atonce,whichmakesthecriticalunprovablesentence (∀x)A(x)constructivelytrue,i.e.provablebymeansofthe 

(ω)-ruleinterpretedconstructively. 

Lorenzen(1974,p.222)putitlikethis: 21 

ω-incompleteness[...]demonstratesthatnotallconstructivelytrue 

propositionsarelogicallydeduciblefromtheaxioms. Thisshould 

comeasnosurprise. Auniversalproposition (∀x)A(x)isconstructivelytruewhen 

A(N)forall Nistrue. Butinorderlogicallyto 

deducetheuniversalproposition (∀x)A(x),wemustfirstdeduce A(x) 

withafreevariable x.Soweshouldhaveexpected ω-incompleteness. 

ButPeanoarithmeticis ω-completeifwerestrictourselvestoaddition. 

ThepointofGödel’sproofwastodemonstratethatPeanoarithmetic 

withonlyadditionandmultiplication(withoutthehigherformsof 

inductivedefinition)alreadyshowsthe ω-incompletenessthatwasto 

beexpectedingeneral. 

Itisofrealsignificanceherethatitwasnoneotherthan(Hilbert,1931) 

who—probablystillunawareofGödel’sresult 22 —employedthe (ω)-rule 

asameansofimprovinghisoldprojectoffoundingarithmeticonaxiomatic 

grounds. So,ourclaimthatGödel’stheoremsdidnotdestroybutrefine 

Hilbert’soptimisminthesuggestedsemi-formalwayissoundalsofroma 

historicalperspective.Andusingtheconceptofsemi-formalismagain,we 

canextendthisoptimismyetfurtherbyclaimingthatfull-formalsystems 

21 TranslationbyK.R.Pavlovicin(Lorenzen,1987,p.240–241). 

22 SeeBernays’remarksin(Hilbert,1935,p.215)butalsoFeferman’scommentaryin 

(Gödel,1986,pp.209–210).


suchasPeanoandRobinsonarithmeticareconsistentsimplybecausetheir 

axiomsareprovableinthearithmeticalsemi-formalismand,moreover,even 

initsconstructivevariant.This,infact,istheusualmodel-theoreticargument: 

ifatheoryisinconsistent,thenitdoesnothaveamodel, 

inarelativesetting: 

ifPeanoarithmeticisinconsistent,thensoisthearithmeticalsemiformalism. 

Inthefirstcasetheconsequentisprecluded“byfiat”. Inthesecondcase 

onedoesnotneedtousesuchtricks,becauseitwasactuallyprovedthatthe 

rulesofsemi-formalismdonotevaluatearithmeticalsentencesincorrectly. 

4 

Now,shouldweperhapsfollowZermelofurtheranddiscardthenarrower 

conceptofprooftotallybysayingthateverythingtrueisprovable?While 

thedangerofthefirstextremeliesinthefactthatthenarrower,limited 

methodscanandeventuallywillfailbecauseoftheirlimitedness,theshortcomingofZermelo’salternativeisthatitissafetothepointofbecoming 

totallyidle. Theproblemsofsettheoryareaparticularlygoodexample 

ofsuchasituation. Letmeillustrateitverybrieflywiththehelpofthe 

conceptofcontinuum. 23 

Continuumhashadanintricatehistoricaldevelopment,fromthePythagoreandefinitionofproportionbymeansofareciprocalsubtraction,throughtheEuclidiantheoryofpointsconstructiblebymeansofarulerandcompass,totheCartesianideaofnumbersasrootsofpolynomials.Bygrasping 

realnumbersasarbitrary(Cauchy)sequences,ratherthanassequencesthat 

areinsomesenselaw-like,Cantorbelievedhimselftohavewonthewhole 

gamebysimple“fiat”. Butthiswasnomoresubstantiatedthanitwould 

havebeenfortheGreekstodefinerealnumbersaspointsconstructibleby 

whatevermeans,orforusnowtosaythateverythingtrueisprovable.Obviously,thiswoulddisposeofproblemslikethequadratureofthecircle,the 

axiomatizabilityofarithmetic,orthe“Entscheidungsproblem”,butitwould 

alsodisposeofthewholeofmathematics—insofarasitisunderstoodas 

anenterpriseofsolvingproblemssomehowrelatedtohumanlivesrather 

thanasapurescienceindulgedinforitsownsake. Hence,thereasonfor 

retaininganddevelopingthedifferencebetweenthebroader(andvaguer) 

andthenarrower(morelimited)sphereofmethodsliesinthefactthatit 

23 Foradetailedaccountsee(Kolman,n.d.).


mirrorsthegeneralprocessofexplainingsomethingcomplicatedthrough 

somethinglesscomplicated. 

Settheoryrunsintoproblemsbecauseofitsfailuretokeepthesedifferencesapart. 

Settheoristsbelieve,ontheonehand,thattheContinuum 

Hypothesisiseithertrueorfalsewhetherweknowitornot,but,onthe 

otherhand,theonlyspecificideatheycangiveusaboutitsstandardmodel 

isonelooselyconnectedtoZermelo’sfull-formalism,bywhichitis,however,undecidable,i.e.neithertruenorfalse.So,becausetheonlycriterion 

oftruthistheincompleteandpossiblyinconsistentfull-formalism,wemust 

facethepossibilitythatthestatusofquestionslike“howbigisthecontinuum?” 

maybesimilartothatofquestionslike“howmanyhairsdoes 

Othellohave?”,notbecausewedonotyetknowtheanswer,butbecause 

noanswerisavailable. Thisdeficitdoesnotmakesuchquestionshumanindependent,butonlydeeplyfictitious,thereasonforwhich,again,isnot 

thattheyarestillundecided(suchadecisionisnotdifficulttomake,e.g., 

byendorsing V = L)butbecausenothingreallyimportanthingesonthem. 

Myconclusionmayresemblethepositionof(Feferman,1998,p.7),accordingtowhomtheContinuumHypothesis,unlikeHilbert’sSecondProblem,“doesnotconstituteagenuinedefinitemathematicalproblem,”becauseitisan“inherentlyvagueorindefiniteone,asarepropositionsof 

highersettheorymoregenerally.”Ihaveattempted,however,tobemore 

specificaboutwherethedifferencebetweensettheoryandarithmeticcomes 

from.Theso-callediterativehierarchy,describedinapseudo-constructive 

mannerbyZermelo’saxioms,isnotamodelinthesamesenseinwhichthe 

standardmodelofarithmeticis,becausetheconceptofsubsetisleftunexplained,alongwiththerangeofquantificationandtherespective 

(∀)-rule. 24 

Tosumup:Hilbert’ssolvabilitythesisisnotrefutedbyGödel’sincompletenesstheorems,norbytheContinuumHypothesis;however,theyoblige 

ustorephraseitasfollows:everyproblemis(potentially)solvableifitis 

endowedwithwell-definedtruth-conditions,or,asZermelowouldputit, 

witha“reasonable”conceptoftruth. 

VojtěchKolman 

DepartmentofLogics,FacultyPhilosophy&Arts,CharlesUniversity 

nám.JanaPalacha2,11638Praha1,CzechRepublic 

vojtech.kolman@ff.cuni.cz 

24 OnecanpossiblysaythatsettheoryhasfailedbothofFrege’scriteriaforreference,as 

describedsoinfluentiallybyQuine,namely“tobeistobeavalueofaboundvariable” 

and“noentitywithoutidentity”with“|P(N)| =?”takenasevidence.


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Wittgenstein on Pseudo-Irrationals, 

Diagonal Numbers and Decidability 

Timm Lampert ∗ 

Inhisearlyphilosophyaswellasinhismiddleperiod,Wittgensteinholds 

apurelysyntacticviewoflogicandmathematics. However,hissyntactic 

foundationoflogicandmathematicsisopposedtotheaxiomaticapproachof 

modernmathematicallogic. TheobjectofWittgenstein’sapproachisnot 

therepresentationofmathematicalpropertieswithinalogicalaxiomatic 

system,buttheirrepresentationbyasymbolismthatidentifiesthepropertiesinquestionbyitssyntacticfeatures. 

Itrestsonhisdistinctionof 

descriptionsandoperations;itsaimistoreducemathematicstooperations. 

ThispaperillustratesWittgenstein’sapproachbyexamininghisdiscussion 

ofirrationalnumbers. 

1 Tractarianheritage 

IntheTractatus,TLPforshort,Wittgensteindistinguishesbetweenoperationsandfunctions.AsdoRussellandWhiteheadinthePrincipiaMathematica,PMforshort,heuses“functions”inthesenseof“propositional 

functions”,whicharerepresentablebysymbolsoftheform ϕxwithina 

logicalformalism. Incontrast,theconceptofoperationisWittgenstein’s 

owncreation.AccordingtoWittgenstein,the“basicmistake”ofthesymbolismofPMisthefailuretodistinguishbetweenpropositionalfunctions 

andoperations(WVCp.217,andTLP4.126).Inthisrespect,thesyntax 

ofPMsuffersfromthesamedeficiencyasthesyntaxofordinarylanguage. 

Wittgensteindistinguishesbetweenfunctionsandoperationsbythecriterionofthepossibilityofiterativeapplication,TLP5.25f.: 

(Operationsandfunctionsmustnotbeconfusedwitheachother.) 

Afunctioncannotbeitsownargument,whereasanoperationcan 

takeoneofitsownresultsasitsbase. 

∗ IamgratefultoVictorRodychfordiscussionsandcomments.

104 TimmLampert 

Duetoitspossibleiterativeapplication,anoperationgeneratesaseries 

ofinternallyrelatedelements.Thisseriesisdefinedbyaninitialmember, 

η,andanoperation, Ω(ξ),thatmustbeappliedtogenerateanewmember 

fromapreviousone ξ.Theformofsuchadefinitionis [η,ξ,Ω(ξ)].Thisseries 

isnotdefinedasan“infiniteextension”butbytheiterativeapplicationof 

anoperationthatdeterminesforms. Thenaturalnumbers,forexample, 

aredefinedbytheoperation +1. Startingwith 0asinitialmember,this 

yieldstheseries 0, 0 + 1, 0 + 1 + 1etc.,whichisdenotedby [0,ξ,ξ + 1], 

cf.TLP6.03.AccordingtoWittgenstein’spointofviewnumbersareforms 

definedbyoperations(cf.WVC,p.223).Theyareneitherobjectsdenoted 

bynamesnorclassesorclassesofclassesdescribedbyfunctions. While 

functionsdeterminetheextensionofapropertyindependentofitssymbolic 

representation,operationsdeterminethesyntaxofsymbols.Operationsdo 

notrefertoanythingoutsidethesymbols;theydetermineformal(internal) 

propertiesratherthanmaterial(external)properties. Operationsdonot 

stateanything,butdeterminehowtovarytheformoftheirbases(inputs) 

withoutcontributinganycontent.Incontrast,functions,e.g.,“xishuman,” 

statethattheirargumentshavesomeproperty,whichisnotdeterminedby 

thesymbolofthearguments.Afunctiondeterminesanextensionofobjects, 

namelythe“totality”orclassofobjectsthatsatisfythefunction. 

Operationsareinternallyrelated,theycan“counteracttheeffectofanother”and“canceloutanother”(TLP5.253);theyformasystem.InTLPWittgensteinreconstructssocalled“truthfunctions”suchasnegation,conjunction,disjunctionandimplicationas“truthoperations”.Theyformthe 

systemoflogicaloperations. Likewise,heunderstandsaddition,multiplication,subtractionanddivisionasasystemof“arithmeticoperations”.In 

bothcases,thisforcessignificantchangesinthetraditionalsymbolismof 

logicandarithmetic.Inlogic,heinventshisab-notation,inwhichthetruth 

operatorsarenotrepresentedby ¬, ∧, ∨or →butbyab-operations,which 

assigna-andb-polestoa-andb-poles(cf.,e.g.,CL,letters28,32,NL, 

pp.94–96,102,MN,pp.114–116,andTLP6.1203). Bythisheintendsto 

overcomewithinpropositionallogicthe“basicmistake”ofPMinfailingto 

distinguishsymbolicallybetweenoperationsandfunctions. Inarithmetic 

hedefinesnaturalnumbersbyoperations,cf.TLP6.02–6.04,andindicates 

asymbolismofprimitivearithmeticwhollyrestingonoperations(cf.TLP 

6.24f.). HeexplicitlyopposesthistotheFrege’sandRussell’sprogramto 

reducemathematicstoa“atheoryofclasses”(TLP6.031),theseclasses 

beingdefinedbypropositionalfunctions. 

WittgensteincalledforasymbolismbasedonoperationsasacounterprogramtoFrege’sandRussell’slogicism. 

Thisstillholdsforhismiddle 

period. Insteadofhispeculiarterm“operation,”hefrequentlyusesthe 

commonexpression“law,”andinsteadofthetechnicalterm“propositional

WittgensteinonPseudo-Irrationals 105 

function,”heusesthelessspecificexpressionof“description”. Yet,he 

stillclaimsthatmathematicsisdealingwithsystems,operationsorlaws 

andnotwithtotalities,functionsordescriptions(cf.,e.g.,WVC,p.216f., 

orMS107,p.116). Likewise,heclaimsthat“thefalsitiesinphilosophy 

ofmathematics”arebasedonaconfusionofthe“internalpropertiesof 

aform”,whicharedeterminedbyoperations,and“properties”interms 

ofmaterialpropertiesofdailylife,whichareidentifiedbypropositional 

functions,cf. PGII, §42. Healsocallstheviewthatbasesmathematics 

onfunctionsthe“extensionalview”whereasheprofessesan“intensional 

view”thatidentifiesmathematicalpropertiesbysyntacticpropertiesofan 

adequatesymbolicrepresentation(PGII,VII, §41,RFMV, §34–40). 

InthefollowingwegoontoillustrateWittgenstein’sintensionalviewin 

hisintermediate(1929–1934)discussionofirrationalnumbers.Finally,we 

willapplythisdiscussiontodiagonalnumbers,aswellastothenotionsof 

enumerability,decidabilityandprovability.Weherebywanttoaddresstwo 

challengesfacedbyWittgenstein’sprogram: 

(i)Howtoapplyittootherpartsofmathematicsbesidesprimitivearithmetic? 

(ii)Howtorelateittothebasicnotionsandimpossibilityresultsofmodernmathematicallogic? 

2 Irrationals 

Cauchysequences 

IrrationalsarecustomarilydefinedasequivalenceclassesofidenticalCauchy 

sequences.ACauchysequenceisaninfinitesequenceofrationalnumbers 

a1,a2,...suchthattheabsolutedifference |am − an|canbemadelessthan 

anygivenvalue ǫ > 0whenevertheindices m,naretakentobegreaterthan 

somenaturalnumber k. TwoCauchysequences a1,a2,... and a ′ 1 ,a′ 2 ,... 

areidenticalifandonlyifforanygiven ǫ > 0thereissomenaturalnumber 

ksuchthat |an − a ′ n| < ǫforall ngreaterthan k. Theideabehindthis 

definitionisthatallmethodsapproximatingthe“trueexpansion”ofanirrationalnumbermustonceresultinthesameexpansionuptoacertaindigit. 

Forexample,themethodsillustratedinTables1and2bothapproximate 

thetruedecimalexpansionof √ 2inaplainmanner. 

a1 a2 a3 a4 a5 a6 a7 a8 a9 

x 2 < 2 1 1.25 1.375 1.40625 1.4140525 

x 2 > 2 2 1.5 1.4375 1.421875 

Table1.Method1


a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 

x 2 < 2 1 1.4 1.41 1.414 1.4141 

x 2 > 2 2 1.5 1.42 1.415 1.4142 

Table2.Method2 

Atsomepointthemethodscomeupwithidenticaldecimalexpansions 

uptoacertaindigit. Forexample,from a9onbothsequencesbeginwith 

1.41. Thus,goingfurtherandfurtheroneapproximatesmoreandmore 

“the”expansionoftheirrationalnumber.However,nofinitesequencewill 

everrepresentthe“trueexpansion”,asitisthelimitofallsequencesapproximatingit;the“trueexpansion”isbeyondallfinitesequences—itis 

infinite. 

WithrespecttoWittgenstein’spointofview,itisimportanttonotethat 

thesemethodsofapproximationdonotgeneratethenextdigitsbyiteration. 

Instead,atanystepitmustbecheckedwhetherthesquareoftheresultis 

< 2or > 2. 

Wittgenstein’scritique 

Wittgenstein’smaincritiqueofthedefinitionofirrationalnumbersinterms 

ofCauchysequencesisthatthisdefinitiondoesnotprovideanidentity 

criterion,whichdecidestheidentityoftworealnumbers(PR §§186,187, 

191,195). Theproblemisthat,onthestandardconceptionofirrational 

numbersasinfinitesequencesofrationalnumbers,foranyinfinitesequence 

sthereareinfinitemanysequencesthatareidenticalwith suptoacertain 

digit k.However,thedefinitiondoesnotprovideamethodtospecifysome 

upperboundfor kincomparingtwoarbitraryrealnumbers. Thus,no 

finitecomparisonissufficienttodecidewhethertwoarbitrarysequences 

areidentical. Thedefinitionhasitthatthe“trueexpansion”liesbeyond 

allfinitesequences. Therefore,itprovidesonlyasufficientcriterionfor 

anegativeanswerbutnosufficientcriterionforapositiveanswertothe 

questionofidentifyingarbitraryrealnumbers.Inthisrespect,wehavethe 

samesituationasinthecaseofdeterminingwithinatraditionallogical 

calculuswhethersomeformulaoffirstorderlogicisnotatheorem. 

Onemightreplytothiscritiquethatonecannotclaimthedecidability 

ofthingsthatsimplyarenotdecidable;thenatureoftherealnumbersas 

infinitesequencesimpliesthatonecannotdecideupontheidentityoftwo 

realnumbers. However,infactitisfromthepurporteddefinitionthat 

theproblemarises,anditisnotcarvedinstonethatthisindeedcaptures 

the“nature”ofrealnumbers. AccordingtoWittgenstein’sanalysisthe 

definitionisnothingbutaconsequenceoftheextensionalviewofmodern 

mathematics. Thisspuriouslytakesthedesignationsofrealnumbersby


ordinarylanguageasdescriptionsofeverydayproperties,whichdetermine 

acertainextension. Forexample,inthecaseof √ 2onewronglyanalysestheordinaryexplanationintermsof“thenumberthatwhenmultiplied 

byitselfisidenticalwith 2”asadescriptionofamaterial,non-symbolic 

property. Thispropertyisthenconceivedasbeingsatisfiedbythe“true 

infiniteexpansion”,whichisapproximatedbymultiplyingfinitesequences 

withthemselvesandcomparingtheresultto 2.InordertocometounderstandWittgenstein’spointofview,itiscrucialtorecognizethatthereisan 

alternativetothisconceptionthatreferstoknownmathematics.According 

tothispointofview,realnumbersarenotdefinedbyextensions,butby 

lawsinthesenseofWittgenstein’soperations. 

Wittgenstein’salternative 

InordertocometounderstandWittgenstein’spositiononemustrecognizethatherejectsmethodsofapproximationsuchastheaboveillustrated 

methods1and2.Althoughthesekindsofmethodsofapproximationmight 

becalled“laws,”theyarenot“laws”intermsofoperations.Theyarenot 

operationsbecausetheydonotgenerateasequencebyiteration. Howto 

goondoesnotsimplydependonthepreviousmembersbutonacomparisonbetweenthelastmemberandsomecondition. 

Forexample,ateach 

stageinthedevelopmentofthedecimalexpansionof √ 2,onemustconsiderwhethersquaringthelastmemberisgreaterorsmallerthan 

2.This 

methodisincompatiblewithWittgenstein’spurelysyntacticfoundationof 

mathematicalproperties. Inhisprogram,anysequencemustbedefinable 

byanoperationthatdeterminesnothingbutthesyntaxofthemembersof 

thesequence.Onlyinthiswayisthepropertyconstitutingthesequencereducedtoaninternalpropertyofformsthatcanbeidentifiedbythesymbolic 

featuresofthemembersoftheseries. 

Wittgenstein’s wellknownrejection of“arithmetical experiments”is 

basedonhisrequirementtodefinesequencesbysyntacticmeansalone, 

PR §190: 

Inthiscontextwekeepcomingupagainstsomethingthatcouldbe 

calledan“arithmeticalexperiment”.Admittedlythedatadetermine 

theresult,butIcan’tseeinwhatwaytheydetermineit(cf.,e.g.,the 

occurrencesof 7in π.)Theprimeslikewisecomeoutfromthemethod 

forlookingforthem,astheresultsofanexperiment. Tobesure,I 

canconvincemyselfthat 7isaprime,butIcan’tseetheconnection 

betweenitandtheconditionitsatisfies. —Ihaveonlyfoundthe 

number,notgeneratedit. 

Ilookforit,butIdon’tgenerateit.Icancertainlyseealawinthe 

rulewhichtellsmehowtofindtheprimes,butnotinthenumbers


thatresult. Andsoitisunlikethecase + 1 

1! 

canseealawinthenumbers. 

1 1 , −3! , + 5! etc.,whereI 

Imustbeabletowritedownapartoftheseries,insuchawaythat 

youcanrecognizethelaw. 

Thatistosay,nodescriptionistooccurinwhatiswrittendown, 

everythingmustberepresented. 

Theapproximationsmustthemselvesformwhatismanifestlyaseries. 

Thatis,theapproximationsthemselvesmustobeyalaw. 

TheseriesofprimesisWittgenstein’sparadigmofaseriesthatcannotbe 

generatedbyanoperation. Althoughoperationsareavailabletogenerate 

aninfiniteseriesofprimes,nooperationisknowntogeneratetheprimesin 

acertainorderthatensuresthatallprimesareenumerated.Inhisdetailed 

discussionsofprimesinotherplaces,Wittgensteindrawstheconsequence 

thatwestilllackofaclearconceptof“the”primes. Allwehaveisa 

conceptofwhat“a”primeis,whichallowsustodecidewhetheragiven 

numberisprimeornot(PR §159,161,cf.(Lampert,2008)).Forthesame 

reason,herejectsthedefinitionofarealnumber Pasthedualfractionwith 

an = 1if nisprimeand an = 0otherwise(cf.PGII, §42).Thisdefinition 

doessatisfythedefinitionofrealnumbersbyCauchysequences,butitdoes 

notsatisfyWittgenstein’scriterionofbeingdefinablebyanoperation. In 

thequotedpassage,Wittgensteinemphasizesthatwedohaveamethod 

tolookforthenextprime: wegothroughtheseriesofnaturalnumbers 

anddecideonebyonewhethereachmembersatisfiestheconditiontobe 

divisibleonlyby 1anditself. However,thismethoddoesnotsatisfyhis 

standardsofadefinitionbyoperation.Aslongaswearenotabletoreduce 

thepropertyofbeingaprimetosomeoperationgeneratingtheseriesof 

primesbyiteration,“wecan’tseetheconnection”betweenthemembersof 

theseriesandtheconditiontheysatisfy:wecannot“recognizethelaw”in 

theseries.Theproblemisthesameaswiththeaboveillustratedmethods 

ofapproximating √ 2.Insteadofgeneratingthenextmemberbyiteration, 

wemustdecidewhethersomeconditionissatisfiedornotinordertofind 

thenextmember. 

Wittgenstein’sreferencetotheseriesofprimesasanillustrationofarithmeticalexperimentsdemonstratesthathisconceptofoperationisnotequivalenttothatofprimitiverecursivefunction.Primesaredefinablebyaprimitiverecursivefunction,butnotbyanoperation. 

1 Iterationinthecaseof 

operationsmeansthattheoutputofthe n th applicationofanoperationis 

1 ThequestioninwhatsenseWittgensteincharacterizesrealnumbersas“laws”isthoroughlydiscussedintheliterature(cf. 

(DaSilva,1993), (Frascolla, 1994,pp.85–92), 

(Marion,1998),(Rodych,1999)and(Redecker,2006,ch.5.2)). However,themainreasonwhytheidentificationoflawswithWittgenstein’snotionofoperationsseemedto


itselftheinputofthe n + 1 th applicationoftheverysameoperation. In 

contrast,recursioninthecaseofprimitiverecursivefunctionsmeansthat 

thevalueofaprimitiverecursivefunction fforthesuccessorof n, S(n),is 

definedbyreferringtothevalueoftheverysamefunction ffor n. This 

doesnotimplythatthevaluesof farethemselvestheirarguments.Thisis 

onlytrueincaseofthesuccessorfunction,whichitselfisprimitiverecursive.However,theidentityfunction,e.g., 

I(x) = x,andthezerofunction, 

Z(x) = 0,onwhichthedefinitionofprimitiverecursivefunctionsarebased, 

arefunctionsanddonotdefineaseriesbyiterativeapplication.Thesame 

holdsforprimitiverecursivecharacteristicfunctions. Theyhavetheform 

“f(x) = 0if ϕ(x)and f(x) = 1otherwise.”InWittgenstein’sterms,characteristicfunctionsareaparadigmof“descriptions”andnotofoperations. 

Incontrast,anyiterationbyapplyingoperationshastheform an = Ω ′ �ai 

where �aistandsformemberspreviousto an. Forexample,theseriesof 

Fibonaccinumbersisdefinedby an = an−2 + an−1.Recursioninthecase 

ofprimitiverecursivefunctionsispartofastrategyofdefiningprimitive 

recursivefunctions,whereasoperationsarenotdefinedbyiterationbutappliediteratively.Theyaredefinedbysomepurelysyntacticvariationthatgeneratesaformalseriesofsystematicallyvariedmembersifiterativelyapplied. 

InthecaseofFibonaccinumbers,thisoperationconsistsofadding 

thelasttwomembers.Startingfrom 0and 1,thisgeneratestheseries 0, 1, 

0+1, 1+(0+1), (0+1)+(1+(0+1)), (1+(0+1))+((0+1)+(1+(0+1))) 

etc. 2 

IfnotevenprimitiverecursivefunctionssatisfyWittgenstein’sstandards 

ofapurelysyntacticfoundationofmathematics,thiscausesdoubtswhether 

hisprogrammeisrealizableatall.Likewise,hisrejectionofarithmeticalexperimentsandhisclaimto“recognizethelaw”intheserieshascaused 

trouble.ThedecimalsequencesofirrationalsdonotsatisfyWittgenstein’s 

demandforsequencesthatmanifestlyobeyalaw.DonotirrationalscontradictWittgenstein’sclaimfromtheirverynature?Thus,itseemsunclear 

howWittgenstein’spointofviewcanevendojusticetosuchbasicirrational 

numbersas πand √ 2(cf.,e.g.,(Redecker,2006,p.212)). 

However,theseproblemsonlyariseifoneoverlooksthefactthatthe 

possibilityofdefinitionsbyoperationsdependsonthemodeofrepresentation.Incaseofirrationals,thesyntacticfeaturesofthedecimalsystemareresponsiblefortheir“lawless”representation.However,thiskindofrepresentationisnotessential;itobscurestheirlawfulnatureinsteadofrevealing 

it.InMS107p.91,Wittgensteinwrites(translatedbyT.L.): 

beinsufficienttomostcommentatorsisthatoperationsinWittgenstein’ssensewerenot 

distinguishedsharplyfromthenotionofprimitiverecursivefunctions. 

2 Bracketsaremerelyintroducedtoidentify an−2and an−1.


Theprocedureofextracting √ 2inthedecimalsystem,e.g.,isan 

arithmeticalexperiment,too. However,thisonlymeansthatthis 

procedureisnotcompletelyessentialto √ 2andarepresentationmust 

existthatmakesthelawrecognizable. 

Toseetheconnectionbetweenthemembersofasequencerepresentinga 

realnumberandtheconditionorpropertythatthesememberssatisfy,one 

mustrefertoanequivalencetransformationthatreducesthispropertyto 

aninternalpropertyofforms. Thereisnoequivalencetransformationbetween 

√ 2andadecimalnumber.Thisalreadyshowsthatitisimpossible 

torepresent √ 2bythedecimalsystem;whateverdecimalnumberonegenerates,itcannotbeidenticalwith 

√ 2—referringto“infiniteextensions”is 

justanotherexpressionofthisdeficiency.However,usingtherepresentation 

bycontinuedfractions,itispossibletorepresent √ 2byanoperation,cf. 

MS107,p.126(translatedbyT.L.,cf.MS107,p.99): 

[...] in 1 

2 , 1 

2+ 1 

2 

1 , 

2+ 1 

2+ 1 

2 

recognizeinthedecimaldevelopment. 

etc. onecanrecognizethelawonecannot 

Theconnectionbetweenthepropertyof √ 2as“thenumberthatmultipliedwithitselfisidenticalwith2”anditsdefinitionbyitscontinued 

fractionisduetoequivalencetransformation: 

x 2 = 2 | √ 

x = √ 2 | a = 1 + (a − 1) 

x = 1 + ( √ 2 − 1) | a = 1 1 

a 

x = 1 + 1 1 

√2−1 

x = 1 + 1 √ 

2+1 

√ 2 

2 −12 x = 1 + 1 √ 

2+1 

x = 1 + 1 

1+x 

x − 1 = 1 

1+x 

1 x − 1 = 2+(x−1) 

| 1 

a−b 

= a+b 

a 2 −b 2 

| a = a 

√ 2 2 −1 2 

| x = √ 2, a + b = b + a 

| −1 

| 1 + x = 2 + (x − 1) 

Thus, √ 1 

2−1isrepresentablebytheoperation 2+(x−1) .Startingwith 1−1 

1 

for x−1,theiterativeapplicationofthisoperationyieldstheseries 2+(1−1) , 

2+ 

1 

1 

2+(1−1) 

, 1 

1 

2+ 1 

2+ 

2+(1−1) 

etc. ThisisidenticaltotheseriesWittgenstein 

mentionsifoneeliminates +(1 − 1)byanequivalencetransformation. In


theshortnotationofregularperiodiccontinuedfractions, √ 2isdefinableby 

[1;2].Acontinuedfractionofarealnumberisperiodicifandonlyifthereal 

numberisaquadraticirrational(theoremofLagrange). Thenotationof 

continuedfractionidentifiesacommonpropertyofquadraticirrationalsby 

acommonsyntacticfeature,andthusshowsthatthispropertyisaninternal 

property.Otherirrationalnumbersarerepresentablebyregularcontinued 

fractionsthatarenotperiodicbutstilldefinablebyoperations,suchasthe 

Eulernumber e : [2;1,2,1,1,4,1,1, 6, 1,1,8,...].Anothertypeofirrational 

numbersarenotdefinablebyoperationswithinregularcontinuedfractions 

butwithinirregularcontinuedfractionssuchas 4 

π = 1+ 12 .Further- 

2+ 52 

. .. 

more,thecontinuedfractionrepresentationforanumberisfiniteifandonly 

ifthenumberisrational. Thisshowsthatthismodeofrepresentationrevealsbyitssyntacticpropertiesinternalpropertiesofnumbersthatarenot 

identifiedbythedecimalnumbersystem. Welearnmoreabout“thelaws 

ofnumbers”,theirinternalstructure,byrepresentingtheminthenotation 

ofcontinuedfractions. 

Mathematicalproofsrevealthisinternalstructurebyequivalencetransformations. 

Consider,forexample,thegoldenratio. Itsrepresentationas 

adecimalnumberdoesnotshowitsexceptionalnature. However,byan 

equivalencetransformationresultinginanoperationdefiningacontinued 

fraction,internalpropertiesofthegoldenratioareidentifiedbythesyntacticfeaturesofthisadequaterepresentation.Thisprocedurereducesthe 

propertythattheratiooftwoquantities aand bisidenticaltotheratioof 

thesumofthemtothelargerquantity atoanoperation: 

φ = a 

b 

= a + b 

a 

= 1 + b 

a 

2+ 

3 2 

1 

= 1 + . (1) 

φ 

Bytheoperation 1 + 1 

φ ,theperiodic,regularcontinuedfraction [1;1]is 

defined. Bythisrepresentationitisproventhatthegoldenratiois“the 

mostirrationalandthemostnoblenumber,”becausethesepropertiesare 

identifiedbythelowestpossiblenumbersinaninfiniteregularcontinued 

fraction.Furthermore,bythisrepresentationitisproventhattheratioof 

twoneighbouredFibonaccinumbersconvergestothegoldenratio.Forthe 

Fibonaccinumbersaredefinedby an+1 = an+an−1.Thus,with a = anand 

b = an−1weyieldequation(1).Thesyntaxofcontinuedfractionsprovides 

symbolicconnectionsthatprovecertaininternalrelationsbetweennumbers. 

Thecontinuedfractionrepresentationofanyirrationalnumberisunique. 

Thus,anydefinitionofarealnumberbyanoperation(or“induction”)definingacontinuedfractionsatisfiesWittgenstein’scriterionforrepresentinga 

realnumber,MS107p.89(translationT.L.):


Iwantarepresentationoftherealnumberthatrevealsthenumber 

inaninductionsuchthatIhaveherewiththeonlyproper,unique 

symbol. 

Itisbythispropertyofuniquenessthatthesymbolicrepresentationof 

irrationalsbycontinuedfractionsservesasanidentitycriterion,whichallows 

onetocompareirrationalsandrationalnumbers.Theprincipleisthesame 

asinthecaseofcomparingfractionsbyconvertingthemtofractionswith 

identicaldenominators. Theproblemofdecidingtheidentityofnumbers 

resultsfromadeficiencyintheirrepresentation,allowingforambiguity. 

Thisdoesnotmeanthattheremustbeoneandonlyonepropernotation 

fornumbers. Nordoesitmeanthatcontinuedfractionsare“the”proper 

notationofrealnumbers.Differentinternalpropertiesofnumbers,andherewithdifferenttypesofnumbers,maybeidentifiedbydifferentsystemsof 

representation.Anddifferenttypesofnumbersmaybecomparablewithin 

differentmodesofrepresentation(cf.MS107,p.123).Naturalnumberscan 

becomparedaccordingtotheconventionsofthedecimalsystem,fractions 

arecomparablebyconvertingthemtofractionswithidenticaldenominator,rationalnumbersandquadraticirrationalsarecomparablebyregular 

continuedfractionsetc. Furthermore,newproofsconsistofmakingnew 

symbolicconnections.Theyinventnewpossibilitiesofcomparingnumbers 

andofrevealingtheirinternalrelations.Notallinternalrelationsofanumbertoothernumbersmustberevealedwithinonlyonenotationalsystem. 

Forexample,insteadofrepresenting πbyanirregularcontinuedfraction 

√ 2 

√ 2+ √ 2 

� √ √ 

2+ 2+ 2 

(asquotedabove), 2 

πcanalsoberepresentedby 2 · 2 · 2 ·· · · 

or π 2 2 4 4 6 6 8 8 

2by 1 · 3 · 3 · 5 · 5 · 7 · 7 · 9 · · · ·.Theinternalpropertiesofdifferentnumbersmaycallforoperationsreferringtodifferentmodesofrepresentation. 

Thereneednotbea“systemofirrationalnumbers”inthesenseasthereis 

a“systemofnaturalnumbers”ora“systemofrationalnumbers”(cf.PG 

II, §42,RFM,app.3, §33).Aswehaveseen,onlyquadraticirrationalsare 

definablebyperiodic,regularcontinuedfractions,andanothertypeofirrationalsisnotevendefinablebyregularcontinuedfractions.Differenttypes 

ofirrationalsaredefinablebydifferentkindsofoperationswithindifferent 

modesofrepresentation. 

AccordingtoWittgenstein’sintensionalpointofview,ourmathematicalcomprehensionandknowledgedependsonthesyntaxofmathematical 

representation. Thisisnotduetopsychologicalreasons. Instead,thisis 

becausemathematicalproofsmakesymbolicconnectionsbetweendifferent 

modesofrepresentation,andbecausethesolvabilityofmathematicalproblemsdependsonimposingadequatenotations.Insteadofconcludingfrom 

aspecific,deficientmodeofrepresentationthelawlessnatureofirrational 

numbers,whichmakesitimpossibletodecideupontheiridentityandwhich


invokesmisconceptionssuchas“infiniteextensions”,oneshouldlookforadequaterepresentationsthatrevealtheirlawfulnatureandmakeitpossible 

todecideupontheiridentity.Thisisdonebyreducingtheirpropertiesto 

operationsinsteadofconceptualizingthemintermsoffunctions.Ifsucha 

reductionisnotavailable,thismeansthatonedoesnothaveafullunderstandingofthepropertiesinquestion. 

Wecanthenonlyrefertoavague 

understandingexpressedwithinadeficient,descriptivesymbolism.Onlyby 

imposinganadequateexpressionthatdepictsthosepropertiesbyitssyntacticfeatures,canwebesurethatthosepropertiesareproperlydefined. 

Thisapproachisinconflictwithbasicimpossibilityresultsofmodern 

mathematicallogic,suchasthenon-enumerabilityoftheirrationals,the 

undecidabilityoffirst-orderlogicortheincompletenessoflogicalaxiomatizationsofarithmetics. 

ThisdoesnotmeanthatWittgenstein’spointof 

viewimpliesthattheseresultsarefalseinthesensethattheirnegationis 

true. Instead,hisintensionalviewimpliesthatitdoesnotmakesenseto 

speakof“theirrationals”unlessanoperationisknownthatallowsusto 

generatethembyiteration(andthustoenumerate“theirrationals”).This, 

ofcourse,doesnotmeanthatheclaimsthatsuchanoperationisormust 

beavailable. Likewise,hisintensionalviewimpliesthatonecannotspeak 

ofdecidabilityorprovabilityinanabsolutesense,suchthatonecansayin 

advancethatcertainpropertiesofformulaeofacertainsyntaxarenotdecidableorprovable,independentofthesyntacticmanipulationsthatmight 

beinventedtoidentifythoseproperties. AccordingtoWittgenstein“beingatautology”(“beingtrueinallinterpretations”)or“beingatheorem” 

offirstorderlogicisnotdefinedproperlyunlesssomesortofequivalence 

procedureisinventedthatconvertsfirstorderformulaetoanadequaterepresentationthatidentifiestheirlogicalpropertiesbyitssyntacticproperties. 

Fromthispointofview,itcannotbesaidthatitisimpossibletodefinesuch 

procedures,becausethepropertiesinquestionthataresaidtobeundecidableorunprovablearenotrepresentedproperlyunlesssuchproceduresare 

available. Likewise,fromWittgenstein’spointofviewtheincompleteness 

ofaxiomaticsystemsofarithmeticmeansinthefirstplacethatthosesystemsdonotproperlyrepresentthepropertiesinquestion.Itdoesnotmean 

thatweknowthatacertainpropertyholds,butitsformalrepresentationis 

notderivable.Instead,itmeansthatwehaveadeficientunderstandingof 

thatpropertyexpressedbyaninadequaterepresentation.Inthefollowing, 

wewillshowthatthisconflictbetweenWittgenstein’spointofviewand 

theimpossibilityresultscanallbetracedbacktohisrejectionof“descriptions”intermsofcharacteristicfunctionsasadequateformstorepresent 

realnumbers.


3 Pseudo-Irrationals 

Wittgensteinillustrateshispointofviewbyprovidingseveraldefinitionsof 

pseudo-irrationals. ThesearedefinitionsofirrationalsintermsofCauchy 

sequences. However,contraryto √ 2or πnoreductionstooperationsof 

thesedefinitionsareavailable.Thus,accordingtoWittgensteinthereareno 

irrationalscorrespondingtothosedefinitions.Besidestheabovementioned 

definitionof Pasthedualfraction 0.a1a2 ...with an = 1if nisprimeand 

an = 0otherwise,Wittgensteindiscussesthefollowingdefinitions(cf. PG 

II, §42): 

π ′ :Thedecimalnumber a1.a2a3 ...with anan+1an+2 = 000 

if anan+1an+2 = 777in π;otherwise an = anof π. 

F:Thedualfraction 0.a1a2a3 ...with an = 1if x n + y n = z n issolvable 

for n(1 ≥ x,y,z ≥ 100);otherwise an = 0. 

Allthesedefinitionsareintendedtodefineanirrationalnumberbya 

characteristicfunction.Inthiscase,thedots“...”refertoan“infiniteextension”.Thus,theyareill-definedaccordingtoWittgenstein’sstandards. 

Theydonotidentifyanumberbutdescribeanarithmeticalexperiment. 

Wittgensteinemphasizesthatevenifthecharacteristicfunctionsbecome 

reducibletooperations,thisdoesnotmeanthatthisshowsthatthedefinitionsinfactdefineirrationalnumbers. 

Instead,itmeansthatvague 

definitionsthatdonotidentifynumbersarereplacedwithexactdefinitions 

thatareabletoidentifynumbers.He,forexample,considersthesituation 

whenFermat’stheoremisproven. Duetohisrejectionofdescriptions,he 

doesnotanalysethissituationintermsofcomingtoknowthenumber F 

thatbeforewasonlydescribed.Instead,theproofallowsonetoreplacethe 

pseudo-definitionof F,whichdoesnotidentifyanumber(neitherarational 

noranirrationalone),with F = 0.11,whichisarationalnumber(PGII, 

§42).Before,itwasnotdecidablewhether“F”denotesanumbersuchthat 

F = 0.11ornot;thedefinitionbydescriptionsimplydidnotdefinerules 

todothis.Thisdemonstratesthelackofmeaningthatisgivento“F”by 

thepreviousdefinition.Theproof,ifitisvalid,makesconnectionstoother 

partsofmathematicsthatwerenotrecognizedbeforeandthusgives“F”a 

clearmeaning. 

Cantor’sproofofthenon-enumerabilityofirrationalnumbersisbasedon 

definingadiagonalnumberbyacharacteristicfunction. Givensomeenumerationofdualfractionsbetween 

0and 1,theproofofthenon-enumerabilityof“all”ofthemisbaseduponthefollowingdiagonalnumber 

D: 

D:Thedualfraction 0.a1a2 ...with an = 0ifthe n ′th digitofthe n ′th dual 

fractionis 1;otherwise an = 1.


Tothisdefinition,thesameobjectionsapplyastothedefinitionsof P, π ′ 

or F:Itisadefinitionbydescriptionintermsofacharacteristicfunction. 

Itdescribesanarithmeticalexperimentanddoesnotidentifyanumber, 

whichcanonlybedonebyanoperation. However,suchanoperationis 

notavailable. Thus,itisnotmeaningfultosaythat Disan“irrational 

number”notoccurringintheassumedenumerationofirrationals. This, 

ofcourse,doesnotmeanthatWittgensteinclaimsthat“theirrationals” 

areenumerable. Instead,heobjectstoidentifyingirrationalnumbersby 

non-periodic,infinitedecimalordualfractions.Thiscriteriondoesnotsay 

anythingaboutacertaintypeofnumbers;itonlysayssomethingaboutthe 

deficiencyofthedecimalnotation(PGII, §41).Thisnotationcannotserve 

astheuniquenotationforrealnumbers,asitdoesnotmakeitpossible 

todecideupontheidentityofnumbers.Likewise,Wittgensteinobjectsto 

thepictureofarealnumberasa“point”onthe“line”ofrealnumbers. 

Theseitemsareelementsoftheextensionalview.Theyarisefromtreating 

“isanirrationalnumber”aswellas“isarationalnumber”or“isanatural 

number”asconcepts(propositionalfunctions)identifyingcertainsetsof 

numbers. Thismakesitpossibletoaskaboutthe“cardinality”ofthose 

sets.This,inturn,allowsone 

(i)touse“infinite”asanumberwordandspeakof“theinfinitenumber” 

ofobjectssatisfyingsomeconcept,and 

(ii)tocomparethecardinalityofsetsbycoordinatingtheirelements. 

Finally,fromthisandthemethodofdiagonalizationonecomestospeakof 

setswithacardinalitygreaterthanthatofthesetofnaturalnumbers.First 

andforemost,Wittgenstein’scriticismisthatthisconceptualmachineryis 

ratheranexpressionoftheextensionalviewthanadescriptionofthenature 

ofnumbers(RFM,app.3, §19).Hecutstherootsof(transfinite)settheory 

byconceptualizing“typesofnumbers”intermsof“systems”insteadof 

“sets”.Accordingtohisintensionalpointofview,thecriteriontoidentify 

atypeofnumberisthepossibilitytogeneratethembyanoperation. As 

thisimpliestheirenumerabilityintermsoftheiterativeapplicationofan 

operation,itdoesnotmakesensetospeakoftypesofnumbersthatarenot 

enumerable. 

AccordingtoChurch’sthesis,theconceptofdecidabilityisrepresentable 

byaprimitiverecursivecharacteristicfunction. Thus,onthebasisofan 

enumerationoffirst-orderlogicformulaebytheirGödelnumbers,thepropertyofbeingatheorem(oratautology)isrepresentablebythefollowing 

number: 

T:Thedualfraction 0.a1a2 ...with an = 0if ⊢ ϕn(or |= ϕn);and an = 1 

otherwise.


Onthebasisofdiagonalization,undecidabilityproofsdemonstratethat 

characteristicfunctionssuchastheonedefining Tcannotbeprimitiverecursive. 

FromWittgenstein’spointofview,theseproofsarebasedupon 

aconfusionofmaterialandformalproperties. Asaformalproperty,theoremhood(orbeingatautology)isnotrepresentablebyacharacteristic 

function. Instead,thesepropertiesareonlyrepresentedadequatelybya 

sharedsyntacticpropertyinanidealnotation. Thisisillustratedbythe 

representationoftautologiesviatruthtablesordisjunctivenormalformsof 

propositionallogicaswellasbymeansofVenndiagramsinmonadicfirst 

orderlogic.Wittgenstein’sconceptioncallsforequivalencetransformations 

toidentifythetruthconditionsoflogicalformulaebymeansofsyntactic 

propertiesoftheirproperrepresentation.Thisconceptiondiffersfromthe 

traditionalsemanticsoffirst-orderlogic.Presuminganendlessenumeration 

ofinterpretations ℑ1, ℑ2,...,eachbeingeitheramodeloracounter-model 

ofaformula A,onemightrepresentthetruthconditionof Aaccordingto 

theseinterpretationsbythefollowingnumber: 

θ(A):Thedualfraction 0.a1a2 ...with an = 0if ℑn |= Aand an = 1 

otherwise. 

Onthecontrary,Wittgenstein’sapproachcallsforarepresentationof 

thetruthconditionsofaformula Athatallowsonetoidentifythetruth 

conditionsof Awithoutdecidingwhethersingleinterpretationsaremodels 

orcounter-modelsof A.Furthermore,theproperrepresentationoffirstorderformulaeshouldrevealtheinternalrelationsofnon-equivalentlogical 

formulaebymakingitpossibletogeneratethesystemoftruthconditions 

byoperations.TohaveanideaofwhatWittgensteinenvisages,onemight 

thinkofasystematicgenerationofreduceddisjunctivenormalformsofthe 

Quine–McCluskeyalgorithm, 3 thatrepresentallpossibletruthfunctions 

ofpropositionallogic. Likewise,thetaskoffirstorderlogicistodefine 

analogousdisjunctivenormalformsandproceduresfortheiruniquereductionwithinfirstorderlogic.Toclaimthatthisisimpossiblepresumesthe 

extensionalviewthatisrejectedbyWittgenstein’sendeavour. 

Likewise,Gödelrepresents“xisaproofof y”byaprimitiverecursive 

function xByindefinition45ofhisincompletenessproof(cf.(Gödel,1931, 

p.358)).Onthisbasis,heexpresses“xisprovable”by ∃yyBxindefinition 

46. ThisisincompatiblewithWittgenstein’sclaimthattheinternalrelationofbeingprovable(derivable)shouldbedefinedbyoperationsinsteadof 

propositionalfunctions.This,inturn,presumesaproofprocedureinterm 

3 NotethatthereduceddisjunctivenormalformsoftheQuine–McCluskeyalgorithm 

areunique; anyequivalentpropositionalformulaisrepresentedbythesamereduced 

disjunctivenormalform.AmbiguityonlycomesintoplayinthesecondstepoftheQuine– 

McCluskeyalgorithmthatintendstominimizereduceddisjunctivenormalforms.


ofequivalencetransformationstoanadequatesymbolismthatmakessuch 

adefinitionpossible,insteadofaproofprocedureintermsoflogicalderivationsfromaxioms.Thelackofsuchadefinitionmeansadeficiencyinthesyntacticrepresentationoftheformulaeinquestion.AccordingtoWittgenstein’spointofview,theconclusionthatmustbedrawnfromGödel’sincompletenessproofistolookforaformalrepresentationofarithmeticthat 

isnotbasedupontheconceptofpropositionalfunction,whichisatthe 

heartofanylogicalformalization. 

Wittgenstein’sintensionalreconstructionofmathematicsisnotmeant 

tobea“refutation”oftheextensionalviewofmodernmathematicallogic. 

Instead,firstandforemostitintendstoproposeadecisivealternativeconceptualizationofmathematicsthatradicallydiffersinitsfoundations.Accordingtohim,thefruitofthisendeavourshouldbeaclarificationofthephilosophicalproblemsofmodernmathematicsthatwillhavethesameinfluenceontheincreaseofmathematicsassunshinehasonthegrowthof 

potatoshoots(PG,II, §25). 

TimmLampert 

UniversityofBerne 

timm.lampert@philo.unibe.ch 

Abbreviations 

CL Wittgenstein,L.(1997).CambridgeLetters,Oxford:Blackwell. 

MN Wittgenstein,L.(1979). NotesdictatedtoG.E.MooreinNorway,in: 

Notebooks1914–1916,pp.108–119.Oxford:Blackwell. 

MS107 Wittgenstein,L.(2000)Manuscript107accordingtovonWright’scatalogue. 

PublishedinWittgenstein’sNachlass: theBergenElectronic 

Edition.London:OxfordUniversityPress. 

NL Wittgenstein,L.(1979). NotesonLogic,in: Notebooks1914–1916, 

pp.93–107.Oxford:Blackwell. 

PG Wittgenstein,L.(1974). PhilosophicalGrammar,Blackwell: London 

1974. 

PR Wittgenstein,L.(1975)PhilosophicalRemarks,Oxford:Blackwell. 

RFM Wittgenstein,L.(1956)RemarksontheFoundationsofMathematics, 

Oxford:Blackwell. 

TLP Wittgenstein, L. (1994). Tractatus Logico-Philosophicus. London: 

Routledge. 

WVC Waismann, F. (1979) Wittgenstein and the Vienna Circle, Oxford: 

Blackwell.


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Wittgenstein’sphilosophyofmathematics(pp.93–99).Vienna:Hölder–Pichler– 

Tempsky. 

Frascolla,P.(1994).Wittgenstein’sphilosophyofmathematics.Routledge:London. 

Gödel,K.(1931). ÜberformalunentscheidbareSätzederPrincipiaMathematica 

undverwandterSysteme.MonatsheftefürMathematikundPhysik,38,173–198. 

Lampert,T. (2008). Wittgensteinontheinfinityofprimes. Journalforthe 

HistoryandPhilosophyofLogic,29,63–81. 

Marion,M.(1998).Wittgensteinandfinitism.Synthese,105,141–176. 

Redecker,C.(2006).Wittgensteinsphilosophiedermathematik.Frankfurt:Ontos. 

Rodych,V. (1999). Wittgensteinonirrationalsandalgorithmicdecidability. 

Synthese,118,279–304.

What is the Definition of ‘Logical Constant’? 

Rosen Lutskanov ∗ 

Thedesignofthispaperistomotivateandintroduceinformallyadefinitionofthenotionof‘logicalconstant’whichdoesnotpresupposethe 

analytic/syntheticdistinction.Tothisend,I’mgoingto 

1 

1.exploretheoriginofthisnotion; 

2.showwhyitisimportanttodefineit; 

3.reviewsomeparadigmatic(butostensiblyunsatisfactory)allegeddefinitions; 

4.hintatthetruerelationbetweenthenotionsof‘logicalconstant’and 

‘analyticity’; 

5.makemanifesttheimplicitrenderingofanalyticitywhichisnestedin 

theclassicaldefinitionsoflogicalconstants; 

6.discussthestrictlyalternativeconstrualofanalyticityprominentin 

present-dayphilosophyoflogic; 

7.providesketchydefinitionoflogicalconstantsthatdeviatesfromthe 

firstbutremainstruetothesecond. 

ItwasBolzano,whointhedistant1837wasprobablythefirsttosuggest 

thatthereareconceptsbelongingtologicalone:accordingtohisownexample,thefactthatthequestion“whethercorianderimprovesone’smemory”obviouslydoesnotconcernlogicatall,suggeststhatitisnotaboutcorianderbutstudiessomethingdifferent(Hodges,2006,p.42).Inhisownview, 

thelogic’ssubjectmatterisexhaustedbythe‘logicalideas’whichaffectthe 

∗ IwouldliketoexpressmydeepgratitutetotheorganizersofLOGICA2008forthe 

granttheyhaveawardedmeforparticipationintheconference.

120 RosenLutskanov 

logicalformofpropositions(or‘sentencesinthemselves’)andaretobesingledoutbytheconditionthattheirvariancemodifiesthetruth-valueofany 

expressioncontainingthem(Siebel,2002,p.590).ThencameFregewhoin 

his“Begriffsschrift”(1879)hadaclear-cutdivisionofsymbols,employedin 

formallanguages,intotwokinds: ‘thosethatcanbetakentomeanvariousthings’(variablearguments)and‘thosethathaveafullydeterminate 

sense’(constantfunctions). Asfaraslogicisconcerned,thesecondkind 

ofsymbolscorrespondstoBolzano’s‘logicalideas’becauseitrepresents 

thosepartsoftheformalexpressionthathavetoremaininvariantunder 

replacement(Frege,1960,p.13).FinallyourstoryreachesRussell,whoin 

1903introducedthenowfamiliarterm‘logicalconstant’assubstitutefor 

Bolzano’s‘logicalideas’andFrege’s‘logicalfunctions’. Accordingtohim 

thesearethenotionsaccountableforthetruthofallpropositionswhichwe 

viewasapriorijustified.Buthedidnotprovideaformalcharacterization, 

onlythefollowingdeliberatelyconfusingexplanation:“logicalconstantsare 

allnotionsdefinableintermsofthefollowing:Implication,therelationofa 

termtoaclassofwhichitisamember,thenotionofsuchthat,thenotion 

ofrelation,andsuchfurthernotionsasmaybeinvolvedinthegeneralnotionofpropositionsoftheaboveform”(Russell,1903,p.3).Thereasonfor 

suchstrikingobscurityisthefactthathethoughtthat“logicalconstants 

themselvesaretobedefinedonlybyenumeration,fortheyaresofundamentalthatallthepropertiesbywhichtheclassofthemmightbedefined 

presupposesometermsoftheclass”(Russell,1903,pp.8–9). 

2 

LaterRussell’sinventionsurvivedthedemiseofhislogicism,althoughits 

introductionwasinitiallymotivatedaspartoftheattempttoshowthat 

allmathematicalnotionsarereducibletothenotionsoflogic(exemplified 

bythe‘logicalconstants’). Todaywearegenerallyinclinedtoclaimthat 

“logicalconceptiswhatcanbeexpressedbyalogicalconstantinalanguage”hencethequestion“Whatislogic?”istobeansweredbyanswering 

thequestion“Whatisalogicalconstant?” (Hodes,2004,p.134). Onthe 

otherhand,wejustcannotaffordtreatingthenotionoflogicalconstant 

asindefinableasRusselldid,sincepresentlywehaveatourdisposalalternativelistsoflogicalconstantsimposingonusdifferentconceptionsabout 

thesubjectmatteroflogic.Famously,QuinedidhisbesttoexpelRussell’s 

“relationofatermtoaclassofwhichitisamember”fromthelistoflogicalconstants,claimingthatthetheoryofthe‘∈’-relationisnotlogicbut 

“settheoryinsheep’sclothing”(Quine,2006,p.66).Thisexcommunication 

ofset-membershipfromtheprovinceoflogicisthesoledifferencebetween 

Quine’snominalisticpreferenceforfirst-orderlogicandRussell’sontologi-


callyexuberanttypetheory. Inthefaceofthismanifestdiscrepancy,we 

havetoadmitthattheonlywaytoprovideamotivatedchoiceoflogical 

frameworkistoexhibitjustifieddefinitionoftheterm‘logicalconstant’and 

toshowhowourconceptionoflogicstemsoutofit. Theage-old‘laundrylist’comprisingthevenerablemembersofthefamilyoflogicalnotions 

isnotenough. So,whichdefinitionsof‘logicalconstant’arecurrentlyin 

circulation? 

3 

Luckily,wehaveplentyofanswersofthistoilsomequestion;regrettably, 

noneofthemfaredverywell.ThefirstattempttoproviderigorousdefinitionwasprovidedbyCarnapinhis“LogicalSyntaxofLanguage”whichwas 

subsequentlysimplifiedbyTarski. Hisdefinitionwasfoundedontheconceptsof‘premiss-class’and‘range’(Spielraum): 

twopremiss-classeswere 

saidtobe‘equipollent’ifeachofthemisconsequenceoftheotherandthe 

rangewasdefinedasclassofpremiss-classes Mwiththepropertythateach 

premiss-classwhichisequipollenttoapremiss-classbelongingto Malso 

belongsto M.ThenCarnapexplainedthattherange Mofaproposition p 

represents“theclassofallpossiblecasesinwhich pistrue”or“thedomain 

ofallpossibilitiesleftopenby p”(Carnap,1959,p.199).Inthissetting,it 

seemsnaturaltodefinethe‘logicaljunctions’assimpleset-theoreticaloperationsonranges:by’supplementary’rangeofagivenrange 

M1wemean 

arange M2comprisingthosepremiss-classesthatdon’tbelongto M1;then 

foraproposition p1withrange M1wecandefineits‘negation’ p2asthe 

propositionwhoserangecoincideswiththesupplementaryrangeof p1.In 

thesamevein,wecandefinethe‘disjunction’oftwopropositions p1and p2 

asanotherproposition p3whoserangeistheunionoftherangesof p1and 

p2(Carnap,1959,p.200). AyearlaterTarskishowedthatthedefinition 

canbesimplifiedbysubstituting‘content’for‘range’(thecontentof pis 

theclassofallnon-analyticconsequencesof p):then p2isnegationof p1iff 

theyhaveexclusivecontentsand p3isdisjunctionof p1and p2iffitscontentisproductofthecontentsof 

p1and p2(Carnap,1959,p.204).These 

attempteddefinitionsofCarnapandTarskiwerenotconceivedassatisfactory,probablybecausetheyfoundedtheconceptualapparatusoflogicon 

theconceptualapparatusofsettheory. Thisisnotanepistemologically 

flawlessmove:theoperationofsentencenegationseemsmorefamiliarthan 

theintricateoperationofclasscomplementation;thatiswhythefirstisnot 

tobedefinedbymeansofthesecond. 

MaybethisisthereasonwhylaterTarskitookanothercourse. In 

hisfamouslecture“Whatarelogicalnotions”(1966)heproposedthenow 

classicaldefinition:logicalarejustthesenotionswhichareinvariantunder


allpermutationsoftheuniverseofindividualsontoitself. Thisdefinition 

provokedseverecriticismsbecauseittreatsaslogicalpropertiesallcardinalityfeaturesofthedomainofdiscourse. 

Anotherpainfuldefectwas 

exposedbyMcGeewhodefinedanoperationof‘wombatdisjunction’(∪W) 

suchthat‘p ∪W q’istrueif‘p ∨ q’istrueandtherearewombats(there 

isanelementofthedomainofthemodelwhichsatisfiesthepredicate‘is 

wombat’)andfalseotherwise(Feferman,1997,p.9–10). Clearly,wombat 

disjunctionisinvariantunderarbitrarypermutations,butitishardtoadmitthatitislogicalnotion—inordertoestablishthetruthorfalsityof 

anypropositioncontainingessentialoccurrencesofwombatdisjunctionwe 

needtocorroborateaspecificempiricalassumptionconcerningtheexistence 

(ornon-existence)ofwombats. Thereareseveralwell-knownattemptsto 

rectifyTarski’sdefinitionbyreplacing‘invarianceunderarbitrarypermutations’with‘invarianceunderarbitrarybijections’(Mostowski,1957),‘rigidinvarianceunderarbitrarybijections’(McCarthy,1981),and‘invarianceunderarbitraryhomomorphisms’(Feferman,1997).Asfarasweknow,noone 

oftheseattemptsisabletodiscriminateproperlybetweenthelogicaland 

theempirical(Mostowski’scriterionqualifies‘unicorn’aslogicalnotion)or 

thelogicalandthemathematical(Feferman’scriterionrenders‘thereexist 

infinitelymany’asbelongingtologic). Thatiswhy,wecanrecapitulate 

thispartofthediscussionbynoticingthat“itseemsinevitabletoconclude 

thattheseproposalsinspiredbyTarski... donotevenmeettheminimal 

requirementofextensionaladequacy”(Gomez-Torrente,2002,p.20). 

Athirdvariantfordefinitionofthenotionoflogicalconstantstemsfrom 

theworksofGentzen. Hisfollowerswereinclinedtoclaimthatlogical 

constantsaretobeidentifiedsolelybytheintroductionandelimination 

rulesgoverningtheirinferentialuses.Conjunction,forexample,isnothing 

butthispartofourlexiconthatfeaturesininferenceslike 

A, B 

A ∧ B 

and 

A ∧ B 

A,B . 

ThisbrightideawasshatteredbyPrior,whoprovidedhisinfamous tonkcounterexampledealingwithanewparticle‘tonk’governedbythefollowing 

rules: 

A 

Atonk B 

( tonk-Int) and 

Atonk B 

B 

( tonk-Elim). 

Theintroductionof‘tonk’allowsshowingtheformallanguageinquestion 

tobeinconsistent:justsubstitute‘¬A’for‘B’andapplysuccessively(tonk- 

Int)and(tonk-Elim). Thiswasintendedtomeanthatnotanysetofintroductionandeliminationrulesdefinesalogicalconstant:somethingmore 

hadtobeadded. Themysteriousadditionalingredientwaslateridentifiedas‘conservativity’(Belnap)or‘harmony’(Dummett). 

InDummett’s


ownexplanation,“Letuscallanypartofadeductiveinferencewhere,for 

somelogicalconstant c,ac-introductionruleisfollowedimmediatelybya 

c-eliminationrulea‘localpeakfor c’. Thenitisarequirement,forharmonytoobtainbetweentheintroductionrulesandeliminationrulesfor 

c, 

thatthelocalpeakfor cbecapableofbeingleveled,thatis,thatthereisa 

deductivepathfromthepremisesoftheintroductionruletotheconclusion 

oftheeliminationrulewithoutinvokingtherulesgoverningtheconstant 

c”(Dummett,1991,p.248). AsDummetthimselfreadilyacknowledged, 

“Theconservativeextensioncriterionisnot,however,tobeappliedtomore 

thanasinglelogicalconstantatatime.Ifwesoapplyit,weallowforthe 

priorexistence,inthepracticeofusingthelanguage,ofdeductiveinference, 

sincethereareanumberoflogicalconstants”but“theadditionofjustone 

logicalconstanttoalanguagedevoidofthem... cannotyieldaconservativeextension”since“ifdeductiveinferenceisevertobesaidtobeableto 

increaseourknowledge,thenitmustsometimesenabletorecognizeastrue 

astatementthatweshouldnot,withoutitsuse,beenablesotorecognize” 

(Dummett,1991,p.220).Thisdifficultyseemsinsurmountable:wecanuse 

thelevelingoflocalpeakstechniquetoidentifyasingleparticleaslogical 

constant,butitisnotpossibletorelyonthesamestrategytodelineatethe 

realmoflogicalnotions. 

4 

Uptothispointwehavereviewedthreeparadigmaticattemptstoprovide 

definitionofthenotionoflogicalconstant.Itappearsthatnoneofthemis 

materiallyadequate: 

(i)Carnap’sset-theoreticapproachconstruedlogicalnotionsusingprecisemathematicalmethodsbutdidnotevenposethequestionwhich 

operationsonpremiss-classesaretobeviewedasbelongingtologic; 

(ii)Tarski’smodel-theoreticapproachcouldnotsingleouttheclassof 

logicalconstantsandexperiencedseriousdifficultieswithborderline 

casessuchasnon-existentobjectsandmathematicalentities; 

(iii)Dummett’sproof-theoreticapproachprovidedjustifiedcriteriaforlogicalityofsingleconnectives(‘intrinsicharmony’)butcouldn’tachieve 

generallyapplicablestandard(for‘totalharmony’). 

Butthematerialinadequacyisnotthesoleoreventhegravestshortcoming 

ofthesepurporteddefinitions. Theyallweredevisedwithaneyeonthe 

notionscurrentlyrecognizedas‘logical’butwerenotcouchedinabroad 

theoreticalframework,clarifyingtheirinterplaywithsomeparticularrenderingofthenotionofanalyticity. 

Ifweturnbackweshallseethatthe


adventoflogicalconstantswasnecessitatedbythefactthatthe‘analytic 

program’(theattempttoidentify‘logicaltruth’and‘analyticaltruth’)was 

essentialpartofthe‘logicistprogram’:atruthisanalyticifitcanbereducedtogenerallogicallawsanddefinitions. 

Inanutshellthisreduction 

establishesthatonlylogicalconstantsoccuressentiallyinitandthelogicalconstantsweredrivenoutonstagesimplytoprovideatouchstonefor 

terminationofthisreductiveprocedure.Thatiswhy,“Thequestion‘What 

isalogicalconstant?’ wouldbeunimportantwereitnotfortheanalytic 

program”(Hacking,1994,p.3). Nowweareabletoperceivewherethe 

realproblemlies:ontheonehand,whenwetrytodefinelogicalconstants 

anddologic,wesilentlypresupposethatitispossibletodiscriminaterigorouslybetweenanalyticallytrue(truebyvirtueoflinguisticconventions) 

andsyntheticallytrue(truebyvirtueofbrutemattersoffact);ontheother 

hand,whenwetrytomakesenseofwhatwearedoinganddophilosophy 

oflogic,weovertlyblurtheanalytic/syntheticdistinction.Inthefollowing 

twoparagraphsI’lldomybesttoexplainwhythisdouble-mindednessisso 

crucialinthepresentcontext. 

5 

Whenwedomathematicallogic,weinvariablyandunwittinglysticktothe 

‘Viennese’orthodoxy.Thewayformallanguagesarepresentedandlogical 

symbolsareemployedwasmodeledupontheparadigmofWittgenstein’s 

Tractatus. Letusrememberthathis“fundamentalidea”wasthatwhile 

allotherwordsstandforobjects,“thelogicalconstantsarenotrepresentatives”(Wittgenstein,1963,prop.4.0312). 

Thisconceptionwasthesole 

basisoftheideathatthe‘real’propositionsareempiricallycontentful‘picturesofreality’,whilethepropositionsoflogicarerepresentationallyidle 

‘tautologies’(Wittgenstein,1963,prop.4.462).Carnaprehearsedthesame 

lineofthoughtinhisworksonformalsemantics:hestartedwiththesuggestionthat“wemustdistinguishbetweendescriptivesignsandlogicalsigns 

whichdonotthemselvesrefertoanythingintheworldofobjects,butserve 

insentencesaboutempiricalobjects”(Carnap,1958,p.6)andconcluded 

thatitispossibletoclassifyanysentenceas‘L-sentence’(thatis,‘logical’ 

=‘analytic’=‘trueorfalseonlogicalgrounds’)or‘F-sentence’(‘factual’ 

=‘synthetic’=‘trueorfalsebyvirtueoffactsoftheworld’). Although 

developedindifferentsetting,Tarski’smodel-theoreticapproachtoformal 

semanticsreiteratesthesamestepswhicharemirroredinthetwotypesof 

clausesinhisrecursivetruthdefinition: ontheoneside,wehaveabase 

clauseintroducingavaluationfunctionthatassignstruth-valuestoatomic 

sentencesinthemodel(heresentencesreceivetruth-valuesonextra-logical 

reasons;ifwehaveinmindsomeparticularinterpretationofthelanguage


wecansaythattheyare‘trueorfalsebyvirtueoffactsoftheworld’); 

ontheotherside,wehaverecursiveclausewhichdeterminesinwhatway 

thetruth-valuesofcomplexsentencesbuiltfromatomiconesandlogical 

constantsdependonthetruth-valuesalreadyassignedtoatomicsentences 

(heresentencesreceivetruth-valuesonintra-logicalreasons,thedefinitional 

sub-clausesfortheparticularlogicalconnectivesareanalyticallytruelinguisticconventionsfixingthemeaningoflogicalvocabulary).Finally,ifwe 

takealookattherivalproof-theoreticapproachchampionedbyDummett, 

wewouldseethesamepattern.Theinsistenceon‘conservativity’indealingwithintroductionandeliminationrulesforlogicalconstantscouldbe 

motivatedonlybytheideaofthepurelytautologouscharacteroflogically 

validinferences.Thelocalpeaksshouldbeinprinciple‘levelable’,precisely 

becausethemanipulationwithlogicalvocabularyaddsnosubstantivenew 

informationabouttheworld—inshort,becauselogicisanalyticandhas 

nothingtodowithsentences,truebyvirtueoffactsoftheworld. 

6 

Whenwedophilosophyoflogic,weareoftensaidcompletelydifferent 

things,incompatiblewiththeideathatlogicaltruth(conceivedasaparadigmaticcaseofanalyticity)istobedemarcatedfromfactualtruth.Starting 

withWittgensteinagain,weseethatallhislaterdevelopment—from 

“SomeRemarksonLogicalForm”whereheadmitsthat“wecanonlyarriveatacorrectanalysisbywhatmightbecalled,thelogicalinvestigationofthephenomenathemselves”(Wittgenstein,1993b,p.30)to“OnCertainty”wherehedeniedthepossibilitytodistinguishfromtheoutsetlogical 

fromempiricalpropositionsbecause“theriver-bedofthoughtsmayshift” 

(Wittgenstein,1993a,p.15)—canbeseenasrejectionoftheprevious 

sharpdivisionofalllocutionsintovacuouslytrue‘tautologies’andmeaningful‘picturesofreality’.Tarskihimself,asearlyas1930,wascommitted 

tothesamelineofthought:inanoteofCarnap’sdiary,datedFebruary22, 

1930weread:“8–11withTarskiataCafe.Aboutmonomorphism,tautology,hewillnotgrantthatitsaysnothingabouttheworld;heclaimsthatbetweentautologicalandempiricalstatementsthereisonlyameregradualandsubjectivedistinction”(Mancosu,2005,pp.328–329).Severalyearslater,in“Ontheconceptoffollowinglogically”,weread:“Atthefoundationofourwholeconstructionliesthedivisionofalltermsofalanguageinto 

logicalandextra-logical.Iknownoobjectivereasonswhichwouldallowone 

todrawaprecisedividinglinebetweenthetwocategoriesofterms... the 

divisionoftermsintologicalandextra-logicalexertsanessentialinfluence 

onthedefinitionalsoofsuchtermsas‘analytic’and‘contradictory’;yetthe 

conceptofananalyticsentence... tomepersonallyseemsrathermurky


(Tarski,2002,pp.188–189).Stilllater,in1944Tarskiconfessedinaletter 

toMortonWhitethatheisinclinedtothinkthat“logicalandmathematical 

truthsdon’tdifferintheiroriginfromempiricaltruths—bothareresults 

ofaccumulatedexperience... [andwehavetobepreparedto]rejectcertain 

logicalpremises(axioms)ofourscienceinexactlythesamecircumstances 

inwhichIamreadytorejectempiricalpremises(e.g.,physicalhypotheses)” 

(White,1987,p.31).Itwouldnotbestrange,ifthesewordssoundfamiliar: 

thesamecritiqueswereformulatedbyQuine,whometCarnapinPrague 

in1933andforcedhimtoadmittheuntenabilityoftheanalytic/synthetic 

distinction: “Isthereadifferenceinprinciplebetweenlogicalaxiomsand 

empiricalsentences? He[Quine]thinksnot. PerhapsI[Carnap]seeka 

distinctionjustforitsutility,butitseemsheisright: gradualdifference: 

theyaresentenceswewanttoholdfast”(Quine,2004,p.55).In“Truthby 

Convention”(Quine,1936)stressedthatsomeanalyticallytruestatements 

—definitionalconventions—canbeoverthrownforempiricalreasons,and 

in“Twodogmasofempiricism”(1951)introducedthefieldmetaphorthat 

obliteratescompletelytheanalytic/syntheticdistinction,makingevident 

thatitis“follytoseekaboundarybetweensyntheticstatements,which 

holdcontingentlyonexperience,andanalyticstatements,whichholdcome 

whatmay”(Quine,1961,p.50).Generally,thedestructionofthisdistinctionwaseffectedinHarvard: 

fromthe1940disputesofCarnap,Tarski 

andQuine,toWhite’searly“Theanalyticandthesynthetic:anuntenable 

dualism”(1950),Quine’sground-braking“Twodogmasofempiricism”and 

Goodman’sreflectiveequilibriumtheorydevelopedin“Thenewriddleof 

induction”(1954). 

7 

Thedefinitionsoflogicalconstantswehavediscussedwereshowntobemotivatedbytheuntenableassumptionthatwearecapableofdiscriminating 

rigorouslybetweenanalytic(truebyvirtueoflinguisticconventions)and 

synthetic(truebyvirtueofmattersoffact)propositions. Itseemstome 

thatitisjustifiedtosearchforadefinitionoflogicalconstantsthatconforms 

tothemainstreamphilosophyoflogic,afortioriadefinitionwhichdoesnot 

presupposetheanalytic/syntheticdistinction. Needlesstosay,everything 

Icansuggestonthistopicuptothepresentmomentissketchyandinconclusive.Firstofall,Iadmitthatlogicisconcernedwiththecodificationof 

inferentialpracticeswhicharegenerally‘outthere’beforewetrytoimpose 

normativerestrictionsonthem. ThesepracticesproducewhatBrandom 

calls‘materialinferences’—inferencesthatarenotjustifiedwithrecourse 

tothefeaturesoflogicalvocabularybutseemasimmediatelyacceptable. 

Anychainofmaterialinferencescanbecalledan‘argument’—thissug-


geststhatingeneralthematerialinferencesareseriallyorderedandaimat 

something—theclaimthatneedstobeestablishedastrueorfalse. Any 

argumentcanbemodelednaturallyinaslightmodificationoftheframeworkdevelopedinGuptaandBelnap’s“RevisionTheoryofTruth”(Gupta 

&Belnap,1993).Letusconsideraformallanguage Landamodel M0that 

assignstosomesentencesin Lthevalue‘true’:thesearethe‘axioms’(in 

theirancientinterpretationas‘sentencesproposedforconsideration’)that 

wetemporarilyacceptastrue.Thenthesetofpossiblematerialinferences 

withpremisestruein M0definesajump-operatorcorrelatingwithitanothermodelof 

L(letusdesignateitas‘M1’)containingallthosesentences 

thathavetobeacceptedastrueonthebasisofthebootstrapmodel M0 

(ingeneral,wedonotsupposethatthejumpoperatorismonotone:some 

previouslyacceptedsentencescanberefutedatlaterstages). Thesame 

procedurecanbeappliedagainandagainwhichgivesrisetoindefinitely 

extendibleseriesofmodels M0, M1, M2,etc. whichweshallcall‘anargument’(whosepremisesaretheaxioms,definedby 

M0). Inthecourse 

ofanytypicalargument Athereshallbesentencesthatatsomestageof 

itsdevelopment(say Mn)receiveconstantinterpretation(thesearethefixpointsofthejumpoperator);forthosethatareevaluatedastrueinall 

successivestages(Mn+1, Mn+2,etc.) weshallsaythattheyare‘rendered 

stablytrue’(bytheargument A). Now,insteadofasingleargument,let 

usconsiderabunchofarguments A1, A2,etc.andsupposethatthereisa 

classofsentences,classifiedasstablytruebyanyoneofthem;Ipropose 

thesesentencestobecalled‘renderedvalid’(bythesetofarguments A1, 

A2,etc.).Mysuggestionistoequatelogicalitywiththejustdefinedconcept 

of‘validity’;inthiswayweremainfairtosomeoftraditionallyrecognized 

distinctivefeaturesoflogicaltruth: 

(i)itistopic-neutral(becauseitisnotrelativetoaparticularargumentativesetting); 

(ii)itisnecessary(becauseitinevitablyshowsupinanytrainofreasoning 

belongingtothegeneralargumentativesetting); 

(iii)itisanalytic(becausegivenavalidsentenceandasetofargumentative 

premiseswecandemonstratebymeansofanalysisoftheaccepted 

materialinferencesthatitisgenuinepropositionoflogic). 

Moreover,thisdivisionofsentencesintoanalytic(renderedvalid)andsynthetic(notrenderedvalid)cannotbedrawnfromtheoutsetbecauseingeneralthequestion“isthesentencesrenderedvalidbythesetofarguments 

A1, A2,etc.?”isnotdecidable. 

Afterwehavesecuredaworkablenotionoflogicality,wecanaskourselvesagain:whatisalogicalconstant? 

Theansweristhatalexicalunit


istobetreatedaspieceoflogicalvocabularywhenitisinvariablyinterpretablecomponentofsomesetofvalidsentences.Thenwecanhuntdown 

thelogicalconstantsusingthe‘inverse’logicalapproachdevelopedbyvan 

Benthemwhosuggestedthatinsteadofchoosingsomepredefinedsetoflogicalconstantsandaskingwhattypesofinferencesarevalidatedbythem,wecantakesomeintuitivelyconvincingsetof(material)inferencesthatvalidateparticularpropositionsandsearchforthespecificconstantsthatare 

accountableforthem.Thismethodologicalshiftfrompredefinednormative 

accountsoflogicalitytopurelydescriptiveexplorationsofinferentialpracticeswasnamed“Copernicanrevolutioninlogic”(Benthem,1984,p.451). 

WhatI’vetriedtodohere,wastoshowthattherevolutionmustgoon... 

RosenLutskanov 

InstituteforPhilosophicalResearch,BulgarianAcademyofSciences 

6PatriarchEvtimiiBlvd.,Sofia1000,Bulgaria 

rosen.lutskanov@gmail.com 

http://www.philosophybulgaria.org/en/Sekcii/Logika/Sastav.php 

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Epistemic Logic with Relevant Agents 

Ondrej Majer Michal Peliˇs ∗ 

Theaimofepistemiclogicsistoformalizeepistemicstatesandactionsof 

(possiblyhuman)rationalagents.Atraditionalmeansforrepresentingthese 

statesandactionsemploystheframeworkofmodallogics,whereknowledge 

correspondstosomenecessityoperator.Modalaxioms(K,T,4,5,...)then 

correspondtostructuralpropertiesoftheagent’sknowledge. Employing 

strongmodalsystemssuchas S5leadstorepresentationsofagentswho 

aretooidealinmanyrespects—theyarelogicallyomniscient,theyhave 

aperfectreflectionoftheirbothpositiveandnegativeknowledge(positive 

andnegativeintrospection)etc.Sometimestheserepresentationsarecalled 

epistemiclogicsofpotentialratherthanactualknowledge. 

Frameworksrepresentingonlyperfectagentshavebeenfrequentlycriticized,see(Fagin,Halpern,Moses,&Vardi,2003)and(Duc,2001),and 

somestepstowardsmorerealisticrepresentationshavebeenmade(e.g., 

(Duc,2001)).Wealsoattempttorepresentagentsinanenvironmentmore 

realistically.Ourmotivationisepistemic,weshallconcentrateonanagent 

workingwithexperimentalscientificdata. 

Arealisticagent 

Ouragentisascientistundertakingexperimentsorobservations.Hertypicalenvironmentisanexperimentalsetupandherknowledgeisusually 

experimentaldata(inputsandoutputsofanexperiment/observation)and 

somegeneralizationsextractedfromtheexperimentaldata. 

∗ Workonthistextwassupportedinpartbygrantno.401/07/0904oftheGrantAgency 

oftheCzechRepublicandinpartbygrantno.IAA900090703(Dynamicformalsystems) 

oftheGrantAgencyoftheAcademyofSciencesoftheCzechRepublic.Wewishtothank 

toTimothyChildersforvaluablecomments.

132 OndrejMajer&MichalPeliˇs 

Weassumetheobservations(‘facts’)aretypicallyrepresentedbyatoms 

andtheirconjunctionsanddisjunctions,whilegeneralizations(‘regularities’) 

arerepresentedbyconditionals(andtheircombinations). Aconditional 

issupposedtorecordaregularlyobservedconnectionbetweenthefacts 

representedbytheantecedentandthefactsrepresentedbytheconsequent. 

Itseemstobeclearthatformanyreasonsthematerialimplicationisnot 

anappropriaterepresentationofsuchaconditional.Oneofthemainreasons 

isthatthematerialimplicationmayconnectanytwoarbitraryformulas α, 

β.Forexample, 

1. α → (β → α), 

2.(α ∧ ¬α) → β, 

3. α → (β ∨ ¬β), 

aretautologiesofclassicallogic.Inourepistemicinterpretationthematerial 

implicationwouldmakea‘law’fromeverytwo‘facts’,whichwouldobviously 

maketherepresentationuseless. Ithasotherundesirableproperties. It 

cannotdealwitherrorsinthedata,whichresulttocontradictoryfacts 

(asituationwhichmayverywellhappeninthescientificpracticedueto 

equipmenterrors).Onesucherrorcorruptsalltheremainingdata(froma 

contradictioneverythingfollows—see2). Italsoadmits’laws’whichare 

ofnouseastheirconsequentisatautology(asin3—atautologyfollows 

fromanything) 

Thetautologies1–3arejustexamplesoftheparadoxesofmaterialimplication.Asthese‘paradoxes’werecompletelysolvedonlyinthesystemsof 

relevantlogics,theobviouschoiceforaconditionalforourscientificagent 

isrelevantimplication. 

2 Relationalsemanticsforrelevantlogics 

Ourpointofdeparturewillbethedistributiverelevantlogic RofAnderson 

andBelnap(1975). Themostnaturalwaytointroducerelevantlogicsis 

certainlyprooftheoretical(see,e.g.,(Paoli,2002)).Howeverwewouldlike 

tofollowthemodaltraditioninrepresentinganagent’sepistemicstatesas 

asetofformulasandmaketheagent’sknowledgedependentnotonlyon 

thecurrentepistemicstate,butalsoonthestatesepistemicalternatives. 

Technicallyspeakingwewanttousearelationalsemantics.Thiscannotbe 

astandardKripkesemanticswithpossibleworldsandabinaryaccessibility 

relation,butamoregeneralrelationalstructure. 

FormallyourframeworkwillbebasedontheRoutley–Meyersemantics, 

asdevelopedbyMares(Mares,2004),Restall(Restall,1999),Paoli(Paoli,


2002),andothers,towhichweshalladdepistemicmodalities.Thissemanticshasbeenunderconstantattackforitsseemingunintuitivness,butwe 

believeitfitsverywellourmotivations. 

Wegiveaninformalexpositionofstructuresintherelevantframeand 

definitionofconnectives(forformaldefinitionsseetheappendixA). 

Relevantframe 

Arelevantframeisastructure F = 〈S,L,C,✂,R〉,where Sisanon-empty 

setofsituations(states), L ⊆ Sisanon-emptysetofdesignatedlogical 

situations, C ⊆ S 2 isacompatibility relation, ✂ ⊆ S 2 isarelationof 

involvement, R ⊆ S 3 isanrelevancerelation. 

Amodel Misarelevantframewiththerelation �,where s � ϕhasthe 

samemeaningasinKripkeframes—that scarriestheinformationthat 

theformula ϕistrue(ϕ ∈ sifweconsiderstatestobesetsofformulas). 

Situations Situationsorinformationstatesplaythesameroleaspossible 

worldsinKripkeframes. Weassume,theyconsistofdataimmediately 

availabletotheagent. Likepossibleworlds,wecanseesituationsassets 

offormulas,but,unlikepossibleworlds,situationsmightbeincomplete 

(neither ϕnor ¬ϕistruein s)orinconsistent(both ϕand ¬ϕaretrue 

in s). 

Conjunctionanddisjunction Classical(weak)conjunctionanddisjunction 

correspondtothesituationwhentheagentcombinesdataimmediately 

availabletoher,i.e.datafromhercurrentsituation. Theybehaveinthe 

samewayasinthecaseofclassicalKripkeframes—theirvalidityisgiven 

locally: 

s � ψ ∧ ϕiff s � ψand s � ϕ 

s � ψ ∨ ϕiff s � ψor s � ϕ 

Weakconnectivesaretheonlyoneswhicharedefinedlocally.Thetruth 

ofnegationandimplicationdependsalsoonthedatainsituations,related 

totheactualones,sotheyaremodalbynature. Itispossibletodefine 

strongconjunctionanddisjunctionaswell(seeappendixA). 

Implication Implicationisamodalconnectiveinthesensethatitstruth 

dependsnotonlyonthecurrentsituation,butalsoonitsneighborhood.It 

canbeagainunderstoodinanalogywiththestandardmodalreading.We 

saythatanimplication (ϕ → ψ)holdsnecessarilyinaKripkeframeiff 

inallworldswheretheantecedentholds,theconsequentholdsaswell. In 

otherwords,theimplication (ϕ → ψ)holdsthroughalltheneighborhood


oftheactualworld. Intherelevantcasetheneighborhoodofasituation 

sisgivenbypairsofsituations y, zsuchthat s, y, zarerelatedbythe 

ternaryrelation R.Weshallcall y, zantecedentandconsequentsituations, 

respectively.Wesaythattheimplication (ϕ → ψ)holdsatthesituation s 

iffitisthecasethatforeveryantecedentsituation ywhere ϕ(theantecedent 

oftheimplication)holds, ψ(theconsequentoftheimplication)holdsatthe 

correspondingconsequentsituation z. 

s � (ϕ → ψ) iff (∀y,z)(Rsyzimplies (y � ϕimplies z � ψ)) 

Therelation Rreflectsinourinterpretationactualexperimentalsetups. 

Antecedentsituationscorrespondtosomeinitialdata(outcomeofmeasurementsorobservations)ofsomeexperiment,whiletherelatedconsequent 

situationscorrespondtothecorrespondingresultingdataoftheexperiment. 

Implicationthencorrespondstosome(simple)kindofarule:ifIobserve 

inmycurrentsituation,thatateveryexperiment(representedbyacouple 

antecedent—consequentsituation)eachobservationof ϕisfollowedbyan 

observationof ψ,thenIaccept‘ψfollows ϕ’asarule. 

Logicalsituations Theframeworkwepresentedsofarisveryweak:there 

arejustfewtautologiesvalidinallsituationsandsomeoftheimportant 

ones—thosebeingusuallyconsideredasbasiclogicallawsaremissing. 

Forexamplethewidelyacceptedidentityaxiom (α → α)andtheModus 

Ponensrulefailtoholdineverysituation. 

Thisisconnectedtothequestionoftruthinarelevanceframe(model). 

IfwetakeahintfromKripkeframes,weshouldequatetruthinaframe 

withtruthineverysituation. Butthiswouldgivesusanextremelyweak 

systemwithsomeveryunpleasantproperties(cf.(Restall,1999)).Designers 

ofrelevantlogicstookadifferentroute—insteadofrequiringtruthinall 

situations,theyidentifythetruthinaframejustwiththetruthinall 

logicallywellbehavedsituations. Thesesituationsarecalledlogical. In 

ordertosatisfythe‘goodbehavior’ofasituation litisenoughtorequire 

thatalltheinformationinanyantecedentsituationrelatedto liscontained 

inthecorrespondingconsequentsituationaswell:foreach x,y ∈ S, Rlxy 

implies |x| ⊆ |y|,where |s|isthesetofallformulas,whicharetrueinthe 

situation s. 

Itiseasytoseethatsituationsconstrainedinthiswayvalidateboththe 

identityaxiomand(implicative)ModusPonens. 

Involvement Involvementisarelationresemblingthepersistencerelation 

inintuitionisticlogic—wecanseeitasarelationofinformationgrowth. 

Howevernoteverytwosituationswhichareininclusionwithrespecttothe 

validatedformulasareintheinvolvementrelation.Werequirethatsuchan


inclusionisobservedorwitnessed.Noteverysituationcanplaytheroleof 

thewitness—onlythelogicalsituationscan. 

x ✂ y iff (∃l ∈ L)(Rlxy) 

Negation InKripkemodelsthenegationofaformula ϕistrueataworld 

iff ϕisnottruethere.Assituationscanbeincompleteand/orinconsistent, 

thisisnotanoptionanymore. Negationbecomesamodalconnective 

anditsmeaningdependsontheworldsrelatedtothegivenworldbya 

binarymodalrelation Cknownascompatibility. Informallywecansee 

thecompatiblesituationsasinformationsourcesourscientistwantstobe 

consistentwith. (Imaginethedataofresearchgroupsworkingonrelated 

subjects.) 

Theformula ¬ϕholdsat s ∈ Siffitisnot‘possible’(inthestandard 

modalsensewithrespecttotherelation C)that ϕ: atnosituation s ′ , 

compatiblewith(‘accessiblefrom’)thesituation s,itisthecasethat ϕ 

(either s ′ isincompletewithrespectto ϕor ¬ϕholdsthere). 

s � ¬ϕ iff (∀s ′ ∈ S)(sCs ′ implies s ′ �� ϕ) 

Informallyspeaking,theagentcanexplicitlydenysomehypothesis(a 

pieceofdata)onlyifnoresearchgroupinherneighborhoodclaimsitis 

true. Thisconditionalsohasanormativeside:shehastobeskepticalin 

thesensethatshedenieseverythingnotpositivelysupportedbyanyofher 

colleagues(inthesituationsrelatedtoheractualsituation). 

Ifwewanttograntnegativefactsthesamebasiclevelaspositivefacts,we 

canreadtheclauseforthedefinitionofcompatibilityintheotherdirection: 

theagentcanrelateheractualsituationjusttothesituationswhichdonot 

contradicthernegativefacts. 

Dependingonthepropertiesofthecompatibilityrelationweobtaindifferentkindsofnegations.Weshallshortlycommentonthem.Thecompatibilityrelationisingeneralnotreflexive:inconsistentsituationsarenotself-compatibleandsoreflexivityholdsonlyforconsistent 

situations.Itisclearthatforaninconsistentself-compatiblesituationthe 

clausefornegationwouldnotwork.Ontheotherhand,inconsistentsituationscanbecompatiblewithsomeincompletesituations. 

Noris Ctransitive. Letushavesituations x, y, zsuchthat x � ϕ, 

z � ¬ϕ,and ydoesnotincludeeither ϕor ¬ϕ.Assumethat xCyand yCz. 

Thenaccordingtothedefinitionofnegationitcannotbethat xCz. 

Itisquitereasonabletoassumethat Cissymmetric. Thiscondition 

impliesthatwegetonlyonenegation(otherwisewewouldgetleftandright 

negation)andwegetthe’unproblematic’halfofthelawofdoublenegation 

(if x � ϕ,then x � ¬¬ϕ).


Wealsoassume Cisdirectedandconvergent.Directednessmeansthat 

thereisatleastonecompatiblesituationforeach x ∈ S.Convergencesays 

thatthereisamaximalcompatiblesituation x ⋆ .(SeeappendixA.) 

Maximalcompatiblesituations(withrespectto x)canbeinconsistent 

abouteverythingnotconsideredin x.Fromthesymmetryof Cweobtain 

x✂x ⋆⋆ .Ifweassume,moreover, x☎x ⋆⋆ ,thenwegettheoperation ⋆withthe 

property x = x ⋆⋆ ,i.e.theRoutleystar.Thedefinitionofnegation-validity 

isthenwrittenintheform: 

x � ¬ϕ iff x ⋆ �� ϕ 

TheRoutleystarhasbeenoneofthecontroversialpointsoftheRoutley– 

Meyersemantics,butinourmotivationithasaquitenaturalexplanation:if 

compatiblesituationsrepresentcolleaguesfromdifferentresearchgroupsour 

agentcollaborateswith,thenthemaximalcompatiblesituationcorrespond 

toacolleague(‘boss’)whohasalltheinformationtheothercolleaguesfrom 

thegrouphave.Theniftheagentwantstoacceptsomenegativeclauseshe 

doesnothavetospeaktoeachofthecolleaguesandaskhis/heropinion, 

shejustasksthe‘boss’directlyandknowsthatbossesopinionrepresents 

theopinionsoftheentirecompatibleresearchgroup. 

Thiscompletesourexpositionofrelationalsemanticsforrelevantlogics. 

Wenowmovetoepistemicmodalities. 

3 Knowledgeinrelevantframework 

Therehavebeensomeattemptstocombineanepistemicandrelevantframework(see(Cheng,2000)and(Wansing,2002)),buttheyhaveadifferent 

aimthenourapproach. 

Fromapurelytechnicalpointofviewthereareanumberofwaysto 

introducemodalitiesintherelevantframework—GregRestallin(Restall, 

2000)providesanicegeneraloverview. Aswementioned,therelevant 

frameworkalreadycontainsmodalnotions. Wethereforedecidedtouse 

thesenotionstointroduceepistemicmodalitiesratherthantointroduce 

newones. 

Intheclassicalepistemicframewhatanagentknowsinaworld wis 

definedaswhatistrueinallepistemicalternativesof w,whicharegivenby 

thecorrespondingaccessibilityrelation.Ourideaoftheagentasascientist 

processingsomekindofdatarequiresadifferentapproach. 

Weassumeouragentinhercurrentsituation sobserves(hasadirect 

approachto)somedata,representedbyformulaswhicharetrueat s.Sheis 

awareofthefactthatthesedatamightbeunreliable(oreveninconsistent). 

Inordertoacceptsomeofthecurrentdataasknowledgetheagentrequires 

aconfirmationfromsome‘independent’resources.


Inourapproachresourcesaresituationsdealingwiththesamekindof 

dataavailableinthecurrentsituation.Aresourceshallbemoreelementary 

thanthecurrentsituation,i.e.,itshouldnotcontainmoredata(aresource 

isbelow sinthe ✂-relation). Alsothedatafromtheresourceshouldnot 

contradictthedatainthecurrentsituation(aresourceiscompatiblewith s). 

Definition4(Knowledge). 

s � Kϕ iff (∃x)(sC ✁ xand x � ϕ), 

where sC ✁ xiff sCxand x ✂ sand x �= s. 

Inshort, ϕisknowniffthereisanresource(‘lower’compatiblesituation 

differentfromtheactualone)validating ϕ. 

Weallowedouragenttodealwithinconsistentdatainordertogetamore 

realisticpicture.However,theagentshouldbeabletoseparateinconsistent 

data. Themodalityweintroducedprovidesuswithsuchanappropriate 

filter. Letusassumeboth ϕand ¬ϕarein s(e.g.,ouragentmightreceivesuchinconsistentinformationfromtwodifferentsources).Theagent 

considersboth ϕand ¬ϕtobepossible,butneitherofthemisconfirmed 

informationasaccordingtothedefinition,nosituationcompatibleto scan 

containeither ϕor ¬ϕ. 

Basicproperties 

Itistobeexpectedthatoursystemblocksalltheundesirablepropertiesof 

bothmaterialandstrictimplication. Moreover,weruledoutthevalidity 

ofsomeofthepropertiesof‘classical’epistemiclogicsthatwehavecriticized,inparticular,bothpositiveandnegativeintrospection,aswellas 

someclosureproperties. 

Letushavearelevantframe F = 〈S,L,C,✂,R〉.Recallthatthetruthin 

theframe Fcorrespondstothetruthinthelogicalsituationsof F(underany 

valuation). Wewillalsousethestrongernotionoftruthinallsituations 

of F(underanyvaluation). Fromtheviewpointofourmotivationthe 

latternotionismoreinterestingasouragentmighthappentobeinother 

situationsthanthelogicalones. 

Ourapproachmakesthe‘truthaxiom’Tvalid.Foranysituation s ∈ S, 

if ϕisknownat s(s � Kϕ),thenthereisa✂-lowercompatiblewitness 

with ϕtrue,whichmakes ϕtobetrueat saswell.Thus,formula 

Kα → α 

isvalid. 

TheaxiomKandthenecessityrule,commontoallnormalepistemic 

logics,fail.First,letusassumethat ϕisvalidformula.Thenecessityrule


( ϕ 

)wouldimplythevalidityof Kϕ. |= ϕmeansthat ϕistrueinevery 

Kϕ 

logicalsituation l. However,for l � Kϕaconfirmationfromadifferent 

resourceisrequired,theremustbeasituation xsuchthat x � ϕand lC✁x, which,ingeneral,doesnotneedtobethecase. 

Second,inourinterpretationthevalidityofaxiomKisnotwellmotivatedanddoesnothold. 

Kisinfacta‘distributionofconfirmation’: If 

animplicationisconfirmedthentheconfirmationoftheantecedentimplies 

theconfirmationoftheconsequent. 

�|= K(α → β) → (Kα → Kβ) 

Introspection Aswedefinedknowledgeasindependentlyconfirmeddata, 

theepistemicaxioms4and5correspondinourframeworktoa‘second 

orderconfirmation’ratherthantointrospection.Itiseasytoseethatboth 

axiomsfail. 

�|=Kα → KKα, 

�|=¬Kα → K¬Kα 

Necessityandpossibility Wedonotintroducepossibilityusingthestandarddefinition 

Mϕ def 

≡ ¬K¬ϕ.Ourideaofepistemicpossibilityisthatour 

agentconsidersallthedataavailableatthecurrentsituationaspossible.If 

weintroduceformally s � Mϕas s � ϕ,thenitfollowsfromtheTaxiom 

thatinallsituationsnecessityimpliespossibility: 

(∀s ∈ S)(s � Kϕ → Mϕ) 

Howeverforthestandarddualpossibilitythisisnottrue. 

�|= Kϕ → ¬K¬ϕ 

Letuscommentontherelationofnegationandnecessityinourframework. 

Thereisadifferencebetween s �� Kϕand s � ¬Kϕ. Theformer 

simplysaysthat ϕisnotconfirmedatthecurrentsituation s,whilethelattersaysthat 

ϕisnotconfirmedinthesituationscompatiblewith s.From 

thispointofviewitisuncontroversialthatboth Kϕ(confirmationinthe 

currentsituation)and ¬Kϕ(thelackofconfirmationinthecompatiblesituations)mightbetrueinsomesituation 

s(thenecessaryconditionisthat 

sisnotcompatiblewithitself). 

Closureproperties ItiseasytoseethatthemodalModusPonens 

Kα K(α → β) 

Kβ


doesnothold(forthereasonsgiveninthesectiononKaxiom).However, 

itsweakerversion 

Kα K(α → β) 

β 

holdsnotonlyinlogicalsituations,butinallsituations.If Kαand K(α → 

β)aretrueinany s ∈ S,then s � β.AxiomTandtheassumption Rsss 

arecrucialhere. 

Contradictioninoursystemisnon-explosive: ϕand ¬ϕmightholdina 

contradictorysituation,whichneednotbeconnectedtoanysituationwhere 

ψholds. 

�|= (ϕ ∧ ¬ϕ) → ψ 

Ontheotherhand,theknowledgeofcontradictionimpliesanything(asa 

contradictionisneverconfirmed): 

|= K(ϕ ∧ ¬ϕ) → ψ 

Modaladjunctionalsodoesnothold—if Kαand Kβaretruein s,then 

obviously (α∧β)istruetherebecauseofthetruthaxiombut K(α∧β)does 

notneedtobetruein s.(Ifeachof αand βisconfirmedbysomeresource, 

therestillmightbenoresourceconfirmingtheirconjunction.) 

4 Conclusion 

Weintroducedasystemofepistemiclogicbasedontheframeworkofrelevantlogic.Wegaveanepistemicinterpretationoftherelationalsemanticsforrelevantlogicsanddefinedepistemicmodalitiesmotivatedbythisinterpretation. 

Insteadofintroducingadditionalrelationsintotheframework, 

wearguedinfavorofusingmodalitiesbasedontherelationsalreadycontainedintheframe. 

Thewholeprojectisataninitialstage:thereismuchtobedoneboth 

technicallyandintheareaofinterpretation.Inparticularweshalldevelop 

inamoredetailtheepistemicinterpretationofourframework,givean 

axiomatizationofoursystem,andcharacterizeitsformalproperties. 

A Relevantlogic R 

Therearemoreformalsystemsthatcanbecalledrelevantlogic.Fromthe 

proof-theoreticalviewpoint,allofthemareconsideredtobesubstructural 

logics(see(Restall,2000)and(Paoli,2002)). Herewepresenttheaxiom 

systemand(Routley–Meyer)semanticsfrom(Mares,2004)withsomeelementsfrom(Restall,1999).


Syntax 

Weusethelanguageofclassicalpropositionallogicwithsignsforatomic 

formulas P = {p,q,... },formulasbeingdefinedintheusualway: 

Axiomschemes 

1. A → A 

ϕ ::= p | ¬ψ | ψ1 ∨ ψ2 | ψ1 ∧ ψ2 | ψ1 → ψ2 

2. (A → B) → ((B → C) → (A → C)) 

3. A → ((A → B) → B) 

4. (A → (A → B)) → (A → B) 

5. (A ∧ B) → A 

6. (A ∧ B) → B 

7. A → (A ∨ B) 

8. B → (A ∨ B) 

9. ((A → B) ∧ (A → C)) → (A → (B ∧ C)) 

10. (A ∧ (B ∨ C)) → ((A ∧ B) ∨ (A ∧ C)) 

11. ¬¬A → A 

12. (A → ¬B) → (B → ¬A) 

Stronglogicalconstants ⊗(groupconjunction,fusion)and ⊕(groupdisjunction)aredefinablebyimplicationandnegation: 

Rules 

• (A ⊕ B) def 

≡ ¬(¬A → B) 

• (A ⊗ B) def 

≡ ¬(¬A ⊕ ¬B) 

AdjunctionFrom Aand Binfer A ∧ B. 

ModusPonensFrom Aand A → Binfer B.


Routley–Meyersemantics 

An R-frameisaquintuple F = 〈S,L,C,✂,R〉,where Sisanon-emptysetof 

situationsand L ⊆ Sisanon-emptysetoflogicalsituations.Therelations 

C ⊆ S 2 , ✂ ⊆ S 2 ,and R ⊆ S 3 wereintroducedinsection2,herewesumup 

theirproperties. 

Propertiesoftherelation R Thebasicpropertyof R: 

if Rxyz, x ′ ✂ x, y ′ ✂ y,and z ✂ z ′ , then Rx ′ y ′ z ′ . 

Thismeansthattherelation Rismonotonicwithrespecttotheinvolvement 

relation. 

Moreoveritisrequiredthat: 

(r1) Rxyzimplies Ryxz; 

(r2) R 2 (xy)zwimplies R 2 (xz)yw,where R 2 xyzwiff 

(∃s)(Rxysand Rszw); 

(r3) Rxxx; 

(r4) Rxyzimplies Rxz ⋆ y ⋆ . 

Propertiesoftherelation C Compatibilitybetweentwostatesisinherited 

bythestatesinvolvedinthem(’lessinformativestates’): 

If xCy, x1 ✂ x,and y1 ✂ y,then x1Cy1. 

Moreover,werequirethefollowingproperties: 

(c1)(symmetricity) xCyimplies yCx; 

(c2)(directedness) (∀x)(∃y)(xCy); 

(c3)(convergence) (∀x)(∃y(xCy)implies (∃x ⋆ )(xCx ⋆ and 

∀z(xCzimplies z ✂ x ⋆ ))); 

(c4) x ✂ yimplies y ⋆ ✂ x ⋆ ; 

(c5) x ⋆⋆ ✂ x. 

Model R-model MisaR-frame Fwithavaluationfunction v: P → 2 S . 

Thetruthofaformulaatasituationisdefinedinthefollowingway: 

• s � piff s ∈ v(p), 

• s � ¬ϕiff s ⋆ �� ϕ,


• s � ψ ∧ ϕiff s � ψand s � ϕ, 

• s � ψ ∨ ϕiff s � ψor s � ϕ, 

• s � (ϕ → ψ)iff (∀y,z)(Rsyzimplies (y � ϕimplies z � ψ)). 

Aswealreadysaid,thetruthofaformulainamodelandinaframe, 

respectively,isdefinedastruthinalllogicalsituationsofthismodel/frame. 

Asusual, R-tautologiesareformulastrueinallrelevantframes.Whenever 

ϕisaR-tautology,wewrite |= ϕandsaythat ϕisavalidformula. 

Thecondition(r1)validatestheimplicativeversionofModusPonens 

(axiomschema3).Itdoesnotvalidatetheconjunctiveversion (A ∧ (A → 

B)) → B,whichrequires(r3). 

(r2)correspondstothe‘exchangerule’ (A → (B → C)) → (B → 

(A → C)),whichisderivablefromtheaxiomsgivenabove. 

(r4)validatescontraposition(axiomschema12).Ifweworkwithoutthe 

Routleystar,thiscanberewrittenas: 

Rxyzimplies (∀z ′ Cz)(∃y ′ Cy)(Rxy ′ z ′ ). 

Directednessandconvergenceconditionsarenecessaryforthedefinition 

oftheRoutleystar.From(c1)weobtainthevalidityof (A → ¬¬A)and 

fromthelastcondition(c5)wegettheaxiomschema11. 

OndrejMajer 

InstituteofPhilosophy,AcademyofSciencesoftheCzechRepublic 

Jilská1,11000Praha1 

majer@site.cas.cz 

http://logika.flu.cas.cz 

MichalPeliˇs 

InstituteofPhilosophy,AcademyofSciencesoftheCzechRepublic 

Jilská1,11000Praha1 

pelis@ff.cuni.cz 

http://logika.flu.cas.cz 

References 

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discovery. InE.Kawaguchi,H.Kangassalo,H.Jaakkola,&I.Hamid(Eds.), 

InformationmodellingandknowledgebasesXI(pp.136–159).Amsterdam:IOS 

Press. 

Duc,H.N. (2001). Resource-boundedreasoningaboutknowledge. Unpublished 

doctoraldissertation, FacultyofMathematicsandInformatics, Universityof 

Leipzig.


Fagin,R.,Halpern,J.,Moses,Y.,&Vardi,M.(2003).Reasoningaboutknowledge. 

Cambridge,MA:MITPress. 

Mares,E.(2004).Relevantlogic.Cambridge:CambridgeUniversityPress. 

Mares,E.,&Meyer,R. (1993). Thesemanticsof r4. JournalofPhilosophical 

Logic,22,95–110. 

Paoli,F.(2002).Substructurallogics:Aprimer.Dordrecht:Kluwer. 

Restall,G. (1993). Simplifeidsemanticsforrelevantlogics(andsomeoftheir 

rivals).JournalofPhilosophicalLogic,22,481–511. 

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rivals).JournalofPhilosophicalLogic,24,139–160. 

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computation: The1994Moragaproceedings.CSLILectureNotes(Vol.58,pp. 

463–477).Stanford,CA:CSLI. 

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PhilosophicalLogic,31,591–612.

Betting on Fuzzy and Many-valued Propositions 


Peter Milne 

Ina1968article,‘ProbabilityMeasuresofFuzzyEvents’,LotfiZadehproposedaccountsofabsoluteandconditionalprobabilityforfuzzysets(Zadeh, 

1968).Where Pisanordinary(“classical”)probabilitymeasuredefinedon 

a σ-fieldofBorelsubsetsofaspace X,and µAisafuzzymembershipfunctiondefinedon 

X,i.e.afunctiontakingvaluesintheinterval [0,1],the 

probabilityofthefuzzyset Aisgivenby 

� 

P(A) = µA(x)dP. 

X 

Thethingtonoticeaboutthisexpressionisthat,inaway,there’snothing 

“fuzzy”aboutit.Tobewelldefined,wemustassumethatthe“levelsets” 

{x ∈ X : µA(x) ≤ α}, α ∈ [0,1], 

are P-measurable. Theseareordinary,“crisp”,subsetsof X. Andthen 

P(A)isjusttheexpectationoftherandomvariable µA.—Thisisentirely 

classical.Ofcourse,youmayinterpret µAasafuzzymembershipfunction 

butreallywehave,ifyou’llpardonthepun,inlargemeasurelostsightof 

thefuzziness. 

Soyoumightask: 

•isthistheonlywaytodefinefuzzyprobabilities? 

Theanswer,Ishallargue,isyes. 

DefiningconditionalprobabilityZadehoffered 

P(A|B) = P(AB) 

, when P(B) > 0, 

P(B)

146 PeterMilne 

where 

Onemightwonder: 

∀x ∈ X µAB(x) = µA(x) × µB(x). 

•isthistheonlywaytodefineconditionalprobabilities? 

Theanswer,Ishallsuggest,isno,itisnottheonlywaybutitistheonly 

sensibleway. 

Zadehassignsprobabilitiestosets.WhatIofferhere,usingDutchBook 

Arguments,isavindicationofZadeh’sspecificationswhenprobabilityis 

assignedtopropositionsratherthansets.(Buttranslationbetweenpropositiontalkandsetandeventtalkisstraightforward.It’sjustthatproposition 

talkfitsbetterwithbettingtalk.) 

2 Betsandmany-valuedlogics 

Iapply“theDutchBookmethod”,asJeffPariscallsit(Paris,2001),to 

fuzzyandmany-valuedlogicsthatmeetasimplelinearitycondition.Ishall 

callsuchlogicsadditive. 

Additivity 

Foranyvaluation vandforanysentences Aand B 

v(A ∧ B) + v(A ∨ B) = v(A) + v(B) 

where‘∧’and‘∨’theconjunctionanddisjunctionofthelogicinquestion. 

Additivityiscommon:theGödel,Łukasiewicz,andproductfuzzylogics 

arealladditive,asareGödelandŁukasiewicz n-valuedlogics. 

InordertoemployDutchBookarguments,weneedabettingscheme 

suitablysensitivetotruth-valuesintermediatebetweentheextremevalues 

0and 1.Settingouttheclassicalcasetherightwaymakesonegeneralization 

obvious. 

Ratherthanbettingodds,whicharealgebraicallylesstractable,weuse, 

asisstandard,a“normalized”bettingschemewithfairbettingquotients. 

Classically,withabeton Aatbettingquotient pandstake S: 

•thebettorgains (1 − p)Sif A; 

•thebettorloses pSifnot-A. 

Taking 1fortruth, 0forfalsity,and v(A)tobethetruth-valueof A,wecan 

summarisethisschemelikethis: 

thepay-offtothebettoris (v(A) − p)S.


Andnowweseehowtoextendbetstothemanyvaluedcase:weadoptthe 

sameschemebutallow v(A)tohavemorethantwovalues.Thesloganis: 

thepay-offisthelargerthemoretrue Ais. 1 

Usingthisbettingscheme,weobtainDutchBookargumentsforcertainseeminglyfamiliarprinciplesofprobability,seeminglyfamiliarinthat 

formallytheyrecapitulateclassicalprinciples. 

• 0 ≤ Pr(A) ≤ 1; 

• Pr(A) = 1when |= A; 

• Pr(A) = 0when A |=; 

• Pr(A ∧ B) + Pr(A ∨ B) = Pr(A) + Pr(B). 

Here ∧and ∨aretheconjunctionanddisjunction,respectively,ofanadditivefuzzyormany-valuedlogic. 

Otherprinciplesthatmayormaynotbeindependent,dependingonthe 

logic: 

• Pr(A) + Pr(¬A) = 1when v(¬A) = 1 − v(A); 

• Pr(A) ≥ xwhen,underallvaluations, v(A) ≥ x; 

• Pr(A) ≤ xwhen,underallvaluations, v(A) ≤ x; 

• Pr(A) ≤ Pr(B)when A |= B. 

I’llshowhowtwooftheargumentsgoasthere’saninterestingconnection 

withthestandardDutchBookargumentsusedintheclassical,two-valued 

case. 

Welet xrangeoverthepossibletruth-values(whichalllieintheinterval 

[0,1]).Clearly,forgiven p,wecanchooseavalueforthestake Sthatmakes 

Gx = (x − p)S 

negative,forallvaluesof xintheinterval [0,1],if,andonlyif, pislessthan 

0orgreaterthan 1.Hence 

0 ≤ Pr(A) ≤ 1. 

1 Thesuggestedpay-offschemeis,ofcourse,onlythemoststraightforwardwaytoimplementtheslogan. 

Onecoulddistorttruthvalues: takeastrictlyincreasingfunction 

f : [0, 1] 2 → [0, 1]with f(0) = 0, f(1) = 1,andtakepay-offstobegivenby (f(v(A))−p)S. 

Analogously,Zadehcouldhavetaken � 

X f(µA(x))dPtodefinedistortedprobabilities.— 

Andthepointisthatsuch“probabilities”aredistortedforwhen fisnottheidentity 

functionitmaybethat P(A) < ceventhough µA(x) > c,forall x ∈ X.


Sofarsogood,buthere’sthecutebit: 

Gx = xG1 + (1 − x)G0, 

so Gxisnegativeforallvaluesof x ∈ [0,1]if,andonlyif, G1and G0areboth 

negative.Fromtheclassicalcase,weknowthatthenecessaryandsufficient 

conditionforthelatteristhat plieoutsidetheinterval [0,1].Itsufficesto 

lookattheclassicalextremestofixwhatholdsgoodforalltruth-valuesin 

theinterval [0,1]. 

Nextweconsiderfourbets: 

1.abeton A,atbettingquotient pwithstake S1; 

2.abeton B,atbettingquotient qwithstake S2; 

3.abeton A ∧ B,atbettingquotient rwithstake S3; 

4.abeton A ∨ B,atbettingquotient swithstake S4. 

Weassumethatforallallowedvaluesof v(A)and v(B), 

v(A ∧ B) + v(A ∨ B) = v(A) + v(B) and v(A ∧ B) ≤ min{v(A),v(B)}. 

Then,where x, y,and zarethetruth-valuesof A, Band A ∧ Brespectively,thepay-offis 

Gx,y = (x − p)S1 + (y − q)S2 + (z − r)S3 + ((x + y − z) − s)S4. 

Thiscanberewrittenas 

Gx,y = zG1,1 + (x − z)G1,0 + (y − z)G0,1 + (1 − x − y + z)G0,0. 

Theco-efficientsareallnon-negativeandcannotallbezero.Thus Gx,yis 

negative,forallallowable x, y,and z,justincase G1,1, G1,0, G0,1,and 

G0,0areallnegative.FromthestandardDutchBookargumentforthetwovalued,classicalcase,weknowthistobepossibleif,andonlyif, 

p+q �= r+s. 

Hence 

Pr(A ∧ B) + Pr(A ∨ B) = Pr(A) + Pr(B). 

3 Theclassicalexpectationthesisforfinitely-many-valued 

Łukasiewiczlogics 

AsaninitialvindicationofZadeh’saccount,wefindthatinthecontext 

ofafinitely-many-valuedŁukasiewiczlogic,allprobabilitiesareclassical 

expectations. Thatis,theprobabilityofamany-valuedpropositionisthe


expectationofitstruth-valueandthatapropositionhasaparticulartruthvalueisexpressibleusingatwo-valuedproposition. 

Sointhissetting,in 

analogywithZadeh’sassignmentofabsoluteprobabilitiestofuzzysets,all 

probabilitiesareexpectationsdefinedoveraclassicaldomain. 

InallŁukasiewiczlogics,conjunctionanddisjuctionareevaluatedbythe 

functions max{0,x + y − 1}and min{1,x + y},respectively. 

EmployingŁukasiewicznegationandoneormoreofŁukasiewiczconjunction,disjunction,andimplication,onecandefineasequenceof 

n + 1 

formulasofasinglevariable, Jn,0(p),Jn,1(p),... ,Jn,n(p),whichhavethis 

property(Rosser&Turquette,1945):inthesemanticframeworkof (n+1)valuedŁukasiewiczlogicitisthecasethatforeveryformula 

A,forall k, 

0 ≤ k ≤ n,andforeveryvaluation v, 

v(Jn,k(A)) = 1,if v(A) = k 

n ; 

v(Jn,k(A)) = 0,if v(A) �= k 

n . 

Inthesemanticframeworkof (n + 1)-valuedŁukasiewiczlogic,forallsentences 

A, 

|= Jn,0(A) ∨Ł Jn,1(A) ∨Ł · · · ∨Ł Jn,n(A) and 

Jn,i(A) ∧Ł Jn,j(A) |=, 0 ≤ i < j ≤ n. (*) 

Fromtheprobabilityaxioms,wehave,forallsentences A,that 

� 

Pr(Jn,i(A)) = 1. 

0≤i≤n 

Thepropositionsoftheform Jn,i(A)aretwo-valued,so, (n + 1)-valued 

Łukasiewiczlogicreducingtoclassicallogiconthevalues 0and 1,thelogic 

ofthesepropositionsisclassical. Thus,whenrestrictedtothesepropositionsandtheirlogicalcompounds,theprobabilityaxiomsgiveusaclassical, 

finitelyadditive,probabilitydistribution. Whatweshownextisthatthis 

classicalprobabilitydistributiondeterminestheprobabilitiesofallpropositionsinthelanguage. 

Theorem2(ClassicalExpectationThesis).Intheframeworkof (n + 1)valuedŁukasiewiczlogic, 

Pr(A) = 1 

n 

� 

0≤i≤n 

iPr(Jn,i(A)). 

Proof.From(*)andthetwo-valuednessofthe Jn,i(A)’swehave 

A =�= (A ∧Ł Jn,0(A)) ∨Ł (A ∧Ł Jn,1(A)) ∨Ł · · · ∨Ł (A ∧Ł Jn,n(A)).


Fromourprobabilityaxiomsitfollowsthatlogicallyequivalentpropositions 

mustreceivethesameprobability,so 

Pr(A) = � 

Pr(A ∧Ł Jn,i(A)). (†) 

0≤i≤n 

Weconsidertwobets,oneon A ∧Ł Jn,k(A)atbettingquotient pand 

stake S1,theotheron Jn,k(A)atbettingquotient qwithstake S2. The 

pay-offsare: 

G = k 

n 

G �= k 

n 

� � 

k 

= − p S1 + ((1 − q)S2) when Ahastruth-value 

n k 

n , 

= −pS1 − qS2 when Ahastruth-valueotherthan k 

n . 

Setting S2 = − k 

nS1givesapay-off,independentofthetruth-valueof � 

A,of − p S1,whichcanbemadenegativebychoiceof S1provided 

� qk 

n 

p �= qk 

n .Ontheotherhand,forarbitrary S1and S2,when p = qk 

n thetwo 

pay-offsare 

� 

k 

G k 

= = (1 − q) 

n n S1 

� 

+ S2 when Ahastruth-value k 

, and 

n 

� 

k 

G k 

�= = −q 

n n S1 

� 

+ S2 when Ahastruth-valueotherthan k 

n . 

Thesecannotbothbenegative.Hence 

Substitutingin(†),weobtain: 

Twocomments 

Pr(A ∧Ł Jn,k(A)) = k 

n Pr(Jn,k(A)). 

Pr(A) = 1 

n 

� 

0≤i≤n 

iPr(Jn,i(A)). 

Firstly,havingbeenobtainedbyanindependentDutchBookargument,the 

ClassicalExpectationThesismayseemtobeanadditionalprinciple.Infact 

itisnot;itisderivablefromouraxiomsforprobability.Toshowthiswehave 

tointroduceapropositionalconstant,introducedintoŁukasiewiczlogicby 

Słupeckiinordertoobtainexpressivecompleteness(Słupecki,1936).


Inthesemanticsof (n + 1)-valuedŁukasiewiczlogic,inwhichallformulasareassignedvaluesintheset 

� 0, 1 2 n−1 

n , n ,..., n ,1� ,thepropositional 

constant thasthisinterpretation: 

underallvaluations v, v(t) = 

n − 1 

n . 

Let t1bethe (n − 2)-fold ∧Ł-conjunctionof twithitself. For 1 < k ≤ n, 

let tkbethe (k − 1)-fold ∨Ł-disjunctionof t1withitself. v(t1) = 1 

n and 

v(tk) = k 

n .Sincewehave 

tk ∧Ł t1 |=, 1 ≤ k < n, and 

|= tn, 

fromourprobabilityaxiomsweobtain: 

Pr(tk) = k Pr(t1), 1 ≤ k ≤ n, and 

Pr(tn) = 1, 

hence 

Pr(tk) = k 

, 1 ≤ k ≤ n. 

n 

Usingthe ti’swecanderivetheClassicalExpectationThesis.(I’llskipthe 

detailshere.) 

Secondly,theDutchBookargumentfortheClassicalExpectationThesis 

goesthroughwithanynotionofconjunctionforwhich v(A&B) = v(A) 

when v(B) = 1and v(A&B) = 0when v(B) = 0. Also,the Jn,i(A)’s 

beingtruth-functional,theClassicalExpectationThesisholdsgoodofevery 

propositioninthesemanticframework,notjustthoseexpressibleusingthe 

Łukasiewiczconnectives. 

4 Theextensiontoinfinitelymanytruth-values(asketch) 

Foranyrationalnumber xintheinterval [0,1],thereisaformula φ(p)ofa 

singlepropositional-variable p,constructedusingŁukasiewicznegationand 

anyoneormoreofŁukasiewiczconjunction,disjunction,orimplication, 

suchthat,underanyvaluationtakingvaluesin [0,1], v (φ(A/p)) = 0if 

v(A) ≤ xand v (φ(A/p)) > 0otherwise(McNaughton,1951). 

EmployingtheGödelnegation, 2 then,wehave, 

2 TheGödelnegationis,tobesure,notusuallytakentobepartofthevocabulary 

oftheŁukasiewiczlogics. Semantically,however,itcanbedefinedintheŁukasiewicz 

fuzzy/many-valuedframeworksastheexternalnegationthatmaps 0to 1andallother 

valuesto 0.


•foreachinterval [0,x]with xrational,aformula J [0,x](A)thattakes 

thevalue 1underanyvaluation vforwhich v(A) ≤ xandotherwise 

takesthevalue 0; 

•foreachhalf-openinterval (x,y]withrationalendpoints xand y, x < 

y,aformula J (x,y](A)thattakesthevalue 1underavaluation vwhen 

v(A) ∈ (x,y]andotherwisetakesthevalue 0. 

Givenastrictlyincreasing,finitesequence x0,x1,... ,xn−1ofrational 

numbersintheopeninterval (0,1),considerthefamilyof n + 1bets: 

� 

2≤i≤n 

•abeton Aatbettingquotient qwithstake S; 

•abeton J [0,x1](A)atbettingquotient p1withstake S1; 

•abeton J (xi−1,xi](A)atbettingquotient piwithstake Si, 1 

•abeton J (xi,1](A)atbettingquotient pnwithstake Sn. 

xi−1 Pr(J (xi−1,xi](A)) ≤ Pr(A) ≤ 

≤ x1 Pr(J [0,x1](A)) + � 

2≤i≤n 

xi Pr(J (xi−1,xi](A)), 

where xn = 1.Sobytakingfinerandfinerpartitionswecanmoreclosely 

approximatetheprobabilityof Afromaboveandbelow.Thismaynotquite 

dotofix Pr(A)exactly. Forthatwemayalsoneedtheprobabilitiesofat 

mostacountableinfinityof(two-valued)statementsoftheform 

v(A) ≤ x 

where xisanirrationalnumber. 3 

Withtheseinhand,wethenfindthat 

Pr(A) = 

� 1 

0 

xdFA(x), 

where FAistheordinary,“classical”distributionfunctiondeterminedby 

theprobabilitiesofthe J [0,x](A)’s, J (x,y](A)’sandhowevermany v(A) ≤ x’s 

with xirrationalwehaveused. 

Byintroducingacountablyinfinitefamilyoflogicalconstants,wecan 

derivethisclassicalrepresentationfromthepreviouslygivenprinciplesof 

probabilitytogetherwiththeprinciple 

3 RecallZadeh’sassumptionregardingthe P-measureabilityof“levelsets”.


•foranyproposition Alogicallyconstrainedtotakeonlythevalues 0 

and 1andforrationalvaluesof xintheinterval [0,1], Pr(tx ∧ A) = 

xPr(A), 

where txtakesthevalue xunderallvaluations v. 

Thereallyneatfeatureofinfinitelymany-valuedŁukasiewiczlogicsis 

thatthisprincipleisderivablefromthebasicprinciples 

• 0 ≤ Pr(A) ≤ 1; 

• Pr(A) = 1when |= A; 

• Pr(A) = 0when A |=; 

• Pr(A ∧Ł B) + Pr(A ∨Ł B) = Pr(A) + Pr(B). 

5 Conditionalprobabilities 

Intheclassicalsetting,abeton Aconditionalon Bisabetthatgoesahead 

if,andonlyif, Bistrueandisthenwonorlostaccordingastowhether 

Aistrueornot. Thepay-offsforsuchaconditionalbetwithstake Sat 

bettingquotient pare: 

•thebettorgains (1 − p)Sif Aand B; 

•thebettorloses pSifnot-Aand B; 

•thebettorneithergainsnorlosesifnot-B. 

Wecansummarisethisbettingschemelikethis: 

v(B)(v(A) − p)S. 

Andso,aswithordinarybets,wenowknowonewaytoextendthe 

schemeforconditionalbetsonclassical,two-valuedpropositionstomanyvaluedpropositions. 

AstraightforwardDutchBookargument,whichagainpiggy-backson 

theproofinthetwo-valuedcase,thentellsusthat 

where 

Pr(A ∧× B) = Pr(A|B) × Pr(B) 

v(A ∧× B) = v(A) × v(B). 

—Allowingforthechangeofsetting,justwhatZadehsaid.


Youcan,ifyouaresominded,generalizetheclassicalschemeusingany 

many-valuedorfuzzyconjunctionthatis“classicalattheextremes”: 

(v(A ∧ B) − v(B)p)S. 

ADutchBookargument—inallessentials,thesameDutchBookargument 

—willthendeliver: 

Pr(A ∧ B) = Pr(A|B) × Pr(B). 

However, Pr(·|B)satisfiestheaxiomsforanabsoluteprobabilitymeasure 

onlywhentheproductconjunction, ∧×isused. 4 

PeterMilne 

DepartmentofPhilosophy,UniversityofStirling 

StirlingFK49LA,UnitedKingdom 

peter.milne@stir.ac.uk 

References 

McNaughton,R.(1951).Atheoremaboutinfinite–valuedsententiallogic.Journal 

ofSymbolicLogic,16,1–13. 

Paris,J.(2001).Anoteonthedutchbookmethod.InG.DeCooman,T.Fine, 

&T.Seidenfeld(Eds.),ISIPTA’01,Proceedingsofthesecondinternationalsymposiumonimpreciseprobabilitiesandtheirapplications,Ithaca,NY,USA(pp. 

301–306).Maastricht:ShakerPublishing.(Aslightlyrevisedversionisavailable 

on-lineathttp://www.maths.manchester.ac.uk/∼jeff/papers/15.ps) 

Rosser,J.,&Turquette,A.(1945).Axiomschemesfor m–valuedpropositional 

calculi.JournalofSymbolicLogic,10,61–82. 

Słupecki,J.(1936).DervolledreiwertigeAussagenkalkül.Comptesrenduesdes 

séancesdelaSociétédesSciencesetLettresdeVarsovie,29,9–11. 

Zadeh,L.A.(1968).Probabilitymeasuresoffuzzyevents.JournalofMathematicalAnalysisandApplications,23,421–427. 

4 Beyondtheclassical,two-valuedcase,productconjunctionrequiresthattherebean 

infinityoftruth-values.

Inferentializing Consequence 

Jaroslav Peregrin ∗ 

Theproofofcorrectnessandcompletenessofalogicalcalculusw.r.t.a 

givensemanticscanbereadastellingusthatthetautologies(or,moregenerally,therelationofconsequence)specifiedinamodel-theoreticwaycan 

beequallywellspecifiedinaproof-theoreticway,bymeansofthecalculus 

(asthetheorems,resp. therelationofinferabilityofthecalculus). Thus 

weknowthatbothfortheclassicalpropositionalcalculusandfortheclassicalpredicatecalculustheoremsandtautologiesrepresenttwosidesofthe 

samecoin.Wealsoknowthattherelationofinferenceasinstitutedbyany 

ofthecommonaxiomsystemsoftheclassicalpropositionalcalculuscoincideswiththerelationofconsequencedefinedintermsofthetruthtables; 

whereasthesituationisalittlebitmorecomplicatedw.r.t.theclassical 

predicatecalculus(thecoincidenceoccursifwerestrictourselvestoclosed 

formulas;otherwise ∀xFxisinferablefrom Fxwithoutbeingitsconsequence).Andofcoursewealsoknowcaseswhereaclassoftautologiesofasemanticsystemdoesnotcoincidewiththeclassoftheoremsofanycalculus. 

(Theparadigmaticcaseisthesecond-orderpredicatecalculuswith 

standardsemantics.) 

Thismaymakeusconsidertheproblemof“inferentializability”.Which 

semanticsystemsare“inferentializable”inthesensethattheirtautologies 

(theirrelationofconsequence,respectively)coincidewiththeclassoftheorems(therelationofinferability,respectively)ofacalculus?Oneanswer 

isready:itisifandonlyifthesetoftautologiesisrecursivelyenumarable. 

Butthisanswerisnotveryinformative,indeedsayingthatthesetisrecursivelyenumerableisonlyreiteratingthatitconicideswiththeclassof 

theoremsofacalculus. Moreover,payingdueattentiontothetermssuch 

as“calculus”and“inference”showsusthatitispossibletorelatethemto 

various“levels”,wherebytheproblemofinferentializabilitybecomesquite 

nontrivial. 

∗ WorkonthispaperwassupportedbythegrantNo.401/07/0904oftheCzechScience 

Foundation.

156 JaroslavPeregrin 

1 Consequence 

Consequence,astheconceptisusuallyunderstood,amountstotruth-preservation,i.e., 

Aisaconsequenceof A1,... ,Aniffthetruthofallof A1,...,An 

bringsaboutthetruthof A,i.e.,iffanytruthvaluationmappingallof 

A1,... ,Anon 1mapsalso Aon 1. 1 Itisobviousthatthe“any”fromthe 

previoussentencecannotmean“anywhatsoever”(ofcoursetheredoesexistafunctionmappingallof 

A1,... ,Anon 1and Aon 0!),itmustmean 

somethinglike“anyadmissible”.Hencetheremustbesomeconceptofadmissibilityinplay:somemappingsofsentencesof 

{0,1}willbeadmissible, 

othersnot. But,ofcourse,thatifwetakethesentencestobesentences 

ofameaningfullanguage,suchadivisionofvaluationsisforthcoming: if 

A1,... ,AnareFidoisadogandEverydogisamammal(hence n = 2), A 

isFidoisamammal,thenthevaluationmappingtheformertwosentences 

on 1andthelatteroneon 0isnotadmissible—itisnotcompatiblewith 

thesemanticsofEnglish. 

Henceweassumethatanysemanticsofanylanguageprovidesforthe 

divisionofthesentencesofthelanguageintotrueandfalse,therebydividingthespaceofthemappingsofthesentenceson 

{0,1}intoadmissible 

andinadmissible.(InfactImaintainamuchstrongerthesis,namelythat 

anysemanticscanbereduced tosuchadivision,butIamnotgoingto 

argueforthisthesishere—Ihavedonesoelsewhere,see(Peregrin,1997).) 

Therebyitalsoestablishestherelationofconsequence,astherelationof 

truth-preservationforalladmissiblevaluations. Ifweusethesentences 

S1,S2,... ofthelanguageinquestiontomarkcolumnsofthefollowing 

tableusingallpossibletruth-valuationsasitsrows,wecanlookatthe 

delimitationoftheadmissiblevaluationsasstrikingoutrowsofthetable. 

1 See(Peregrin,2006). 

S1 S2 S3 S4 · · · 

v1 0 0 0 0 · · · 

v2 1 0 0 0 · · · 

v3 0 1 0 0 · · · 

v4 1 1 0 0 · · · 

v5 0 0 1 0 · · · 

v6 1 0 1 0 · · · 

. 

. . . . . ..

InferentializingConsequence 157 

Amoreexactarticulationofthesenotionsyieldsthefollowingdefinition: 

Definition5.Asemanticsystemisanorderedpair 〈S,V 〉,where Sisaset 

(theelementsofwhicharecalledsentences)and V ⊆ {0,1} S .Theelements 

of {0,1} S arecalledvaluations (of S). (Avaluationwillbesometimes 

identifiedwiththesetofallthoseelementsof Sthataremappedon 1by 

it.)Theelementsof Varecalledadmissiblevaluationsof 〈S,V 〉,theother 

valuations(i.e.theelementsof {0,1} S \ V)arecalledinadmissible. The 

relationofconsequenceinducedbythissystemistherelation |=definedas 

follows 

X |= Aiff v(A) = 1forevery v ∈ Vsuchthat v(B) = 1forevery B ∈ X. 

2 VarietiesofInference 

Nowconsiderthestipulationofaninference, A1,... ,An ⊢ A(forsome 

elements A1,... ,An, Aof S).Suchastipulationcanbeseenasexcluding 

certainvaluations:namelyallthosethatmap A1,...,Anon 1and Aon 0. 

(Thus,forexample,theexclusionsintheabovetablemightbetheresult 

ofstipulating S1 ⊢ S2.)Henceifwecallthepairconstitutedbyafiniteset 

ofelementsof Sandanelementof Saninferon,wecansaythatinferons 

excludevaluationsandaskwhichsetsofvaluationscanbedemarcatedby 

meansofinferons. 

Definition6.Aninferon(over S)isanorderedpair 〈X,A〉where Xisa 

finitesubsetof Sand Aisanelementof S. Aninferonissaidtoexclude 

anelement vof {0,1} S iff v(B) = 1forevery B ∈ Xand v(A) = 0. An 

orderedpair 〈S, ⊢〉suchthat Sisasetand ⊢isafinitesetofinferons 

(i.e.abinaryrelationbetweenfinitesubsetsof Sandelementsof S)willbe 

calledaninferentialstructure.Aninferentialstructureissaidtodetermine 

asemanticsystem 〈S,V 〉iff Visthesetofallandonlyelementsof {0,1} S 

notexcludedbyanyelementof ⊢.Asemanticsystemiscalledaninferential 

systemiffitisdeterminedbyaninferentialstructure. 

Nowanobviousquestioniswhichsemanticsystemsareinferential.But 

beforeweturnourattentiontoit,wewillconsidervariouspossiblegeneralizationsoftheconceptofinference. 

First,letaquasiinferondifferfrom 

aninferoninthatitssecondcomponentisnotasinglestatement,buta 

finitesetofstatements. Aquasiinferonwillexcludeeveryvaluationthat 

mapseveryelementofitsfirstcomponenton 1andeveryelementofits 

secondcomponenton 0. (Ofcoursetheconceptofquasiinferondefined 

inthiswayiscloselyconnectedwiththeconceptofsequentasintroduced


by(Gentzen,1934)and(Gentzen,1936). 2 ) Second,letasemiinferondifferfromaninferoninthatitsfirstcomponentisnotnecessarilyfinite. 

A 

semiquasiinferonwillbeaquasiinferonwithbothitsfirstanditssecond 

componentnotnecessarilyfinite. Third,letaprotoinferential structure 

beaninferentialstructurewithitssecondcomponentnotnecessarilyfinite 

(andthinkoftheconceptsofprotosemiinferential,protoquasiinferentialand 

protosemiquasiinferentialstructureanalogously). 

Inthefollowingdefinition,weabbreviatetheprefixes,whichhavealready 

becomesomewhatmonstrous: 

Definition7.Anelementof Pow(S) × Pow(S)iscalledanSQI-onover 

S.ItiscalledaQI-onifitisanelementof FPow(S) × FPow(S)(where 

FPow(S)isthesetofallfinitesubsetsof S),itiscalledanSI-onifitis 

anelementof Pow(S) × SanditiscalledanI-onifitisanelementof 

FPow(S) ×S. 3 Theorderedpair 〈S, ⊢〉where ⊢isasetofSQI-ons(QI-ons, 

SI-ons,I-ons)willbecalledaPSQI-structure(PQI-structure,PSI-structure, 

PI-structure). ItiscalledanSQI-structure(QI-structure,SI-structure,Istructure)iff 

⊢isfinite.AnSQI-on 〈X,Y 〉issaidtoexcludeanelement v 

of {0,1} S iff v(B) = 1forevery B ∈ Xand v(A) = 0forevery A ∈ Y.A 

(P)(S)(Q)I-structure 〈S, ⊢〉issaidtodetermineasemanticsystem 〈S,V 〉iff 

Visthesetofallandonlyelementsof {0,1} S notexcludedbyanyelement 

of ⊢.Asemanticsystemiscalleda(P)(S)(Q)I-systemiffitisdetermined 

bya(P)(S)(Q)I-structure. 

Summarizingtheconceptsintroducedinthisdefinition,wehavethefollowingtable: 

〈S, ⊢〉isa... iff ⊢isa... ⊢thusbeingasubsetof 

I-structure afinitesetofI-ons FPow(S) × S 

QI-structure afinitesetofQI-ons FPow(S) × FPow(S) 

SI-structure afinitesetofSI-ons Seq(S) × S 

PI-structure asetofI-ons FPow(S) × S 

SQI-structure afinitesetofSQI-ons Pow(S) × Pow(S) 

PQI-structure asetofQI-ons FPow(S) × FPow(S) 

PSI-structure asetofSI-ons Seq(S) × S 

PSQI-structure asetofSQI-ons Pow(S) × Pow(S) 

2 Foranexpositionofsequentcalculusanditsrelationshiptothemorestraightforwardly 

inferentialapproachasembodiedinnaturaldeductionsee,e.g.,(Negri&Plato,2001). 

3 Throughoutthewholepaperweidentifysingletonswiththeirrespectivesingleelements; 

henceweoftenwritesimply vinsteadof {v}.


Ouraimnowistofindcriteriaofthevariouslevelsofinferentializability. 

Beforewestateandprovetheoremscrucialinthisrespect,weintroduce 

somemoredefinitions. 

3 CriteriaofInferentializability 

Definition8.Let Ubeasetofvaluationsofasemanticsystem 〈S,V 〉 

(i.e.asubsetof {0,1} S ). T(U)(thesetof U-tautologies)willbethesetof 

allthoseelementsof Swhicharemappedon 1byallelementsof U;and 

analogously C(U)(thesetof U-contradictions)willbethesetofallthose 

elementsof Swhicharemappedon 0byallelementsof U.Let Xand Ybe 

subsetsof S.Theclustergeneratedby Xand Y, Cl[X,Y ],willbetheset 

ofallthevaluationsthatmapallelementsof Xon 1andallelementsof Y 

on 0.Generally, Uisaclusteriffitcontains(andhenceisidenticalwith) 

Cl[T(U),C(U)].Acluster Uiscalledfinitaryiffboth T(U)and C(U)are 

finite,itiscalledinferentialiff C(U)isasingleton. 

Nowitisclearthatasemanticsystem 〈S,V 〉isaPSQI-systemiff {0,1} S \ 

V isaunionofclusters.(HenceeverysemanticsystemisaPSQI-system, 

foreverysinglevaluationconstitutesacluster.)Thereasonisthatasystem 

isaPSQI-systemifitsinadmissiblevaluationsaredeterminedbyasetof 

SQI-onsandwhatanSQI-onexcludesisaclusterofvaluations.Ifweuse 

specifickindsofSQI-ons,suchasSI-ons,wewillhaveaspecifickindof 

clusters,likeinferentialclusters;andifweallowforonlyafinitenumberof 

SQI-ons,wewillhavetocountwithonlyfiniteunions.Thisyieldsusthe 

factssummarizedinthefollowingtable: 

〈S,V 〉isa... iff {0,1} S \ Visaunionof... 

PSQI-system clusters 

PSI-system inferentialclusters 

PQI-system finitaryclusters 

SQI-system afinitenumberofclusters 

PI-system finitaryinferentialclusters 

SI-system afinitenumberofinferentialclusters 

QI-system afinitenumberoffinitaryclusters 

I-system afinitenumberoffinitaryinferentialclusters 

Theorem3.Asemanticsystem 〈S,V 〉isaPSI-systemiff V contains 

every v ∈ {0,1} S suchthatforevery A ∈ C(v)thereisa v ′ ∈ V suchthat 

T(v) ⊆ T(v ′ )and A ∈ C(v ′ ).


Proof.Asemanticsystem 〈S,V 〉isaPSI-systemsystemiff {0,1} S \ V is 

aunionofinferentialclusters.ThisistosaythatitisaPSI-systemifffor 

every v ∈ {0,1} S \ V thereisaset X ⊆ T(v)andasentence A ∈ C(v) 

suchthatnovaluation v ′ suchthat X ⊆ T(v ′ )and A ∈ C(v ′ )isadmissible. 

Inotherwords, 〈S,V 〉isaPSI-systemiffforevery v �∈ V thereisaset 

X ⊆ T(v)andasentence A ∈ C(v)suchthat V doesnotcontainany v ′ 

suchthat X ⊆ T(v ′ )and A ∈ C(v ′ ). Bycontraposition, 〈S,V 〉isaPSIsystemiffthefollowingholds:givenavaluation 

v,ifforeveryset X ⊆ T(v) 

andeverysentence A ∈ C(v)thereisav ′ ∈ V suchthat X ⊆ T(v ′ )and 

A ∈ C(v ′ ),then v ∈ V.Thisconditioncanobviouslybesimplifiedto:given 

avaluation v,ifforeverysentence A ∈ C(v)thereisav ′ ∈ V suchthat 

T(v) ⊆ T(v ′ )and A ∈ C(v ′ ),then v ∈ V. 

Theorem4.Asemanticsystem 〈S,V 〉isaPQI-systemiff V contains 

every vsuchthatforeveryfinite X ⊆ T(v)andfinite Y ⊆ C(v)thereisa 

v ′ ∈ V suchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ). 

Proof.Asemanticsystem 〈S,V 〉isaPQI-systemsystemiff {0,1} S \ Visa 

unionoffiniteclusters.ThisistosaythatitisaPQI-systemiffforevery 

v ∈ {0,1} S \ V therearefinitesets X ⊆ T(v)and Y ⊆ C(v)suchthatno 

valuation v ′ suchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ )isadmissible. Inother 

words, 〈S,V 〉isaPQI-systemiffforevery v �∈ V therearesets X ⊆ T(v) 

and Y ⊆ C(v)suchthat Vdoesnotcontainany v ′ suchthat X ⊆ T(v ′ )and 

Y ⊆ C(v ′ ).Bycontraposition, 〈S,V 〉isaPQI-systemiffthefollowingholds: 

givenavaluation v,ifforeverysets X ⊆ T(v)and Y ⊆ C(v)thereisa 

v ′ ∈ Vsuchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ),then v ∈ V.Thisconditioncan 

obviouslybesimplifiedto:givenavaluation v,ifforeveryfinite X ⊆ T(v) 

andfinite Y ⊆ C(v)thereisav ′ ∈ Vsuchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ), 

then v ∈ V. 

Weleaveouttheproofofthefollowingtheorem,asitisstraightforwardly 

analogoustotheproofsoftheprevioustwo. 

Theorem5.Asemanticsystem 〈S,V 〉isaPI-systemiff Vcontainsevery 

vsuchthatforeveryfinite X ⊆ T(v)andevery A ∈ C(v)thereisa v ′ ∈ V 

suchthat X ⊆ T(v ′ )and A ∈ C(v ′ ). 

Hencewehavenecessaryandsufficientconditionsforasemanticsystem 

beingaPSI-,aPQI-,oraPI-system.Unfortunately,wedonothavesuch 

conditionsforitsbeinganSQI-,anSI-,aQI-,oranI-system.However,we 

areabletoformulateatleastausefulnecessaryconditionforitsbeingan 

SQI-system.


Theorem6.Asemanticsystem 〈S,V 〉isanSQI-systemonlyif Vcontains 

no vsuchthatforeveryfinite X ⊆ T(v)andfinite Y ⊆ C(v)thereisa 

v ′ �∈ V suchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ). 

Proof.Asemanticsystem 〈S,V 〉isaPQI-systemiff {0,1} S \ V isafinite 

unionofclusters.HenceifitisaPQI-system,theremustexistafiniteset 

Iandtwocollections 〈X i 〉i∈I, 〈Y i 〉i∈Iofsubsetsof Ssothat 

{0,1} S \ V = � 

i∈I 

Cl[X i ,Y i ]. 

Thisisthecaseiff Vequalsthecomplementof � 

V = � 

i∈I 

Cl[X i ,Y i ]. 

Butas Cl[X i ,Y i ] = {v : X i ⊆ T(v)and Y i ⊆ C(v)}, 

Cl[X i ,Y i ] = {v : X i �⊆ T(v)or Y i �⊆ C(v)} = 

Cl[X 

i∈I 

i ,Y i ],henceiff 

= {v : X i ∩ C(v) �= ∅or Y i ∩ T(v) �= ∅} = 

= {v : X i ∩ C(v) �= ∅} ∪ {v : Y i ∩ T(v) �= ∅} = 

= � 

x∈X i 

= � 

x∈X i 

{v : x ∈ C(v)} ∪ � 

{v : y ∈ T(v)} = 

Cl[∅, {x}] ∪ � 

y∈Y i 

y∈Y i 

Cl[{y}, ∅]. 

NowusingthegeneralizeddeMorgan’slawsayingthat 

� � � � 

= 

j∈I j∈J 

f∈F j∈f + 

Z j 

i 

f∈F j∈I 

Z j 

f(j) 

where F = IJ ,wecanseethat 

V = � � 

Cl[f(j), ∅] ∩ � 

Cl[∅,f(j)] 

j∈f − 

where Fisthesetofallfunctionsmappingevery i ∈ Ionanelementof f(i) 

of Xi ∪Yi,and f + ,and f − ,respectively,arethesetsofallthoseelementsof 

Ithataremappedby fonelementsof Xi,and Yi,respectively.Itfurther 

followsthat 

V = � 

Cl[X f ,Y f ] 

f∈F 

where X f = {f(j) : j ∈ f + }and Y f = {f(j) : j ∈ f − }.Asboth f + and f − 

arefinite,thismeansthat Visaunionoffiniteclusters.Itfollowsthatfor


every v ∈ V therearefinitesets X ⊆ T(v)and Y ⊆ C(v)suchthatevery 

valuation v ′ suchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ )isadmissible. Inother 

words,forevery v ∈ Vtherearesets X ⊆ T(v)and Y ⊆ C(v)suchthat V 

containsevery v ′ suchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ).Bycontraposition: 

givenavaluation v,ifforeveryset X ⊆ T(v)and Y ⊆ C(v)thereisa 

v ′ �∈ Vsuchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ),then v �∈ V.Thisconditioncan 

obviouslybesimplifiedto:givenavaluation v,ifforeveryfinite X ⊆ T(v) 

andfinite Y ⊆C(v)thereisav ′ �∈ Vsuchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ), 

then v �∈ V. 

4 AHierarchyofSemanticSystems 

Letusintroducesomemoredefinitions. 

Definition9.Asemanticsystem 〈S,V 〉iscalled 

•saturatediff Vcontainsevery vsuchthatforevery A ∈ C(v)thereis 

a v ′ ∈ Vsuchthat T(v) ⊆ T(v ′ )and A ∈ C(v ′ ); 

•compactiff Vcontainsevery vsuchthatforeveryfinite X ⊆ T(v)and 

finite Y ⊆ C(v)thereisav ′ ∈ Vsuchthat X ⊆ T(v ′ )and Y ⊆ C(v ′ ); 

•co-compactiffVcontainsno vsuchthatforeveryfinite X ⊆ T(v)and 

finite Y ⊆ C(v)thereisav ′ �∈ Vsuchthat X ⊆ T(v)and Y ⊆ C(v ′ ). 

•compactlysaturatediff V containsevery vsuchthatforeveryfinite 

X ⊆ T(v)andevery A ∈ C(v),thereisav ′ ∈ Vsuchthat X ⊆ T(v ′ ) 

and A ∈ C(v ′ ). 

Giventhese,wecanrephrasethetheoremswehaveprovedinthefollowingway: 

Theorem7.Asemanticsystem 〈S,V 〉is 

•alwaysaPSQI-system; 

•aPSI-systemiffitissaturated; 

•aPQI-systemiffitiscompact; 

•anSQI-systemonlyifitisco-compact; 

•aPI-systemiffitiscompactlysaturated; 

Moreover,easycorollariesofthetheoremsarethefollowingnecessary 

conditionsforasystembeinganSI-,aQI-andanI-system: 

Corollary2.Asemanticsystem 〈S,V 〉is 

•anSI-systemonlyifitissaturatedandco-compact;


•aQI-systemonlyifitiscompactandco-compact; 

•anI-systemonlyifitiscompactlysaturatedandco-compact. 

Thekindsofsemanticsystemswehaveintroducedcanbearrangedinto 

thefollowingdiagram,wherethearrowsindicatecontainmentinthesense 

thatanarrowleadsfromaconcepttoadifferentoneiftheextensionofthe 

formerincludesthatofthelatter. 

SPQI-system =semanticsystem[Σ1] 

PQI-system[Σ2] SPI-system[Σ3] SQI-system[Σ4] 

PI-system[Σ5] QI-system[Σ6] SI-system[Σ7] 

I-system[Σ8] 

Diagram1 

Whatwearegoingtoshownowisthatalltheinclusionsareproper. 

Thesymbolsinbracketsfollowingeachkindtermisthenameofasemantic 

systemwhichwillwitnesstheproperness.Thesystemsarethefollowing(S 

issupposedtobeaninfiniteset): 

• Σ1 = 〈S, {v ∈ Pow(S) : T(v)isfinite}〉; 

• Σ2 = 〈S, {∅}〉; 

• Σ3 = 〈S, {v ∈ Pow(S) : C(v)isfinite}〉; 

• Σ4 = 〈S,Pow(S) \ {S}〉; 

• Σ5 = 〈S, {S}〉; 

• Σ6 = 〈{A,B}, {{A}, {B}}〉; 

• Σ7 = 〈S, {v ∈ Pow(S) : C(v) = A}〉forafixed A ∈ S; 

• Σ8 = 〈{A,B}, {{A,B}, {B}}〉. 

ToshowthattheydofitintotheveryslotsofDiagram1wherewehave 

putthem,letusfirstgiveonemoredefinition: 

Definition10.Avaluationiscalledfullifitmapseverysentenceon 1. 

(Inotherwords,thefullvaluationis S.) Avaluationiscalledemptyifit 

mapseverysentenceon 0.(Inotherwords,theemptyvaluationis ∅.)


Σ1isnotsaturated,for Vdoesnotcontainthefullvaluation, f,though 

forevery A ∈ C(f)thereisav∈ V suchthat T(f) ⊆ T(v)and A ∈ C(v). 

(Asthereisno A ∈ C(v),thisholdstrivially.Itfollowsthatnosystemnot 

admittingthefullvaluationissaturated.) HenceitisnotaPSI-system. 

Itisnotcompact,because V doesnotcontainthefullvaluation,butfor 

everyfinitesubset Xof T(f)itcontainsav ′ suchthat X ⊆ T(v ′ )(whereas 

Y ⊆ C(v ′ )foreveryfinitesubset Yof C(f)holdstrivially);henceitisnot 

aPQI-system. Moreover,itisnotco-compact,for V containstheempty 

valuation,whereasas Vcannotcontainanyvaluationmappingonlyafinite 

numberofsentenceson 0,thereis,foreveryfinitesubset Yof S,av ′ �∈ V 

suchthat Y = C(v ′ ).HenceitisnotanSQI-system. 

Σ2isaPQI-system,foritisdeterminedbytheinfinitesetofQI-ons 

{〈{A}, ∅〉 : A ∈ S}.However,itisnotsaturated,for Vdoesnotcontainthe 

fullvaluation,henceitisnotaP(S)I-system.Alsoitisnotco-compact,for 

Vcontainstheemptyvaluation,whereasforeveryfinitesubset Yof Sthere 

isav ′ �∈ V suchthat X ⊆ C(v ′ )(whereasthat Y ⊆ T(v ′ )foreveryfinite 

subset Yof T(f)holdstrivially);henceitisnota(S)QI-system. 

Σ3isaPSI-system,foritisdeterminedbytheinfinitesetofSI-ons 

{〈X,A〉 : X ⊆ Sand Xisinfinite}. However,itisnotcompact,because 

V doesnotcontaintheemptyvaluation,butforeveryfinitesubset Yof 

Sitcontainsav ′ suchthat Y = C(v);henceitisnotaP(Q)I-system. 

Moreover,itisnotco-compact,for V containsthefullvaluation,whereas 

foreveryfinitesubset Xof Sthereisav ′ �∈ V suchthat X = T(v ′ ),hence 

itisnotanS(Q)I-system. 

Σ4isanSQI-system,foritisdeterminedbytheSQI-on 〈S, ∅〉.However, 

itisnotsaturated,for V doesnotcontainthefullvaluation,henceitis 

nota(P)SI-system. Itisnotcompact,because V doesnotcontainthe 

fullvaluation,butforeveryfinitesubset Xof Sitcontainsav ′ suchthat 

X = T(v ′ );henceitisnota(P)QI-system. 

Σ5isaPI-systemforitisdeterminedbytheinfinitesetofI-ons {〈∅, {A}〉 : 

A ∈ S}.Butitisnotco-compact,for Vcontainsthefullvaluation,whereas 

foreveryfinitesubset Xof Sthereisav ′ �∈ Vsuchthat X = T(v ′ ),hence 

itisnota(S)(Q)I-system. 

Σ6 is a QI-system for it is determined by the finite set of QI-ons 

{〈∅, {A,B}〉, 〈{A,B}, ∅〉}. Butitisnotsaturated,forthesupervaluation 

of Vistheemptyvaluation,henceitisnota(P)(S)I-system. 

Σ7isanSI-systemforitisdeterminedbythesingleSI-on 〈S \ {A},A〉. 

Butitisnotcompact,for Vcontains,foreveryfinitesubset Yof S \ {A}, 

a v ′ suchthat T(v ′ ) = Yand C(v ′ ) = A.Henceitisnota(P)(Q)I-system. 

Σ8isanI-system,foritisdeterminedbytheI-on 〈∅, {B}〉.


5 Consequencerevisited 

Ifwhatweareinterestedinistherelationofconsequence,thenourclassificatoryhierarchybecomesexcessivelyfine-grained.Inparticular,weare 

goingtoshowthatforevery(P)(S)QI-systemthereexistsa(P)(S)I-system 

withthesamerelationofconsequence. Todothisletusdefineaconcept 

introducedby(Hardegree,2006): 

Definition11.Let Ubeasetofvaluationsoftheclass Sofsentences. 

Thesupervaluationof Uisthevaluationsuchthat T(v) = T(U). 

ThenextlemmashowsthatourTheorem3isequivalenttooneofHardegree’sresults: 

Lemma1.Asemanticsystem 〈S,V 〉isa(P)(S)QI-systemiff V contains 

supervaluationsofallitssubsets. 

Proof.Thisfollowsdirectlyfromthefactthat 〈S,V 〉isa(P)(S)QI-system 

iffitissaturated,foritcanbeeasilyseenthatitissaturatediff Vcontains 

supervaluationsofallitssubsets. 

Lemma2.Extendingadmissiblevaluationsofasemanticsystembysupervaluationsdoesnotchangetherelationofconsequence. 

Proof.Let 〈S,V 〉beasemanticsystemand |=therelationofconsequence 

inducedbyit.Let vbeasupervaluationofasubsetof Vandlet |= ∗ bethe 

relationofconsequenceinducedby 〈S,V ∪ {v}〉.Supposethetworelations 

donotcoincide;thenthereisasubset Xof Sandanelement Aof Sso 

that X |= A,butnot X |= ∗ A. Thismeansthatitmustbethecasethat 

v(B) = 1forevery B ∈ Xand v(A) = 0,butthatevery v ′ ∈ V suchthat 

v ′ (B) = 1forevery B ∈ Xisboundtobesuchthat v ′ (A) = 1.Butas v ′ is 

thesupervaluationofan U ⊆ V,elementsof Umapallelementsof Xon 1, 

whereasatleastoneofthemmaps Aon 0;whichisacontradiction. 

ThisgivesusthefollowingreducedversionofDiagram1: 

PS(Q)I-system =semanticsystem 

P(Q)I-system S(Q)I-system 

(Q)I-system 

Diagram2 

Hencefromtheviewpointofconsequence,wehavefourtypesofsemantic 

systems:


•SystemsthatareneitherP(Q)I,norS(Q)I.Thesearesystemsofthe 

kindof Σ1and Σ3. 

•P(Q)I-systemsthatarenot(Q)I-systems.Examplesare Σ2and Σ5. 

•S(Q)I-systemsthatarenot(Q)I-systems.Examplesare Σ4and Σ7. 

•(Q)I-systems.Systemsofthekindof Σ6and Σ8. 

Consequenceasinducedbythetruthtablesofclassicalpropositionallogic 

orbythemodeltheoryoftheclassicalfirst-orderpredicatelogic,ofcourse, 

fallintothelastcategory.Indeedanylogicthathasastronglysoundand 

completeaxiomatizationmusttriviallybelonghere. Butevenamongthe 

semanticsystemsstudiedbylogicianstherearesomethatfalloutsidethis 

range((Tarski,1936)madethisintoadeeppoint—consequence,according 

tohim,cannotbeingeneralcapturedintermsofinferentialrules). 

FromDiagram2wecanseethattherearetwowaystogobeyondthe 

boundariesofI-systems:wemayeitheralleviatetherequirementoffinitenessofantecedentsofinferences,oralleviatetherequirementoffiniteness 

ofthewholerelationofinference. The ω-rule,whichisoftendiscussedin 

connectionwiththeformalizationofarithmetic,isanexampleoftheformer 

way;theaxiomschemeofinduction,thatcomprisesaninfinityofconcrete 

axioms,istheexampleofthelatter. 

Foramorespecificexample,considerthelanguageofPeanoarithmetic 

withthesingleadmissiblevaluationdeterminedbytheintendedinterpretationwithinthestandardmodel(letmecallthissystemtruearithmetic,TA).Asitturnsout,thissystemisaPQI-system.Indeed,itcanbedeterminedbythePQI-structurewhoserelationofinferenceconsistsoftheI-ons 

oftheform 〈∅,A〉foreverytruesentence AplustheQI-onsoftheform 

〈{B}, ∅〉foreveryfalsesentence B.(WeknowthatitisnotanI-system,as 

weknowthatthetruthsofTAarenotrecursivelyenumerable.) Callthe 

singleadmissiblevaluationofthesystem t. 

Ifweextendthe(single-element)setofadmissiblevaluationsofTAby 

thefullvaluation,itbecomessaturated(indeedthesupervaluationofevery 

subsetofthesetofitsadmissibletruthvaluationswillbeadmissible:the 

supervaluationoftheemptysetaswellasthesingletonofthefullvaluation 

isthefullvaluation,whereasthesupervaluationofthetworemainingsetsis 

thevaluation t).HencethissystemisaPI-system(indeed,itisdetermined 

bythePI-structuretherelationofinferenceofwhichconsistsoftheI-onsof 

theform 〈∅,A〉foreverysentence Atrueaccordingto tplustheI-onsofthe 

form 〈{B},C〉foreverysentence Bfalseaccordingto t,andeverysentence 

C)buthasthesamerelationofconsequenceastheprevioussystem.


6 Furthersteps 

Ihopetohaveshownhowwecansetupausefulframeworkforasystematic 

confrontationofprooftheoryandsemantics,especiallyofinferenceand 

consequence;andthatIhavealsoindicatedthatthisframeworkletsusprove 

somenontrivialandinterestingresults. However,itshouldbeaddedthat 

tobringresultsimmediatelyconcerningtheusualsystemsofformallogic, 

ourclassificatoryhierarchywillhavetobemadestillmorefine-grained. 

Thepointisthatwhileweonlydistinguishedbetweensystemsthatare 

determinedbystructureswithafinitenumberof(S)(Q)I-ons(i.e.(S)(Q)Isystems)andthosewherethefinitenessrequirementisalleviated(theP(S) 

(Q)I-systems),wewouldneedtoconsidersystemsinbetweenthesetwo 

extremes. Theusualsystemsofformallogiccanbeconsideredasgeneralizingoverinferential(asopposedtopseudoinferential)structuresintwo 

steps. First,theyallowforaninfinitenumberof(S)(Q)I-ons,whichare, 

however,instancesofafinitenumberofschemata.(Thisis,ofcourse,possibleonlywhenwe,unlikeinthepresentpaper,takeintoaccountsomestructuringofthesetofsentencesandconsequentlyofthesentencesthemselves—ifweconsiderthesentencesasgeneratedfromavocabularybya 

setofrules.)ThiscanbeaccountedforintermsofparametricSQI-ons,or 

p(S)(Q)I-ons.p(S)(Q)I-systems,then,fallinbetween(S)(Q)I-systemsand 

P(S)(Q)I-system.ThusforexamplethesemanticsystemofPAisap(Q)Isystem,fortheinfinityofitsaxiomsistheunionofinstancesofafinite 

numberofaxiomschemas. ThesemanticsystemofTAisapSI-system, 

forweknowthatwecanhaveitssoundandcompleteaxiomatizationifwe 

extendtheaxiomaticsystemwiththeomega-rule,whichis,inourterminology,apSI-on.Secondtheyallowforinfinitesetsof(S)(Q)I-onsthatare 

generatedbyafinitenumberofmetainferentialrulesfromsetsofinstances 

offinitenumberofschemata. 

JaroslavPeregrin 



peregrin@ff.cuni.cz 

http://jarda.peregrin.cz 

References 

Gentzen,G.(1934).Untersuchungenüberdaslogischeschliessen1.MathematischeZeitschrift,39,176-210. 

Gentzen,G.(1936).Untersuchungenüberdaslogischeschliessen2.MathematischeZeitschrift,41,405-431.


Hardegree,D.M.(2006).Completenessandsuper-valuations.JournalofPhilosophicalLogic,34,81-95. 

Negri,S.,&Plato,J.von.(2001).Structuralprooftheory.Cambridge:Cambridge 


Peregrin,J. (1997). Languageanditsmodels. NordicJournalofPhilosophical 

Logic,2,1-23. 

Peregrin,J.(2006).Meaningasaninferentialrole.Erkenntnis,64,1-36. 

Tarski,A. (1936). Überdenbegriffderlogischenfolgerung. ActesduCongrés 

InternationaldePhilosophiqueScientifique,7,1-11.

Meaning and Compatibility: 

Brandom and Carnap on Propositions 

Martin Pleitz 

1 Brandom’sandCarnap’ssemantics 

RobertrandominhisLockeLectures 1 hasdevelopedanewformalsemanticsthatisbasedentirelyontheoneprimitivenotionoftheincoherenceof 

setsofsentences(Brandom,2008,pp.117–175). Alanguage,i.e.asetof 

atomicsentences,isformallyinterpretedbyanincoherencepartitionofits 

power-set.Theincoherencepartitionmustsatisfyonlyoneconditionthat 

Brandomcalls“persistence”,i.e.,allsetscontaininganincoherentsubset 

mustthemselvesbeincoherent. Theincompatibilityoftwosentencesis 

thendefinedastheincoherenceoftheirunion. Twosentencesarecalled 

incompatibility-equivalent(or I-equivalentforshort)ifandonlyiftheyare 

incompatiblewiththesamesetsofsentences.Inthatcasethetwosentences 

aresaidtobesynonymous.Therefore,thepropositionexpressedbyasentencecanberepresentedbytheincompatibility-setofthesentence,i.e.,by 

thesetofsetsofsentencesthatareincompatiblewithit(Brandom,2008, 

pp.123ff.).Thisformalframeworkisasemanticsbecausethebasicnotion 

ofincoherencesufficestogiveanaccountofthemeaningofsentences. 

Morethansixtyyearsearlier,RudolfCarnaphadalreadyproposedto 

representpropositionsassetsofsetsofsentences,althoughinadifferentway. 

LikeBrandom,Carnapbuildsalanguagefromasetofatomicsentences. 

Tothislanguageareaddedtheconnectivesofpropositionallogic. 2 Carnap 

thendefinesastate-descriptionasasetsuchthatforeachatomicsentence, 

eitherthesentenceoritsnegationisanelementofit(Carnap,1947/1956, 

1 BrandomdescribesincompatibilitysemanticsatlengthinthefifthofhisJohnLocke 

Lectures,whichheheld2006inOxfordand2007inPragueandthathavebeenpublished 

as(Brandom,2008).Thebasicideaissketchedfirstin(Brandom,1985),andmentioned 

repeatedlyin(Brandom,1994). 

2 Theirmeaningisgivenbytheusualtruth-tables.

170 MartinPleitz 

p.9). Twosentencesaresaidtobe L-equivalentifandonlyiftheyare 

membersofthesamestate-descriptions.Inthatcasethetwosentencesare 

saidtobesynonymous.Thus,thepropositionexpressedbyasentencecan 

berepresentedbyitsrange,i.e.,bythesetofstate-descriptionsthatitis 

anelementof(Carnap,1947/1956,pp.7–32,p.181). 3 Carnapintendshis 

state-descriptionsto“representLeibniz’possibleworldsorWittgenstein’s 

possiblestatesofaffairs”(Carnap,1947/1956,p.9).Butstate-description 

semanticsdifferscruciallyfromthemainstreamofpossibleworldssemantics. 

Ingeneral,possibleworldsareseenasbasicobjects. 4 State-descriptions,by 

contrast,areset-theoreticalconstructionsfromlinguisticentities.Therefore 

state-descriptionsemanticsisaprecursoroftheoriesthatreducepossible 

worldstomaximallycompatiblesetsofsentences. 5 

Thefactthatstate-descriptionsarereductionistpossibleworldsbrings 

outafirstsimilaritybetweenBrandom’sandCarnap’sformalsemantics: 

Botharesemanticswithouttheworld. Meaningisnotmodeledasarelationbetweenlanguageandessentiallynon-linguisticobjects.Rather,bothincompatibilitysemanticsandstate-descriptionsemanticsrepresentmeaningbyset-theoreticalconstructionsbuiltfromlinguisticobjectsalone. 

6 It 

istheaimofmytalktoshowthatthesimilaritiesbetweenBrandom’sand 

Carnap’ssemanticsaremorethansuperficial.Ihopethiscomparisonwill 

shedsomelightonboththeories. 

2 Modifyingstate-descriptionsemantics 

Beforestartingthecomparison,afinaladjustmentmustbemadetoCarnap’stheory.Carnapcannotachievehisownaims,becausehistheorydoes 

3 Strictlyspeaking,thisistrueonlyforatomicsentences. Fornon-atomicsentences,we 

havetolaydowntherecursivedefinitionofwhatitmeansforasentencetoholdin(i.e.to 

betrueat)astate-description. 

4 Onthispoint,radicalmodalrealistslikeDavidLewis,whoholdsthatpossibleworlds 

areconcreteobjects(Lewis,1986,pp.81ff.),andmoderatemodalrealistslikeSaulKripke, 

whotakespossibleworldstobestipulated,havetoagree(Kripke,1972/1980,pp.15ff.). 

5 ReductionismisendorsedbyRobertAdams,AlvinPlantinga,RobertStalnaker,AndrewRoper,PhillipBrickerandMaxwellCresswell((Adams,1979),(Plantinga,1974),(Stalnaker,1979),(Roper,1982),(Bricker,1987),(Cresswell,2006)).Thecleareststatementofthetheorywasperhapsgivenbyoneofitsopponents: 

DavidLewisdescribes 

reductionism—thathecalls“linguisticersatzism”—fromacriticalperspective,butin 

greatdetail(Lewis,1986,pp.142–165). 

6 Asimilaritythatismorethansuperficialconcernsmodality.BrandomandCarnapshare 

aglobalunderstandingofmodality,wherethereisnoequivalentofaKripkeanrelation 

ofaccessibility,andnecessityconformstotheaxiomsofS5(Brandom,2008,pp.129ff., 

141ff.),(Carnap,1947/1956,pp.10,174f.,186).Forademonstrationthatincompatibility 

semanticscanbemodifiedtoaccommodateotherconceptsofnecessity(asitturnsout, 

ofB),see(Peregrin,2007),aswellas(Göcke,Pleitz,&vonWulfen,2008)and(Pleitz& 

vonWulfen,2008).

MeaningandCompatibility:BrandomandCarnaponPropositions 171 

notruleoutstate-descriptionsthatcontainsentencesthatintuitivelyare 

incompatible. ThiscanbeshowntoanexampletakenfromMeaningand 

Necessity. 

Considertheatomicsentences“Scottishuman”and“Scottisarational 

animal”.AccordingtoCarnap’sdefinition,therewillbeastate-description 

containingthefirstandthenegationofthesecond.Thereforethetwosentenceswillnotbe 

L-equivalentandhencenotsynonymous.Buttheyshould! 

CarnapstipulatestheEnglishwords“human”and“rationalanimal”to 

meanthesame(Carnap,1947/1956,p.4f.). And,onlyafewpageslater, 

heexplicitlystatesthatthepredicates“human”and“rationalanimal”are 

coextensionalineverystate-description(Carnap,1947/1956,p.15).Thisis 

notasmalltechnicalpoint. AsDavidLewishaspointedout,anytheory 

thatreducespossibleworldstosetsofsentencesmustrelyonaprimitive 

modalnotion,becausethesentencesthatrepresentpossibleworldsmust 

becoherentinasensethatexceedsmerelogicalconsistency. Lewisgives 

theexampleof“thepositiveandnegativechargeofpointparticles”(Lewis, 

1986,p.154). 

Asapreliminarysolution,letusmodifystate-descriptionsemanticsby 

rulingoutthosestate-descriptionsthatcontainatomicsentencesthatintuitivelyareincompatible. 

Thismodificationwillturnouttoplayacrucial 

roleinthecomparisonofstate-descriptionsemanticstoincompatibilitysemantics(Sections5–7).Thereforeitisimportanttonotethatthejustificationofthismodificationofstate-descriptionsemanticsisindependentof 

theenterpriseofcomparingittoincompatibilitysemantics:Thecriticism 

ofCarnap’stextisimmanent,andLewis’sargumentisperfectlygeneral. 

3 Mutualsimulation 

Withthismodificationinplace(Section2),thecomparisonofBrandom’s 

andCarnap’ssemanticscanstart. Wehaveseenthat,inincompatibilitysemantics,incompatibility-setsexplicate 

7 propositions,and I-equivalence 

explicatessynonymy.Instate-descriptionsemantics,rangesexplicatepropositions,and 

L-equivalenceexplicatessynonymy(Section1). Sothereare 

twoquestions:Whatistherelationoftheobjectsrepresentingpropositions, 

i.e.ofincompatibility-setsandranges? Whatistherelationofthecriteriaofsynonymy,i.e.of 

I-equivalenceand L-equivalence?Iwillanswerthe 

firstquestionwithaninformalrecipefortransformingincompatibility-sets 

intorangesandviceversaandthesecondwithatheoremwhichsaysthat 

I-equivalenceand L-equivalencearecoextensional. 

BothBrandomandCarnaprepresentpropositionsbysetsofsetsofsentences. 

But,atleastsuperficially,rangesandincompatibility-setsdiffer. 

7 Forthisuseof“explicate”,cf.(Peregrin,2007,p.13).


Intuitively,therangeofasentencewillcontainonlysetsofsentencescompatiblewithit,whiletheincompatibility-setofasentencewillcontainonly 

incompatiblesetsofsentences. Butnevertheless,wecanmovebackand 

forthfreelybetweenincompatibility-setsandranges.Theincompatibilitysetofasentencecanberepresentedbyitscomplement,i.e.thecompatibility-setofthesentence. 

Thecompatibility-setinturncanbestratified 

intoabundleofmaximallycoherentsetsofsentencescontainingthesentence,whichisthesameastherangeofthesentence.(Stratificationistheprocedureoftakingoutofthecompatibility-setallsetsofsentencescontainedinothersetsofsentencesofthecompatibility-set.) 

Totransforma 

rangeintoanincompatibility-set,wejusthavetoretracethosesteps.This 

recipeforthetransformationofincompatibility-setsandrangesisonlyinformal,becausethecomparedconceptsstemfromdifferenttheories. 

As 

yet,incompatibility-setsaredefinedonlyinincompatibilitysemanticsand 

rangesaredefinedonlyinstate-descriptionsemantics. 

Forthesamereason,thetheoremofcoextensionalitycannotyetbe 

statedorprovedineithersemantictheory. 8 WefirsthavetodefinesurrogatesoftheCarnapianconceptsinBrandom’ssemanticsandsurrogatesof 

theBrandomianconceptsinCarnap’ssemantics(AppendicesA&B). 

InBrandom’sincompatibilitysemantics,state-descriptions ∗ canbedefinedasmaximallycoherentsetsofsentences. 

9 Thisgivesusadefinitionof 

L-equivalence ∗ asmembershipinthesamemaximallycoherentsets. The 

theoremofcoextensionalitycannowbestatedandprovedinincompatibilitysemantics. 

Itsaysthattwosentencesare L-equivalent ∗ justincase 

theyare I-equivalent,i.e.,thattwosentencesareincludedinthesamemaximallycoherentsetsjustincasetheyareincompatiblewiththesamesets 

ofsentences(AppendixA).Thishasalreadybeenshowninasimilarway 

byJaroslavPeregrin,inhisCommentsonBrandom’sFifthLockeLecture 

(Peregrin,2007,p.17f.). 

InCarnap’sstate-descriptionsemantics,wecandefineincoherence ∗ and 

incompatibility ∗ onthebasisofmembershipinstate-descriptions. Apair 

ofsentencesisdefinedasincompatible ∗10 ifandonlyifthereisnostatedescriptionthatcontainsboth. 

(Thisconceptofincompatibility ∗ canalreadybefoundunderthenameof“L-exclusiveness”inCarnap’s1942In- 

8 Asyet,therelationof I-equivalenceisdefinedonlyinincompatibilitysemanticsandthe 

relationof L-equivalenceonlyinstate-descriptionsemantics. 

9 Anasterisk( ∗ )indicatesanotionthatisdefinedinaforeignsetting. —State-descriptionscansafelybetreatedasmaximallycoherentsetsbecauseoftheexclusionof 

state-descriptionsthatcontainincompatiblesentences(Section2). 

10 Thisformalnotionofincompatibility ∗ inmodifiedstate-descriptionsemanticsislinked 

toourintuitivenotionofcompatibility,becausewehaveusedtheintuitivenotionof 

compatibilitytoruleoutsomeoftheoriginalstate-descriptions. Inunmodified statedescriptionsemantics,thedefinitionofcompatibility 

∗ entailsthatanytwoatomicsentencesarecompatible 

∗ .


troductiontoSemantics(Carnap,1942/1961,pp.70,94).) Onthebasis 

ofthedefinedBrandomianconcepts,itispossibletostateandprovethe 

theoremofcoextensionalityinstate-descriptionsemantics(AppendixB). 

Insum,itcanbeprovedbothinincompatibilitysemanticsandinstatedescriptionsemanticsthattwosentencesare 

L-equivalentjustincasethey 

are I-equivalent. 

4 Towardsagenuinecomparison 

ItisimportanttoseethatthisdoesnotshowthatBrandom’sandCarnap’s 

semanticsareequivalentsimpliciter.Theresultssofararethatinincompatibilitysemantics, 

I-equivalenceandadefinednotionof L-equivalence ∗ are 

coextensionalandthatinstate-descriptionsemantics, L-equivalenceanda 

definednotionof I-equivalence ∗ arecoextensional.Whyisthisnotenough? 

Speakingmetaphorically,foracomparisontobegenuineitmustbeconductedfromtheoutsideofbothsemantictheories. 

Internalcomparisonis 

alwaysindangerofworkingwithpoorsubstitutes. 

Asboththeoriestrytogiveformalmodelsofmeaning,Isuggestthatthe 

commongroundforagenuinecomparisonisalanguagethatisintuitively 

interpreted.Theintuitiveinterpretationofalanguageisgivenbyatranslationofitstermsinto(afragmentof)naturallanguage. 

11 Wewilltherefore 

havetoworkwithparticularexamplesoflanguages.Onlyifincompatibilitysemanticsandstate-descriptionsemanticsassignthesamerelationsof 

synonymyforeveryintuitivelyinterpretedlanguage,willitbeappropriate 

tosaythatthetwosemantictheoriesareequivalentsimpliciter. 

Therearethreepreconditionsforafaircomparisonofwhatincompatibilitysemanticsandstate-descriptionsemanticsmakeofaparticularintuitively 

interpretedlanguage.Firstly,bothsidesshouldhavesharedintuitionsabout 

therelationsbetweenbasicmodalconcepts. 12 Secondly,theyshouldhave 

11 “Interpretation”isanumbrellatermforanyprocedurethatassociatesmeaningwith 

expressionsofalanguage. Here,inthecontextofformallanguages,wecandistinguish 

thefollowingsensesof“interpretation”: 

1. Interpretation(oftheexpressionsofanyformallanguage)bytranslationintothe 

naturallanguageweuse, 

2. interpretation(ofthesentencelettersofpropositionallogic)bytruthtables, 

3. interpretation(oftheexpressionsofapredicatelogicallanguage)byrangesof 

state-descriptions,i.e.interpretationinstate-descriptionsemantics,and 

4. interpretation(ofsentenceletters)byanincoherencepartition,i.e.interpretation 

inincompatibilitysemantics. 

While2,3and4arethemselvespartofformaltheories,1makesuseofourlanguage,and 

hencecanbecalled“intuitive”or“pre-theoretical”interpretation. 

12 Forthepre-theoreticalconceptsofnecessity,compatibilityandapossibleworld,there 

shouldbeagreementaboutprincipleslikethefollowing: Asentenceisnecessaryifand


sharedintuitionsaboutmodalityinparticularcases,aswell. 13 Thirdly,we 

willneedasharedformalframework,inthefollowingsense:Theintuitive 

interpretationofaparticularlanguagemust,ineachtheory,uniquelydetermineaformalinterpretationofthelanguage.ForBrandom,theformal 

interpretationofalanguageisgivenbyitsincoherencepartition,andfor 

Carnap,bythesetofstate-descriptions. 14 

5 Filtersonstate-descriptiontables 

Toseehowanincoherencepartitionrelatestoasetofstate-descriptions, 

letustakeanotherlookatstate-descriptionsemanticsinitsoriginalform. 

Thiswillprovideahelpfulcontrasttounderstandformalinterpretationin 

modifiedstate-descriptionsemantics. 

AccordingtoCarnap’soriginaldefinition,astate-descriptionisaset 

ofsentencesthat,foreveryatomicsentence,containseitherthesentence 

oritsnegation. TheideabehindthisdefinitiongoesbacktowhatLudwigWittgensteinwroteabouttruth-tablesintheTractatus. 

Hestates 

thateachrowofatruth-tablerepresentsapossiblestateofaffairs,and 

togethertheyrepresentallpossibilities(e.g.,Tractatus,4.2&4.3). The 

analogywithtruth-tablesprovidesasystematictechniqueforgivingacompletelistofstate-descriptionsofagivenlanguage,whichwemaycalla 

“state-descriptiontable”. Howmustastate-descriptiontablebechanged 

toaccommodatemodifiedstate-descriptionsemantics?ThemodificationintroducedintoCarnap’soriginalframeworkrulesoutthosestate-descriptions 

containingintuitivelyincompatiblesentences.Thisamountstocrossingout 

rowsinthetruth-table-likelistofallpotentialstate-descriptions. Letus 

callalistofallrowsthatarecrossedouta“filteronastate-description 

table”. 15 

onlyifitistrueineverypossibleworld.Twosentencesarecompatibleifandonlyifthere 

isapossibleworldwherebotharetrue.Asetofsentencesrepresentsapossibleworldif 

andonlyifitismaximallycompatible.—InthecontextofthecomparisonofBrandom’s 

andCarnap’ssemantics,itisimportanttorealizethatthejustificationoftheseprinciples 

restsinneithertheory,butinoursharedintuitionsaboutmodalityingeneral. 

13 E.g.,thehistoricalCarnapmightwelldisagreewithBrandomwhetheritreallyisimpossiblethatablackberryberedandripe(cf.(Brandom,2008,p.123). 

Aswewant 

tocomparesemantictheories,notparticularopinionsaboutmodality,wewillhaveto 

abstractawayfromsuchdisagreements. 

14 Weshouldunderstandourintuitionsaboutaparticularlanguageintermsofanintuitivenotionofcompatibility.Intuitivecompatibilitynaturallyleadstoanincoherence 

partitionand,atleastaccordingtothemodificationofCarnap’sframework(Section2), 

itdetermineswhatstate-descriptionsthereare. 

15 ThisnotionseemstobeexactlythesameaswhatPeregrincalls“inadmissiblevaluations”,cf.JaroslavPeregrin,“Inferentializingsemanticsandconsequence”,talkgivenon 

June17,2008,atLogica2008inHejnice.


Themethodoffiltersonstate-descriptiontablescapturestheformalinterpretationofalanguageasgivenbymodifiedstate-descriptionsemantics.So,howdoesthisrelatetoformalinterpretationinincompatibilitysemantics,i.e.toanincoherencepartition?Asanincoherencepartitionobviously 

putsspecificrestrictionsontheadmissibledistributionsoftruth-valuesover 

theatomicsentences,ituniquelydeterminesafilteronthesetofpotential 

state-descriptions.Butdoesafilteruniquelydetermineanincoherencepartition?Theanswertothisquestionisnotobvious,becauseanincoherencepartitiondeterminesthecoherenceofeverysetofsentences,whilethefilterconcernsonlysetsofmaximallength. 

16 Butifaccordingtothefiltera 

(potential)state-descriptioniscoherent,thenbypersistenceallitssubsets 

arecoherent,too.Inparticular,allsubsetsconsistingofunnegatedatomic 

sentenceswillbecoherent. 17 Thereforeafiltergivesusallcoherentsetsof 

atomicsentencesandthusuniquelydeterminesanincoherencepartition. 

Sowehavefoundagenuinesimilarityofincompatibilitysemanticsand 

modifiedstate-descriptionsemantics: Fromanintuitiveinterpretationof 

aparticularlanguage,wereachanincoherencepartitioninBrandom’ssemanticsandafilteron(potential)state-descriptionsinCarnap’smodified 

semantics.Asthereisabijectionbetweenincoherencepartitionsandfilters, 

thereisagenuineequivalencebetweenbothkindsofformalinterpretation. 

6 Atomicsentences 

Nonetheless,theequivalenceofincoherencepartitionsandfiltersonstatedescriptionshingesonsharedintuitionsabouttheconceptofcompatibility. 

HeretherearecontrarytendenciesinBrandomianandCarnapiansemantics. 

Thisisobviousintheexampleofasimplelanguagewhereallsetsof 

sentencesaresaidtobeintuitivelycompatible.Atleastprimafacie,BrandomandCarnapwillmakeentirelydifferentsenseofthislanguage.Carnapwillunderstandcompatibilityaslogicalindependenceandaccordingly 

assignthemaximalnumberofdifferentstate-descriptions. Consequently, 

therewillbedifferentrangesforallatomicsentences;notwoatomicsentencesaresynonymous. 

Brandom,bycontrast,canmakenotmuchsense 

ofthislanguage,becauseinincompatibilitysemantics,thecoherenceofall 

sentencestrivializesalanguage: Allatomicsentenceswillhavethesame 

incompatibility-set,namelytheemptyset.Soallatomicsentenceswillbe 

16 

Whatismore, unlikethesetsdealtwithbyanincoherencepartition, moststatedescriptionscontainnegatedsentences. 

17 

Brandomhasshownthattheadditionofthepropositionalconnectivestoincompatibilitysemanticsisconservative(Brandom,2008,p.127). 

Wethereforedonotneedto 

computetheincompatibility-setsofconjunctionsandnegationstoanswerthequestion 

whetherafilteruniquelydeterminesanincoherencepartition.


I-equivalentandhencesynonymous. 18 WhyareBrandomandCarnapled 

tosodifferentresults? 

Toclarifythisissue,letustakeanotherhistoricalstepbackwards,to 

whatWittgensteinsaysaboutelementarysentences(Elementarsätze)in 

theTractatus. Apartfromtheirsyntacticalatomicity,Wittgenstein’selementarysentenceshavethefollowingproperties: 

PropertyofBase: Anydistributionoftruth-valuesoverallelementarysentencesfixesthetruthvaluesofallsentencesofthelanguage. 

(Tractatus,4.51,5.3) 

Wittgenstein’s elementary sentences are logically independent in the 

sensethattheyhavethefollowingtwoproperties: 

PropertyofNoEntailment: Noelementarysentenceislogicallyentailedbyanyotherelementarysentence.(Tractatus,5.134) 

PropertyofGlobalCompatibility: Notwoelementarysentencesare 

logicallyincompatible.(Tractatus,4.211) 

Wecanaskofeveryformallanguagewhetheritsatomicsentenceshave 

thepropertiesofBase,ofNoEntailmentandofGlobalCompatibility.Inthe 

caseofpropositionallogic,allthreequestionsmustofcoursebeanswered 

positively(that’swhythemethodoftruth-tablesworks). Thedifferences 

ofthethreesemantictheoriesindiscussioncannowbeexpressedinthe 

followingway: 

ThepropertyofBasedoesnothelptodistinguishbetweenthetheories. 

Inallthreesystems,sentencesbuiltonlywiththehelpofpropositional 

logicarebasedontheatomicsentences,whilesentencescontainingquantifiersormodaloperatorsarenot. 

19 ButthepropertiesofNoEntailment 

andGlobalCompatibilityprovidedistinctivecriteria: Accordingtooriginalstate-descriptionsemantics,atomicsentenceshaveboththeproperty 

18 Theexampleofthesimplelanguage L = {p, q, r}wheretheset {p, q, r}iscoherent, 

illustratesthatthenotionsofrange ∗ and L-equivalence ∗ asdefinedinincompatibility 

semanticsinsomecasesarepoorsubstitutesforthenotionsofrangeand L-equivalence 

ofstate-descriptionsemantics(Section4).In L,therewillbeonlyonestate-description ∗ , 

i.e.onlyonemaximallycoherentset,namely {p, q, r},andaccordinglyonlyonerange ∗ , 

namely {{p, q, r}},Consequently, p, qand rare L-equivalent ∗ .So I-equivalenceand Lequivalence 

∗ areindeedcoextensional(Section3).Butallthesame,theexampleofthe 

languageofthreecompatiblesentencesintuitivelycontradictstheequivalenceofincompatibilitysemanticsand(evenmodified)state-descriptionsemantics,becausetothesame 

intuitivelyinterpretedlanguageCarnapwouldassigneightdifferentstate-descriptionsand 

threedifferentranges. 

19 ThepropertyofBaseislostinpredicatelogic(Peregrin,1995,ch.5).Inmodalpropositionallogic,thepropertyofBaseislost,aswell.Thetruth-valuesofallatomicsentences 

donotingeneraldeterminethetruth-valueofsentencesoftheform“necessarily α”, 

becausethemodaloperatorsarenottruth-functional.


ofNoEntailmentandofGlobalCompatibility. 20 Accordingtomodified 

state-descriptionsemantics,atomicsentencesneednothavethepropertyof 

GlobalCompatibility(Section2).Andaccordingtoincompatibilitysemantics,atomicsentencesneedneitherhavethepropertyofGlobalCompatibilitynorthepropertyofNoEntailment.Thusthedifferencebetweenmodified 

state-descriptionsemanticsandincompatibilitysemanticsthatemergedin 

theexampleofthesimplelanguagehingesonthepropertyofNoEntailment. 

21 

Buteventhislastdifferencecouldbesmoothedoutifstate-description 

semanticswasfullymodifiedbygivingupthepropertyofNoEntailment,as 

well.Iseethefollowingreasontodothis.Weallowedatomicsentencesto 

beincompatiblebecausewewantedtoexcludestate-descriptionscontaining 

particlesbearingnegativeandpositivecharge(Section2).Butforasimilar 

reasonwemaywanttoexcludestate-descriptionscontainingwhalesthat 

arenotmammals(andthelike). Thisamountstorestrictingtheclassof 

state-descriptionsfurther,torespectnotonlyintuitiveincompatibilities, 

butintuitiveentailments,aswell. 

7 Holisticnegationandthesimulationoflogicalindependence 

Nonethelessitmayseemahighpricetogiveupallkindsoflogicalindependencebetweenatomicsentences.Soletusturnagaintoincompatibilitysemantics.WhydoesBrandomaccepttheverylowdegreeoflogicalindependence?IseeareasonthatconcernstheholisticcharacterofBrandom’s 

semanticsandcanbeexplainedinthespecialcaseofnegation. 

Toillustratetheholisticcharacterofnegationinincompatibilitysemanticsletusreturntothesimplelanguageofthreecompatiblesentences(Section6),thatmaybetranslatedas“Carnapisaphilosopher”,“Scottis 

aphilosopher”and“Brandomisaphilosopher”. Inthesimplelanguage, 

thethreesentencesarevalidandhence,theirnegationsareincoherent.In 

otherwords,wecannotcoherentlysaythatCarnapisnotaphilosopher. 

Letusnowenlargethesimplelanguagebyaddingthesentences“Carnap 

isaflorist”,“Scottisaflorist”and“Brandomisaflorist”. Letusfurthermorestipulate(somewhatcounterfactually)thatitisincompatibleto 

beaphilosopherandaflorist. Nowwehavereachedadifferentlanguage 

20 Carnap’soriginaldefinitionisthusinfullagreementwithWittgenstein’sideasaboutelementarysentences.Thisisnotsurprising,ashisstate-descriptionsemanticswasinspired 

byWittgenstein’sideasaboutlogicalpossibilities. 

21 Inincompatibilitysemanticstherearenotonlyincompatibilitiesbetweenatomicsentences,but,accordingtothedefinitionofincompatibility-entailment,theremayberelationsofentailmentbetweenatomicsentences. 

Inmodifiedstate-descriptionsemantics 

theremaybeincompatibleatomicsentences,butatleastprimafaciethisisnoreasonto 

assumethatthereareentailmentsbetweenatomicsentences.


wheretherearesixdifferentincompatibility-setsforthesixatomicsentences 

andtheirnegationsarecoherent.Inotherwords,onlytheadditiontothe 

languageofanincompatiblesentencelike“Carnapisaflorist”makesthe 

negationof“Carnapisaphilosopher”coherentlyexpressible. Thisisan 

exampleoftheholisticcharacterofnegationasdefinedbyBrandom:The 

meaningofnon-pdependsonwhatmaybesaidthatisincompatiblewith p. 

Anotherthingwenowhaveachievedisthat,inthesix-sentencelanguage, 

theoriginalthreesentencesarelogicallyindependentofeachother. From 

ourexamplewecanthusreadoffageneralrecipeforsimulatinglogically 

independentatomicsentencesinincompatibilitysemantics.Wejusthaveto 

startwithanevennumberofatomicsentences,andlaydownthateachconsecutivepairofsentencesisincompatible,butthesetofalleven-numbered 

sentencesiscoherent. 22 Thenalleven-numberedsentenceswillbelogically 

independent.So,thoughinincompatibilitysemanticsthereisnolanguage 

whereallatomicsentencesarelogicallyindependent,therearelanguages 

wherehalfofthemare.Inthiscase,wecanalsosimulatestate-descriptions 

thatsatisfyCarnap’soriginaldefinition.Themaximallycoherentsetswill 

be I-equivalenttothosesetsofsentencesthat,foreachoneofthelogically 

independentsentences,containeitherthesentenceoritsnegation. 

Thepossibilityofsimulatinglogicallyindependentsentencesandoriginal 

state-descriptionshelpstobringouttheimportantdifferencebetweenincompatibilitysemanticsandoriginalstate-descriptionsemantics:Brandom 

canmakesenseofalotmorelanguagesthanCarnaporiginallycould.So, 

togiveupthelogicalindependenceofatomicsentencesbroadensthescope 

offormalsemantics. 

8 ThesimilarityofBrandom’sandCarnap’ssemantics 

Insum,therearedeepsimilaritiesbetweenBrandom’sincompatibilitysemanticsandCarnap’smodifiedstate-descriptionsemantics.Notonlydoesthetheoremofcoextensionalityholdinboththeories(Section3),butforeveryintuitivelyinterpretedlanguage,wecanreachequivalentformalinterpretations:anincoherencepartitionorafilteronstate-descriptions(Section5). 

Thisequivalenceholdsifstate-descriptionsemanticsisfullymodified,i.e.,if 

itabandonstherequirementthattherearenoentailmentsbetweenatomic 

sentences(Section6).Andtherearegoodreasonstodothis(Sections6&7). 

Inordertobeabletospellouttheresultofmycomparisonintheform 

ofaslogan,letmeintroducetheconceptofaminimallyincoherentset.Rememberthatamaximallycoherentsetisacoherentsetsuchthat,forevery 

sentencethatisnotamemberofit,thesetplusthatsentenceisincoherent. 

Aminimallyincoherentsetisthemirror-imageofthis,becauseitisdefined 

22 Thesetofallqueer-numberedsentencesmustofcoursebecoherent,aswell.


asanincoherentsetsuchthat,foreverysentencethatisamemberofit, 

thesetminusthatsentenceiscoherent. Becauseofpersistence,acompletelistofminimallyincoherentsetsisanon-redundantwaytospecifyall 

incoherentsets—thatis:anincoherencepartition. 

WiththehelpofthisconceptIcansumupmyresultinthefollowingway: 

Whenweformallyinterpretalanguage,itisoneandthesamethingwhether 

wegiveacompletelistofallmaximallycoherentsetsoracompletelistof 

allminimallyincoherentsets. Thus,Brandom’sincompatibilitysemantics 

amountstothesameasthereductionistversionofpossibleworldssemantics 

turnedinside-out. 

MartinPleitz 

DepartmentofPhilosophy,UniversityofMünster 

Domplatz23,D–48143Münster,Germany 

martinpleitz@web.de 

A Thetheoremofcoextensionalityinincompatibilitysemantics 

Definition12.Aset Sofsentencesof Liscalledastate-description ∗ iff S 

ismaximallycoherent,i.e.,iffforeverysentence α,either αisamemberof 

Sor αisincompatiblewith S. 

Twosentences αand βcalled L-equivalent ∗ ifftheyareincludedinthe 

samemaximallycoherentsets. 

FirstTheoremofCoextensionality.Thesentences αand βare L-equivalent 

∗ iff αand βare I-equivalent. 

Proof.“⇐”: Let αand βbe I-equivalent. Then,bythedefinitionof Iequivalence,theyareincompatiblewiththesamesetsofsentences(i.e. 

Inc(α) = Inc(β)). Butthentheyarecompatiblewiththesamesentences. 

Theythereforearecontainedinthesamemaximallycompatiblesetsof 

sentences, i.e., inthesamestate-descriptions ∗ andconsequentlyare Lequivalent 

∗ . 

“⇒”: Let αand βbenot I-equivalent. Thenthereisaset Xsuchthat 

X ∪{α}iscoherentwhile X ∪{β}isincoherent.Asforeverycompatibleset 

ofsentencesthereisamaximallycompatiblesetofsentencesthatcontains 

it,thereisastate-description ∗ Ssuchthat X ∪ {α} ⊆ S,andtherefore 

α ∈ S. As Scontains X,itcannotinclude β,becausetheincoherenceof 

X ∪ {β}wouldbypersistencetransferto S. Therefore β �∈ S. As α ∈ S 

and β �∈ S, αand βarenot L-equivalent ∗ . 23 

23 Forasimilarproof,cf.(Peregrin,2007,p.17f.).


B Thetheoremofcoextensionalityinstate-description 

semantics 

Definition13.Asetofsentencesisincoherent ∗ iffthereisnostate-descriptionthatitisasubsetof(notethatincoherence 

∗ ispersistent).Apair 

ofsentencesisincompatible ∗ iffthereisnostate-descriptionthatcontains 

both. 

Twosentences αand βare I-equivalent ∗ iff αand βareincompatible ∗ 

withthesamesetsofsentences. 

SecondTheoremofCoextensionality.Thesentences αand βare Iequivalent 

∗ ifftheyare L-equivalent. 

Proof.“⇐”: Let αand βbe L-equivalent. Thentheyareelementsof 

thesamestate-descriptions. Ascompatibility ∗ isdefinedbyrecourseto 

membershipinstate-descriptions, αand βthereforearecompatible ∗ with 

thesamesetsofsentences. Hencetheyareincompatible ∗ withthesame 

setsofsentences.So αand βare I-equivalent ∗ . 

“⇒”:Let αand βbenot L-equivalent.Thenthereisastate-description 

Dsuchthat αbelongsto Dwhile βdoesnotbelongto D.(Orviceversa. 

Forreasonsofsymmetryitsufficestodealwithonecase.)Nowlet δbethe 

conjunctionthatcompletelydescribes D.Then,accordingtothedefinition 

ofcompatibility ∗ , αand δarecompatible ∗ while βand δareincompatible ∗ . 

Consequently, αand βarenot I-equivalent ∗ . 

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Inference and Knowledge 

Dag Prawitz ∗ 

Wesometimesacquirenewknowledgebymakinginferences.Thisfactmay 

beseenassoobviousthatitsoundsstrangetoputasaproblemhowandwhy 

wegetknowledgeinthatway. Nevertheless,thereisnostandardaccount 

oftheepistemicsignificanceofvalidinferences. 

Forthisdiscussion,Ishallassumethataperson’sknowledgetakesthe 

formofajudgementthatshehasgroundsfor–anassumptionrelatedtothe 

ideathatapersonknowsthat p,onlyifshehasgoodgroundsforholding 

theproposition ptobetrue. Thequestionhowinferencesgiveknowledge 

maythenbeput:howmayonegetinpossessionofgroundsforjudgements 

bymakinginferences? Onewouldexpectthatthereisaneasyanswerto 

thisquestionbyjustreferringtohowtheconceptsinvolvedareunderstood. 

But,asIshallargue,thereislittlehopeofansweringthisquestionwhenthe 

notionofvalidinferenceisunderstoodinthetraditionalway. Toaccount 

fortheepistemicsignificanceofvalidinferences,weseemtoneedanother 

approachtowhatitisforaninferencetobevalidandwhatitistomake 

aninference.Ishalldescribeonesuchapproach. 

Givenavalidargumentoravalidinferencefromajudgement Atoa 

judgement B,itmaybepossibleforanagentwhoisalreadyinpossession 

ofagroundfor Atousethisinferencetogetagroundfor B,too. But 

theagentisnotensuredagroundfor B,justbecauseoftheinferencefrom 

Ato Bbeingvalidandtheagentbeinginpossessionofagroundfor A. 

Theagentmaysimplybeignorantoftheexistenceofthisvalidinference, 

inwhichcaseitsmereexistencedoesnotmakeherjustifiedinmakingthe 

judgement B.Aquestionthathastobeansweredisthereforewhatrelation 

theagenthastohavetotheinferencetomakeherjustifiedinmakingthe 

judgement B. 

∗ ManyoftheideaspresentedherewereworkedoutwhileIwasafellowattheInstitute 

ofAdvancedStudiesatUniversitàdiBolognainthespringof2007andwerepresented 

inlecturesgivenatthePhilosophyDepartmentofUniversitàdiBologna.

184 DagPrawitz 

Ishallrestrictmyselfheretodeductiveinferencesandconclusivegrounds, 

andwhenspeakingof“inference”and“ground”,Ishallalwaysmeandeductiveinferenceandconclusiveground. 

1 Theproblem 

Thequestionunderwhatconditionanagent,callher P,getsaground 

fortheconclusionofavalidinferencecanbeformulatedmoreexplicitlyas 

follows,where,forbrevity,Irestrictmyselftothecasewhenthereisonly 

onepremiss: 

Giventhat 

thereisavalidinference Jfromajudgement Atoajudgement B, (a) 

andthat 

theagent Phasagroundfor A, (b) 

whatfurtherconditionhastobesatisfiedinorderforittobethecasethat 

Phasagroundfor B? (d) 

Theproblemisthustostateafurthercondition(c)suchthat(a)–(c) 

imply(d).Obviously,asalreadyremarked,(a)and(b)alonedonotimply 

(d). Hence,weneedtospecifyacondition(c),inotherwordsarelation 

betweenanagent P andaninference J,whichdescribeshowtheagent 

arrivesatagroundfortheconclusionof J. 

2 Afirstattempttofindacondition(c) 

Sincethemereexistenceofavalidinference Jto Bfromajudgement A, 

forwhichtheagenthasaground,isnotsufficienttogiveheragroundfor 

B,onemaythink 1 thattheextraconditionthathastobesatisfiedisthat 

theagent Pknowsthattheinference Jfrom Ato Bisvalid (ck) 

(thesubscriptkfor‘knowing’). 

Butclearlywedonotnormallyestablishthevalidityofaninference 

beforeweuseit.Ifitwereanecessaryconditionalwaystodoso,aregress 

wouldresult.Theargumentusedtoestablishthevaliditywouldneedsome 

inferences,andifthevalidityofthemhadagaintobeestablishedtogive 

theargumentanyforce,therewouldbeaneedofyetafurtherargument 

andsoon.Unlessthereweresomeinferenceswhosevaliditycouldbeknown 

withoutanyargument,wewouldbeinvolvedinanendlessregressoftrying 

1 Thisseemstobetakenforgrantedby,e.g.,John(Etchemendy,1990,p.93).


toestablishthevalidityofinferencesbeforeanycouldbeusedtogeta 

groundforitsconclusion,andhencewewouldneverbeabletoacquire 

knowledgebyinferences. Atleastincasethevalidityofaninferenceis 

definedinamodel-theoreticalway,analogouslytohowlogicalconsequence 

isdefined,itcannotbemaintainedthatknowledgeofvalidityisimmediate 

anddoesnotrequireanargumenttobeknown. 

Iwanttoremarkinpassingthattheregressnotedaboveisdifferentfrom 

thewell-knownBolzano–Lewisregress. 2 Thislatterregresscastdoubtsnot 

onlyonthenecessityofthecondition(ck)butevenonwhetherthecondition 

issufficienttoguaranteetheagentagroundfortheconclusion.Whyshould 

wethink(ck)tobesufficient?Presumablybecausehavingagroundforthe 

judgement Aandhenceknowingthatthepropositionoccurringin A(i.e., 

theoneaffirmedby A)istrueandknowingthattheinferencefrom Ato 

Bisvalidandhencethatitistruthpreserving(inthesensethatifthe 

propositionoccurringin Aistruethensoisthepropositionoccurringin 

B),theagentcaninfer Bbysimplyapplyingmodusponens.Thisgivesher 

agroundfor B,onemaythink.Ifso,thereasonforsayingthattheagent 

hasagroundfor B,whensheknowstheinferencetobevalid,seemstobe 

thatthereisanotherinferencethantheoriginalonefrom Ato B,namely, 

aninferencefromtwopremisses,oneofwhichistheagent’sknowledgeof 

thevalidityoftheoriginalinference.Itisbecauseofthisnewinferencethat 

theagentisclaimedtogetagroundfor B. Butbythesamereasoning, 

whatreallyguaranteestheagenttohaveagroundfor Bisherknowledgeof 

thevalidityofthisnewinference.Inotherwords,thereisathirdinference 

withthreepremisses,oneofwhichistheagent’sknowledgeofthevalidity 

ofthissecondinference,andsoon. 

Alreadyfromthefirstregressdiscussedabovewemustconcludethat(ck) 

isnottherightconditionthatweareseekingtodescribehowwegenerally 

acquireknowledgeorgroundsbyinferences.Atleast,wemustconcludethis 

if“knows”in(ck)meanssomethinglikehavingestablishedbyargument. 

Onemaysuggestthatthereisanotherconceptofknowledgethatisrelevanthere,forinstanceknowledgebasedonimmediateevidenceorimplicit 

knowledgemanifestedinbehaviour,liketheimplicitknowledgeofmeaning 

thatMichaelDummetthascalledattentionto.Idonotwanttodenythat 

theremaybeanotionofvalidityofinferenceforwhichonecanclarifysuch 

aconceptofknowledgesothat(ck)becomesanappropriatecondition.As 

alreadysaid,itwouldcertainlyrequireadeparturefromwhatnowseems 

tobethedominantunderstandingofthevalidityofaninferenceinterms 

oftruthpreservationforallvariationsofthemeaningofthenon-logicalexpressionsinvolvedintheinference.Anyway,lackingaconceptofknowledge 

2 (Bolzano,1837)and(Carroll,1895). Ihavediscussedthisregressmorethoroughlyin 

(Prawitz,2009).


(andofvalidity)thatmakes(ck)appropriate—todeveloponwouldbea 

majortask—Ishallnowleavethisfirstattempttofindacondition(c). 

3 Asecondattempttofindacondition(c) 

Realizingthat(ck)isnottherightcondition,onemayseetheproposalofit 

asanoverreactiontothesimpleobservationfirstmade,viz.thatanagent 

needtostandinsomerelationtotheinference,ifitistoprovideherwitha 

groundforitsconclusion.Ofcourse,themereexistenceofavalidinference 

cannotautomaticallyprovidetheagentwithagroundfortheconclusion, 

onemaysay.Shehastodosomething.Butshedoesnotneedtoestablish 

thevalidityoftheinference.Allthatisneededisthatsheactuallyusesthe 

inference.Thenthevalidityoftheinferencedoesprovidetheagentwitha 

groundfortheconclusion,giventhatshealreadyhasoneforthepremiss. 

Onemaythussuggestthattheconditionsoughtforshouldsimplybe 

Pmakestheinference J,thatis,infers Bfrom A. (c) 

Thereisclearlysomethingrightinthissuggestion. Oneshoulddistinguishbetweenaninferenceact,andaninferenceinthesenseofanargument 

determinedbyanumberofpremissesandaconclusion.Itisfirstwhenan 

agentmakesaninference,i.e.,carriesoutaninferenceact,thatthequestionariseswhethersheisjustifiedinmakingtheassertionthatoccursas 

conclusionoftheinference. 

Butitmustthenbeaskedwhatismeantbymakinganinferenceor 

byinferringaconclusion Bfromapremiss A. Weusuallyannouncethe 

resultofsuchanactverballybysimplyfirstmakingtheassertion A,then 

saying“hence”or“therefore” B,or,inthereverseorder,wefirstmake 

theassertion B,andthensay“since”or“because” A. Aninferenceact, 

lookeduponasverbalbehaviour,canbeseenasakindofcomplexspeech 

actinwhichwedonotonlymakeanassertionbutalsogiveareasonfor 

theassertionintheformofanotherassertionorsomeotherassertionsfrom 

whichitis(implicitlyorexplicitly)claimedtofollow. 

However,ifthisisallthatismeantbyinferringaconclusionfroma 

premiss,thenonecannotexpectthatconditions(a),(b),and(c)together 

withreasonableexplicationsofthenotionsinvolvedaresufficienttoimply 

thatthepersoninquestionhasagroundfortheconclusion B.Toseethis, 

onemayconsiderascenariowhereapersonannouncesaninferenceinthe 

waydescribed,sayasastepinaproof,butisnotabletodefendtheinference 

whenitischallenged.Suchcasesoccuractually,andthepersonmaythen 

havetowithdrawtheinference,althoughnocounterexamplemayhavebeen 

given.Ifitlaterturnsoutthattheinferenceisinfactvalid,perhapsbya 

longandcomplicatedargument,thepersonwillstillnotbeconsideredto


havehadagroundfortheconclusionatthetimewhensheassertedit,and 

theproofthatsheofferedwillstillbeconsideredtohavehadagapatthat 

time. Thiswouldbeasituationinwhichconditions(a),(b)and(c)were 

allsatisfied,but(d)wouldnotbesaidtohold. 

Iftheinferencefrom Ato Bisgenerallyrecognizedasvalid,then,sociologicallyspeakingsotosay,anagentwhohasagroundfor 

Aandmakes 

theassertion B,giving Aasherreason,willcertainlybeconsideredtohave 

agroundforherassertion. Ifinsteadtheinferenceisnotobviouslyvalid 

eventoexperts,theagentisnotconsideredtohaveobtainedagroundfor 

Bbecauseofmakingtheassertion Bandgiving Aasherreason.Butwe 

areofcoursenotsatisfiedwithasociologicaldescriptionofwhenanagentis 

consideredtohaveaground(obtained,e.g.,byaddingasaconditionthat 

thevalidityoftheinferenceshouldbegenerallyrecognizedbyexpertsin 

thefield). 

Itthusstillremainstostatetheappropriateconditionunderwhichavalid 

inferencegivesanagentagroundfortheconclusionofaninference.Onemay 

thinkthatitmustbebasicallyrightthatwegetagroundforajudgement 

byinferringitformotherjudgementsforwhichwealreadyhavegrounds, 

andthathencecondition(c)isrightlystatedasabove.Butthen“toinfer” 

or“tomakeaninference”mustmeansomethingmorethanjuststatinga 

conclusionandgivingpremissesasreasons.Thebasicintuitionis,Ithink, 

thattoinferisto“see”thatthepropositionoccurringintheconclusionmust 

betruegiventhatthepropositionsoccurringinthepremissesaretrue,and 

theproblemishowtogetagripofthismetaphoricuseof“see”. 

4 Thenatureoftheproblem 

Atthispointitmaybegoodtopauseandconsiderinmoredetailthenature 

oftheproblemthatIhaveposed.Ihaveusedthetermgroundinconnection 

withjudgementstohaveanameonwhatapersonneedstobeinpossession 

ofinorderthatherjudgementistobejustifiedorcountasknowledge, 

followingthePlatonicideathattrueopinionsdonotcountasknowledge 

unlessonehasgroundsforthem.ThegeneralproblemthatIhaveposedis 

howinferencesmaygiveussuchgrounds. 

AsIusethetermground,aperson’sjudgementisjustifiedorcountsas 

knowledgewhensheinpossessionofagroundforthejudgement. Consequently,onedoesnotneedtoshowthatoneisinpossessionofagroundfor 

ajudgmentinordertobejustifiedinmakingthejudgement,itisenough 

thatinfactoneisinpossessionofsuchaground.Justificationsmustend 

somewhere,asWittgensteinputsit.Andthepointwheretheymustendis 

exactlywhenonehasgotinpossessionofwhatcountsasajustificationsor 

aground;somethingwouldbewronglycalledground,ifitwasnotenough


thatonetohadgotinpossessionofit,inotherwords,iftherewasyet 

somethingtobeshown,inorderthatone’sjudgementwastobeconsidered 

justified. 

Hence,itisnottheagentPwhohastostateanadequatecondition 

(c)andshowthat(d)holds,i.e.,thatshehasagroundfor B,whenthe 

conditions(a)–(c)aresatisfied;asjustsaid,sheisjustifiedwhensheisin 

possessionofaground,regardlessofwhatshecanshowaboutit.Itisweas 

philosopherswhohavetostateanadequatecondition(c)andthenderive 

(d)from(a)–(c)togiveanaccountoftheepistemicsignificanceofvalid 

inferences. Thepointofmakinginferencesistoacquireknowledge,and 

philosophyoflogicwouldnotbeuptoitstask,ifitcouldnotexplainhow 

thiscomesabout. Toexplainthisistosayunderwhatconditionsavalid 

inferencecansupplyuswithgrounds. 

Sincethefactthatweacquireknowledgebymakinginferenceissucha 

basicfeatureoflogic,oneshouldexpecttheaccountofthisfacttobequite 

simple,oncewehaveunderstoodrightlythekeyconceptsinvolvedhere,in 

particularthenotionsvalidinference,inferringormakinganinference,and 

ground.Whenthesenotionshavebeenexplicatedappropriately,oneshould 

expectittobeasimpleconceptualtruththat(a)–(c)imply(d). 

Whatissurprisingisthatthereisnogenerallyacceptedaccountofthe 

epistemicsignificanceofinferencesandthatpuzzlingproblemsseemtoarise 

whensuchanaccountisattempted. Thisisasignthatourusualunderstandingofthekeyconceptsinvolvedisfaulty. 

5 Logicalconsequenceandlogicallyvalidinference 

Sinceitisvalidinferencesthatallowepistemicprogress,acrucialingredient 

intheaccountmustbetogivethatnotionanadequatemeaning. The 

conceptofvalidinferenceistraditionallyconnectedwiththatoflogical 

consequenceandwithnecessarytruth-preservation.Oftenonesimplysays 

thataninferenceisvalidifandonlyiftheconclusionisalogicalconsequence 

ofthepremisses,whichinturnisequatedwithitbeingnecessarilythe 

casethattheconclusionistrueifallpremissesare. However,Ihavebeen 

followingFregeintakingthepremissesandtheconclusionofanactof 

inferencetobespeechactsinwhichapropositionisjudgedtobetrue,hence 

takingthepremissesandconclusionofaninferencetobejudgements. 3 The 

traditionalideaofinferenceasnecessarilytruthpreservingisthenbetter 

formulatedbysayingthataninferenceisvalidifandonlyitnecessarilyholds 

3 Inmorerecenttime,thepointthatpremissesandconclusionsarenotpropositionsbut 

judgementsorassertionshasespeciallybeenemphasizedbyPerMartin-Löf(Martin-Löf, 

1985);seealsoGöran(Sundholm,1998).IncontrasttoFregeandMartin-Löf,however, 

Ishallalsoconsiderthecasewhenthepremissesandconclusionarejudgementsmade 

underassumptions.


thatwhenallthepropositionsaffirmedinthepremissesaretrue,thensois 

thepropositionaffirmedintheconclusion. 

SinceAlfredTarski’s(1936)revivalofBernardBolzano’s(1837)definitionoflogicalconsequence,ithasbeencommontointerpretthemodal 

notionofnecessityinthiscontextextensionally,sayingineffect,asweall 

know,thataproposition(orsentence) BisalogicalconsequenceofsetaΓ 

ofpropositions(orsentences)ifandonlyifforallvariationsofthecontent 

ofthenon-logicalnotionsoccurringin Bandintheelementsof Γ,itisin 

factthecasethat Bistruewhenalltheelementsof Γare. 

Howdoesthevalidityofaninferencecontribute,togetherwiththeother 

twoconditions(b)and(c),totheagentbeinginpossessionofagroundfor 

theconclusion? Thisisthecrucialquestionthatanyproposednotionof 

validityhastoface.Inparticular,whyshouldthefactthataninferenceis 

truth-preservingcontributetoourgettingagroundfortheconclusion?As 

alreadynoted,anagent’sknowledgethataninferenceistruthpreserving 

wouldcontributetohergettingagroundfortheconclusionoftheinference, 

butsuchknowledgeshouldnotpresumed,andthefact thatitistruth 

preservingisirrelevant. 

Now,nobodysuggeststhatthevalidityofaninferenceistobedefinedin 

termsofjusttruthpreservation.FollowingBolzanoandTarski,themodeltheoreticaldefinitionsaysthataninferenceisvalidifitistruthpreserving 

regardlessofhowthecontentsofnon-logicalexpressionsarevaried. But 

thisdoesnotessentiallychangethesituation. Whyshouldtheadditional 

factthattheinferenceistruth-preservingalsowhenthecontentofthenonlogicalexpressionsisvariedberelevanttoquestionwhethertheagentcan 

seethatthepropositionaffirmedintheactualconclusion(wherethecontent 

isnotvaried)tobetrue?Thesamecanbesaidofvaliditydefinedinterms 

ofnecessarytruthpreservation,ifthenecessityisunderstoodontologically. 

Whyshouldthefactthataninferenceistruthpreservinginotherpossible 

worldshelptheagenttoseethatthepropositionaffirmedintheconclusion 

istrueintheactualworld?Itisdifficulttoseehowanythingbutknowledge 

ofthisfactcouldberelevanthere(andifknowledgeisassumed,itissufficienttoknowthatthetruthofthepropositionsinthepremissesmaterially 

impliesthetruthofthepropositionintheconclusion—i.e.,novariation 

ofcontentisneeded).Therefore,thereseemstobelittlehopethatonecan 

findanappropriatecondition(c)whenvalidityofinferenceisdefinedinthe 

traditionalway. 

Itisdifferentifthenecessityisunderstoodepistemicallyandthisistaken 

tomeanthatthetruthofthepropositionsassertedbythepremisses,which 

theagentisassumedtoknow,somehowguaranteesthattheagentcansee 

thatthepropositionassertedintheconclusionistrue. Suchanepistemic 

necessitycomesclosetoAristotle’sdefinitionofasyllogismasanargument


where“certainthingsbeinglaiddownsomethingfollowsofnecessityfrom 

them,i.e.,becauseofthemwithoutanyfurthertermbeingneededtojustify 

theconclusion.” 4 Itisofcourserighttosaythatthereisanepistemic 

tiebetweenpremissesandconclusioninavalidinference—somekindof 

thoughtnecessity,wecouldsay,thankstowhichtheconclusioncanbecome 

justified. Buttosaythisisnottogomuchbeyondourstartingpoint. It 

stillremainstosayhowthejustificationcomesabout. 

AlthoughtheideaofBolzanoandTarskitovarythecontentofnonlogicaltermsdoesnothelpusindefiningthevalidityofinference,thisideaisstillusefulfordefininglogicalconsequenceandthelogicalvalidityofinference.Oneimportantingredientinourintuitiveideaoflogicalconsequence 

andlogicalvalidityofinferenceis,Ithink,thattheyaretopicneutral, 

andonenaturalwaytoexpressthisistosaythattheyareinvariantunder 

variationsofnon-logicalnotions. 

Isuggestthatwedistinguishbetweenlogicalconsequenceanddeductive 

(oranalytic)consequenceandsimilarlybetweenaninferencebeinglogically 

validanditbeingonly(deductively)valid.Giventhelatterconceptwecan 

easilydefinelogicalvalidityinthestyleofBolzanoandTarski: 

Aninference Jislogicallyvalid,ifandonlyif,foranyvariationof 

thecontentofthenon-logicaltermsoccurringinthepremissesand 

conclusionof J,theresultinginferenceisvalid. 

Thevariationofcontentmaybeproducedbymakingsubstitutionsforthe 

non-logicaltermsinthemannerofBolzanoorbyconsideringassignments 

ofvaluestotheminthemannerofTarski;wedonotneedtogointothese 

technicaldetailshere. 

Thisdefinitionmakesjusticetotheideathatwhetheraninferenceislogicallyvalidisindependentofthemeaningofthenon-logicalterms.However, 

thelogicalvalidityofaninferenceisnownotreducedtotruthpreservation 

ortothetruthofageneralizedmaterialimplicationbuttothevalidityof 

aninferenceundervariationsofthecontentsofnon-logicalterms. 

Someinferencesarelogicallyvalid,inadditiontobeingdeductivelyvalid, 

andthisisaninterestingfeatureofthem,butitisnotafeatureonwhichthe 

conclusivenessoftheinferencehinges.Itthusremainstoanalysedeductive 

validityandbringouthowsuchaninferencemaydeliveragroundforits 

conclusion. 

6 Grounds 

Rationaljudgementsandsincereassertionsaresupposedtobemadeon 

goodgrounds. Itisnotthatanassertionisusuallyaccompaniedbythe 

4 See(Ross,1949,p.287).


statementofagroundforit;inotherwords,thespeakeroftenkeepsher 

groundforherself.Butiftheassertionischallenged,thespeakerisexpected 

tobeabletostateagroundforit.Tohaveagroundisthustobeinastate 

ofmindthatcanmanifestitselfverbally. 

Iamhereinterestedingroundsthatareobtainedbymakinginferences; 

allgroundscanofcoursenotbeobtainedinthisway,andIshallsoonreturn 

tosomeexamples.Whenanassertionisjustifiedbywayofaninference,itis 

commontoindicatethisbysimplystatingtheinferenceinthewaydiscussed 

above,andthepremissesoftheinferencesarethenoftencalledtheground 

fortheassertion.Thiswayofspeakingmaybeacceptableinaneveryday 

context,butitconcealstheproblemthatwearedealingwith,whichis 

probablyonereasonwhytheproblemhasbeensoneglected. Itmakesit 

seemasifoneautomaticallyhasagroundforaconclusionbyjuststatingan 

inferencethatinfacthappenstobevalid—ineffect,itseemsasonemay 

getagroundbysimplystatingthatonehasone.Wehavediscussedabove 

(Section3)whyagroundforaconclusionisnotforthcomingby“inferring” 

itinthissuperficialsense. 

Buttherearealsootherreasonswhyitisnotagoodterminologyto 

usetheterm“ground”forthepremissesofaninference.Thepremissesare 

judgementsorassertionsaffirmingpropositions,andthefactthatonehas 

judgedorassertedthemastruecannotconstituteagroundfortheconclusion,norcanthetruthofthepropositionsaffirmedconstitutesuchaground;atleastnotinthesenseofsomethingthatanagentisinpossessionof,therebybecomingjustifiedinmakingtheassertionexpressedinthe 

conclusion. Itisratherthefact,ifitisafact,thattheagenthasgrounds 

forthepremissesthatisrelevantforherhavingagroundfortheassertion 

madeintheconclusion.Butthegroundsforthepremissesaregroundsfor 

them,notfortheconclusion. ThequestionthatIhaveposedistherefore 

putintheform:giventhegroundsforthepremisses,howdoesonegetfrom 

themagroundfortheconclusion? 

Weareusedtomeetchallengesofaninferencebybreakingitdowninto 

simplersteps,andwhenonesucceedstoreplacetheinferencebyachainof 

sufficientlysimpleinferencesthereisinpracticenomorechallenges. But 

thephilosophicallyinterestingquestionishowonecanmeetachallengeof 

asimpleinferencethatisnotpossibletobreakdownintosimplersteps.It 

istemptingtofallbackatthatpointonwhatourexpressionsmeanorin 

otherwordsonwhatpropositionsitisthatweaffirmtobetrue.However, 

tomymind,itwouldbedubioustosayofalltheseinferencesthatwewant 

todefendbutcannotbreakdownintosimplerinferencesthattheirvalidity 

isjustconstitutiveforthemeaningofthesentencesinvolved. 5 Thelinethat 

5 Insomepreviousworks(e.g.,(Prawitz,1977,1973);cfalsofootnote6)Ihaveidentified 

agroundforajudgementwithaproofofthejudgement,orIhavespokenofgroundsfor


Ishalltakeisinsteadroughlythatthemeaningofasentenceisdetermined 

bywhatcountsasagroundforthejudgementexpressedbythesentence. 

Orexpressedlesslinguistically:itisconstitutiveforapropositionwhatcan 

serveasagroundforjudgingthepropositiontobetrue. Fromthispoint 

ofviewIshallspecifyforeachcompoundformofpropositionexpressible 

infirstorderlanguageswhatconstitutesagroundforanaffirmationofa 

propositionofthatform.Ifonedoesnotlikethislineofapproach,onemay 

anywayagreewithmyspecificationofwhatconstitutesagroundforvarious 

judgements,whichiswhatmattershere. 

Forinstance,aconjunction p&qwillherebeunderstoodasaproposition 

suchthatagroundforjudgingittobetrueisformedbybringingtogether 

twogroundsforaffirmingthetwopropositions pand q. Wemayputthe 

nameconjunctiongrounding,abbreviated &G,onthisoperationofbringing 

togethertwosuchgroundssoastogetagroundforaffirmingaconjunction. 

Ifwedonotwanttotakethisviewofconjunctions,wemaystillagreefor 

otherreasonsthatthereisanoperation &Gsuchthatif βisagroundfor 

affirming pand γisagroundforaffirming q,then &G(β,γ)isaground 

foraffirming p&q. Similarly,wemaytakeitasafurtherconstitutivefact 

aboutconjunctionthatconverselyanygroundforjudgingittobetrueis 

formedbytheoperationofconjunctiongroundingorjustagreetothatfor 

otherreasons. Whatmattershereisthatthereissuchanoperation &G 

suchthatsomethingisagroundforjudging p&qtobetrueifandonlyifit 

canbeformedbyapplying &Gtotwogroundsforjudging ptobetrueand 

judging qtobetrue,respectively. 

Ihavespokenprimarilyofgroundsforjudgementsorassertions. But 

forbrevity,wemayalsospeakderivativelyofagroundforaproposition p 

meaningagroundforthejudgementorassertionthat pistrue. Wecan 

thusstatetheequivalence 

αisagroundfortheconjunction p&qifandonlyif α = &G(β,γ) 

forsome βand γsuch βisagroundfor pand γisagroundfor q. 

Inferencesaremadenotonlyfrompremissesthathavebeenestablishedas 

holdingbutalsofromassumptionsandpremissesthatareestablishedunder 

assumptions.TocoversuchcasesIshallintroducewhatIshallcallopenor 

unsaturatedgroundsbesidesthegroundsthatwehavetalkedaboutsofar 

andthatIshallcallclosedgrounds.Bothclosedgroundsandunsaturated 

groundswillbesaidtobegrounds. 

Anunsaturatedgroundislikeafunctionandisgivenwithanumber 

ofopenargumentplacesthathavetobefilledinorsaturatedbyclosed 

sentencesandhavetakenthemtobevalidarguments.Iprefernottousethatterminology 

now,becauseIwanttotakeproofstobebuiltupbyinferences,andIdonotwanttosay 

thataninferenceconstitutesagroundforitsconclusion—thequestionisinsteadhow 

aninferencecandeliveragroundfortheconclusion.


groundssoastobecomeaclosedground. Somethingisagroundforan 

assertionof Aundertheassumptions A1,A2,... ,Anifandonlyifitis 

an n-aryunsaturatedgroundthatbecomesaclosedgroundforAwhen 

saturatedbyclosedgroundsfor A1,A2,... ,An.Writing α(ξ1,ξ2,... ,ξn)for 

theunsaturatedgroundand α(β1,β2,... ,βn)fortheresultofsaturatingit 

byclosedgrounds βifor Ai,theconditionfor α(ξ1,ξ2,... ,ξn)tobeaground 

for Aundertheassumptions A1,A2,... ,Anisthusthat α(β1,β2,... ,βn)is 

aclosedgroundfor A. 

Groundsarenaturallytypedbythepropositionstheyaregroundsfor. 

Theopenplacesinanunsaturatedground,inotherwords,thevariablesused 

indisplayingtheunsaturatedground,maythenalsobetypedtoindicate 

thetypeofthegroundsthatcansaturatethematthatplace,inotherwords, 

thatcanreplacethevariables. Ishallusuallysupplyvariableswithtypes 

butshallotherwiseomittypeindications. 

Withthesenotionsathand,wecanspecifythataclosedgroundforan 

implication p → qissomethingthatisformedbyanoperationthatwecan 

callimplicationgrounding, → G,appliedtoa1-aryunsaturatedground 

β(ξ p )forjudging qtobetrueundertheassumptionthat pistrue.Theresultofapplyingthisoperationtotheopenground 

β(ξ p ),whichIshallwrite 

→ Gξ p (β(ξ p )),yieldsthusaclosedgroundfor p → q;itcorrespondsonthe 

syntacticalleveltoavariablebindingoperator,andIindicatethisbywriting 

thevariable ξ p behindtheoperator.Ifitisappliedtoan n-aryunsaturated 

groundfor Aundertheassumptions A1,A2,... ,Anwritten α(ξ1,ξ2,...,ξn), 

Ishallwrite → Gξi(α(ξ1,ξ2,...,ξn))toindicatethatitisthe i-thplacein 

theunsaturatedgroundthatbecomesbound,whichthendenotesanunsaturatedgroundfor 

Aundertheassumptions A1,A2,... ,Ai−1,Ai+1,...,An. 

Wehavethustheequivalence 

αisagroundfor p → qifandonlyif α = → Gξ p (β(ξ p )) 

where β(ξ p )isanunsaturatedgroundforjudgingthat qistrue 

undertheassumptionthat pistrue. 

Finallywehavetopayattentiontothefactthatthepremissesofaninferencemaybeanopenjudgement 

A(x1,x2,... ,xm)(possiblyundersome 

openassumptions),bywhichImeanthatitskernelisnotaproposition,but 

apropositionalfunction p(x1,x2,...,xm)definedforadomainofindividualssuchthatforany 

n-tupleofindividuals a1,a2,... ,am, A(a1,a2,... ,am) 

isthejudgementthataffirms p(a1,a2,... ,am).Wemustthereforeconsider 

unsaturatedgroundsthatareunsaturatednotonlywithrespecttogrounds 

butalsowithrespecttoindividualsthatcanappearasargumentsinpropositionalfunctions.Let 

A(x1,x2,...,xm)and Ai(x1,x2,... ,xm)beassertions 

whosepropositionalkernelsarepropositionalfunctionsover x1,x2,... ,xm, 

andlet A(a1,a2,... ,am)and Ai(a1,a2,... ,am)betheassertionsthatarise


whenweapplythecorrespondingpropositionalfunctionstotheindividuals 

a1,a2,...,am. ThenIshallsaythatsomethingisanunsaturated 

groundfortheopenjudgement A(x1,x2,...,xm)undertheassumptions 

A1(x1,x2,...,xm), A2(x1,x2,...,xm),... ,An(x1,x2,... ,xm)ifandonlyif 

itisanunsaturatedground α(ξ1,ξ2,...,ξn,x1,x2,... ,xm)withrespectto n 

closedgroundsand mindividualssuchthatwhensaturatedbytheindividuals 

a1,a2,... ,amitbecomesanunsaturatedgroundfor A(a1,a2,... ,am) 

undertheassumptions A1(a1,a2,... ,am),A2(a1,a2,...,am),... ,An(a1,a2, 

...,am). 

Wecanthenspecifythataclosedgroundforageneralizedproposition 

∀xp(x)issomethingthatisformedbyanoperationthatIshallcalluniversal 

grounding, ∀G,appliedtoanunsaturatedground α(x)forthepropositional 

function p(x). Theresultofapplyingthisoperationtotheopenground 

α(x),whichIshallwrite ∀Gx(α(x)),againindicatingthat xbecomesbound 

bywritingitbehindtheoperator,isthusaclosedgroundfor ∀xp(x).We 

havethustheequivalence 

αisagroundfor ∀xp(x)ifandonlyif α = ∀Gx(β(x) 

where β(x)isanunsaturatedgroundfor p(x). 

Ifweidentifynegatedpropositions, ¬p,with (p → ⊥)where ⊥isa 

constantforfalsehood,forwhichitisspecifiedthatthereisnogroundfor ⊥, 

wehavespecifiedbyrecursionwhatcanbeagroundforsufficientlymany 

formsofpropositionsexpressibleinclassicalfirstorderlanguages,except 

thatwehavesaidnothingaboutgroundsforatomicpropositions. What 

theyarewillofcoursevarywiththecontentoftheatomicpropositions. 

InthelanguageoffirstorderPeanoarithmeticwemaytakeagroundfor 

anidentitybetweentwonumericalterms t = utobeacalculationofthe 

valueof tand ushowingthattheyarethesame.Alternatively,ifwewantto 

analyseacalculationasconsistingofstepseachofwhichhasaground,we 

needtostartfrommorebasicgrounds.Asalreadysaid,allgroundscannot 

beobtainedbyinferences.Theremustinotherwordsbesomepropositions 

like t = tor‘0isanaturalnumber’forwhichitisconstitutivethatthereare 

specificgroundsforthemthatarenotderivedorbuiltupfromsomething 

else. 

Outsideofmathematics,wemayconsiderobservationstatements,and 

forthem,Isuggest,wetakerelevantverifyingobservationstoconstitute 

grounds. Forinstance,agroundforaproposition‘itisraining’istaken 

toconsistinseeingthatitrains;taking“seeing”inaveridicalsense,it 

constitutesaconclusiveground.Itdoesnotseemunreasonabletosaythat 

toknowwhatpropositionisexpressedby“itisraining”istoknow,orat 

leastimpliesthatoneknows,howitlookswhenitisraining,andhencethat 

oneknowswhatconstitutesagroundforthestatement.


Inthecaseofintuitionisticpredicatelogicwehavetosayinadditionwhat 

countsasagroundfordisjunctionsandexistentialpropositions,whichcan 

bedoneinanobviouswayanalogouslytothecaseofconjunction(butwhich 

becomestoorestrictivewhendisjunctionsandexistentialpropositionsare 

understoodclassically—theseformshavetheninsteadtobedefinedin 

termsofotherlogicalconstantsintheusualway). 

ThegroundsthatIhavedescribedareabstractentitiesthatcanbeconstructedinthemindandthatwecanbecomeinpossessionofinthatway.Alternatively,wemaythinkofagroundforajudgementasjustarepresentationofthestateofourmindwhenwehavejustifiedajudgement. 

Thepossessionofagroundforajudgementcanmanifestitselfinthe 

namingofthatobject,andIhaveintroducedanotationfordoingso. An 

alternativewayofdefiningthegroundswouldhavebeentolaydownthese 

waysofdenotinggroundsasthecanonicalnotationforgrounds,making 

adistinctionbetweencanonical andnon-canonical formssincethesame 

groundmaybedenotedbydifferentexpressions. Tobeinpossessionofa 

groundforajudgementcouldthenbeidentifiedwithhavingconstructeda 

termthatdenotesagroundforthatjudgement. 

7 Inferences 

Asthereaderhasalreadyrealized, theprimitiveoperationsintroduced 

abovetospecifythegroundsfortheaffirmationofpropositionsofvarious 

formscorrespondtocertaininferencerules,namelyGentzen’sintroduction 

rulesinthesystemofnaturaldeductionforfirstorderlanguages. Forinstance,conjunctiongroundingcorrespondstotheschemaforconjunction 

introduction. GerhardGentzensawtheintroductionrulesasdetermining 

themeaningofthecorrespondinglogicalconstants. Ihavenotfollowed 

thatideahere, 6 buthaveinsteadseenthespecificationsofwhatconstitutes 

groundsforaffirmingpropositionsofacertainformtobeconstitutivefor 

propositionsofthatform.OnecansaythatIhavecarriedoverGentzen’s 

ideatothedomainofgrounds,sincethegroundsarebuiltupbyprimitive 

groundingoperationsthatcloselycorrespondtohisintroductionrules.More 

precisely,itholdsforeverysuchgroundingoperation Φthatifweforman 

inferenceaccordingtothecorrespondinginferencerule,thenwecanapply 

Φtogroundsforthepremissesofthatinferenceandshallgetasaresulta 

groundfortheconclusionoftheinference. Furthermore,havingdefineda 

domainofgroundsbypresumingtheseprimitivegroundingoperations,we 

candefineotheroperationsonthegroundsofthisdomainthatwillhave 

6 Insomeotherworks(see,e.g.,(Prawitz,1973)andcf.footnote5)IhaveusedGentzen’s 

ideamoredirectlyinadefinitionofvalidargument,sayingthatanargumentwhoselast 

inferenceisanintroductionisvalidifandonlyiftheimmediatesubargumentsarevalid.


similarpropertywithrespecttootherinferences.Thismakesitpossibleto 

givesomesubstancetotheideathataninferenceissomethingmorethan 

juststatingaconclusionandreasonsforit,anideathatwedescribedabove 

(endofSection3)inmetaphoricaltermsas“seeing”thattheproposition 

affirmedintheconclusionistruegiventhatthepropositionsaffirmedinthe 

premissesaretrue.Thementalactthatisperformedinaninferencemay 

berepresented,Isuggest,asanoperationperformedonthegivengrounds 

forthepremissesthatresultsinagroundfortheconclusion,wherebywe 

seethatthepropositionaffirmedistrue. 

Toillustratetheidealetusconsideraninferencethatisvalidbutnot 

logicallyvalid,sayacaseofmathematicalinduction. Howdoweseethat 

itsconclusion,theinductionstatement, A(x)say,istrueforanynatural 

number n? Isitnotreasonabletosaythatweseethisbyoperatingon 

thegivengroundsfortheinductionbaseandtheinductionstep?Westart 

withthegivengroundfortheinductionbase A(0)andthensuccessively 

applythegroundfortheinductionstep.Intheinductionstepwearriveat 

asserting A(n + 1)undertheinductionassumption A(n),anditsgroundis 

thusanunsaturatedgroundthatbecomesaclosedgroundfor A(n+1)when 

saturatedwith nandaclosedgroundfor A(n).Werealizethatbyapplying 

orsaturatingthisground ntimesbythenaturalnumbers 0,1,... ,n − 1, 

andthegroundsthatwesuccessivelyobtainfor A(0),A(1),... ,A(n − 1), 

wefinallygetinpossessionofagroundfor A(n),whichstatementisthus 

seentohold. 

Inaccordancewiththisidea,Ishallseeanindividualinferenceactas 

individuatedbyatleastthefollowingfiveitems(forbrevityIleaveout 

otheradditionalitemsthatmaybeneededtoindividuateaninferencesuch 

ashowhypothesesaredischarged): 

1.anumberofpremisses A1,A2,...,An, 

2.grounds α1,α2,...,αn, 

3.anoperation Φapplicabletosuchgrounds, 

4.aconclusion B,and 

5.anagentperformingtheoperationataspecificoccasion. 

Inlogicweareusuallynotinterestedinindividualactsofthiskindand 

thereforeabstractawayfromtheagent,whichleavesthefouritems1–4 

individuatingwhatIshallrefertoasan(individual)inference. Tomake 

orcarryoutsuchaninferenceistoapplytheoperation Φtothegrounds 

α1,α2,... ,αn. 

Idefineanindividualinferenceindividuatedby1–4tobevalidif α1, 

α2,... ,αnaregroundsfor A1,A2,... ,An,respectively,andtheresultof


applyingtheoperation Φtothegrounds α1,α2,... ,αn,thatis Φ(α1,α2,... , 

αn),isagroundfor B. 

Accordingtothisdefinitionanindividualconjunctionintroduction,given 

withtwopremissesaffirmingthepropositions p1and p2,grounds α1and α2 

forthem,theoperationconjunctiongrounding &G,andtheconclusionaffirmingtheproposition 

p1&p2,istriviallyavalidinferencesince &G(α1,α2) 

isbydefinitionagroundforaffirming p1&p2,giventhat αiisagroundfor 

affirming pi. 

Ifweintroducetwooperations &R1and &R2definedforgroundsforaffirmations 

of propositions of conjunctive form by the equations 

&Ri(&G(α1,α2)) = αi(i = 1or 2),thenanindividualinferenceofthe 

typeconjunctionelimination,givenbyapremissaffirmingaconjunction 

p1&p2,aground αforit,anoperation &Ri,andaconclusionaffirming pi, 

isvalid,sincetheground αforthepremissmustbeoftheform &G(α1,α2) 

where αiisagroundfor pi,andsincetheground αiisbydefinitionthe 

valueoftheoperation &Riappliedto &G(α1,α2). 

Oftenwealsoabstractawayfromthegroundsandfromanyspecific 

premissesandconclusionofaninference,preservingonlyacertainformal 

relationbetweenthem.Wecanthenspeakofaninferenceformdetermined 

onlybythisformalrelationandanoperation Φ.Forinstance,modusponens 

maynowbeseenassuchaninferenceform,individuatedbygivingan 

operation Φ,namelytheoperation → Rdefinedbelow,andbysayingthat 

oneofthepremissesisaffirmingapropositionoftheformofanimplication 

p → qwhiletheotherpremissaffirmstheproposition pandtheconclusion 

affirmstheproposition q. Ifwealsoabstractawayfromtheoperation Φ, 

wegetwhatwemaycallaninferenceschema. 

Ishallsaythatsuchaninferenceformisvalidwhenitholdsforany 

instanceoftheformwithpremisses A1,A2,... ,An,andconclusion Band 

forallgrounds α1,α2,... ,αnfor A1,A2,... ,Anthattheresult Φ(α1,α2, 

...,αn)ofapplyingtheoperation Φinquestionto α1,α2,... ,αnisaground 

for B. Aninferenceschemaisvalidifitcanbeassignedanoperation Φ 

suchthattheresultinginferenceformisvalid. 

Forinstance,modusponensasusuallyunderstoodwithoutspecifyingan 

operationisaninferenceschema,whichisvalid,becausebyassigningtoit 

theoperation → Rdefinedbytheequation 

→ R( → Gξ p (β(ξ p ),α) = β(α), 

wegetavalidinferenceform. Intheequationabove β(α)istheresultof 

saturating β(ξ A )by α.Toseethattheresultinginferenceformisvalid,we 

havetoseethattheresultofapplyingtheoperation → Rtothegrounds 

forthepremissesofaninferenceofthisformisagroundfortheconclusion 

ofthatinference.Supposethat γand αaregroundsforpremissesofthat


inferenceandthatthepremissesareaffirmingthatthepropositions p → q 

and paretrue.Thenbyhowgroundsforimplicationshavebeenspecified, 

γisoftheform → Gξ p (β(ξ p )),where β(ξ p )isanunsaturatedgroundfor 

affirmingthat qistrueundertheassumptionthat pistrue. Thismeans 

thatif β(ξ p )issaturatedbyagroundforaffirmingthat pistrue,theresult 

isagroundfortheaffirmationthat qistrue.Now αisagroundforaffirming 

that pistrue,hence β(α)isagroundforaffirmingthat qistrue,which 

affirmationistheconclusionoftheinference. And β(α)istheresultof 

applyingtheoperation → Rtothegivengroundsforthepremissesofthe 

inferenceaccordingtothedefinitionof → R. 

8 Conclusion 

Itshouldnowbeclearthatiftheconceptsofinference,makinganinference, 

validityofinference,andgroundareunderstoodinthewaydevelopedhere, 

thequestionthatwestartedwithiseasilyanswered.Thegeneralquestion 

washowandwhyweacquireknowledgebymakinginferences,andthis 

wasmorepreciselyformulatedastheproblemtostatetheconditionsunder 

whichanagent Pgetsagroundforajudgementbyinferringitfromother 

judgements.Giventhat 

andthat 

Jisavalidinference 

fromjudgements A1,A2,... ,Antoajudgement B, (a) 

theagent Phasgrounds α1,α2,...,αnfor A1,A2,... ,An, (b) 

theproblemwastostateathirdcondition(c),describingwhatrelation P 

hastohavetotheinference Jinorderthatitshouldfollowfrom(8)–(c) 

that 

Phasorgetsagroundfor B. (d) 

Whenanindividualinferenceisindividuatednotonlybyitspremisses 

andconclusionbutalsobygroundsforthepremissesandanoperation 

applicabletothem,andwhenmakinganinferenceisunderstoodasapplying 

thisoperationtothegrounds,inotherwords,astransformingthegiven 

groundsforthepremissestoagroundfortheconclusion,itbecomespossible 

tostatethethirdconditionthatwehavesoughtforsimplyas 

Pmakestheinference J. (c) 

Istartedoutfromtheconvictionthatthequestionwhyanagentgetsa 

groundforajudgementbyinferringitfrompremissesforwhichshealready


hasagroundshouldbeeasytoanswer,oncetheconceptsinvolvedare 

understoodinanappropriateway.Thisisnowactuallythecase.Whatit 

meansforaninference Jtobevalid,asithasnowbeendefined,issimply 

thattheoperation Φthatcomeswiththeinference Jyieldsagroundforthe 

conclusion Bwhenappliedtothegrounds α1,α2,... ,αnforthepremisses 

A1,A2,... ,An—inshort,that Φ(α1,α2,...,αn)isaground B.Therefore, 

bymakingtheinference J,thatis,byapplyingtheoperation Φtothegiven 

grounds,theagentgetsinpossessionofagroundfortheconclusion. 

Itremainstosaysomethingaboutwhatitisforanagenttobeinpossessionofagroundfortheconclusion. 

Asalreadysaidabove,itmeans 

basicallytohavemadeacertainconstructioninthemindofwhichthe 

agentisaware,andwhichshecanmanifestbynamingtheconstruction. 

Regardlessofwhethertheconstructionisonlymadeinthemindorisdescribed,itwillbepresenttotheagentundersomedescription,whichwill 

normallycontaindescriptionsofanumberofoperations.Itispresupposed 

thattheagentknowstheseoperations,whichmeansthatsheisabletocarry 

themout,whichinturnmeansthatsheisabletoconvertthetermthat 

describesthegroundtocanonicalform. Furthermoretheagentispresupposedtounderstandtheassertionthatshemakesandhencetoknowwhat 

kindofgroundsheissupposedtohaveforit.Itfollowsthatwhenanagent 

hasgotinpossessionofagroundforanjudgementbymakinganinference, 

sheisawareofthefactthatshehasmadeaconstructionthathastheright 

canonicalformtobeagroundfortheassertionthatshemakes. 

However,itdoesnotmeanthattheagenthasprovedthattheconstructionshehasmadeisreallyagroundforherassertion.Aswehavealready 

discussed(Section4),thiscannotbearequirementforherjudgementto 

bejustified.Butiftheinferenceshehasmadeisvalid,thensheisinfact 

inpossessionofagroundforherjudgement,andthisisexactlywhatis 

neededtobejustifiedinmakingthejudgement,ortobesaidtoknowthat 

theaffirmedpropositionistrue. Furthermore,althoughitisnotrequired 

inorderforthejudgmenttobejustified,byreflectingontheinferenceshe 

hasmade,theagentcanprovethattheinferenceisvalid,ashasbeenseen 

inexamplesabove. 

DagPrawitz 

DepartmentofPhilosophy,StockholmUniversity 

10691Stockholm,Sweden 

dag.prawitz@philosophy.su.se 

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A Sound and Complete Axiomatic System of 

bdi–stit Logic 


Caroline Semmling Heinrich Wansing 

In(Semmling&Wansing,2008),bdi–stitlogichasbeenmotivatedand 

introducedsemantically.Thislogiccombinesthebelief,desire,andintention 

operatorsfromBDIlogic(Georgeff&Rao,1998;Wooldridge,2000)with 

theactionmodalitiesfrom d stitlogic,themodallogicofdeliberatively 

seeingtoitthat(Belnap,Perloff,&Xu,2001),(Horty&Belnap,1995). 

Themulti-modalbdi–stitlogicisanexpressivelyrichlogic,whichallows 

aformalanalysisof,forexample,reasoningaboutdoxasticdecisionsand 

beliefrevision,see(Semmling&Wansing,2009),(Wansing,2006a). 

In(Semmling&Wansing,2009),wehavepresentedasoundandcomplete 

tableaucalculusforbdi–stitlogicbasedonthetableaucalculusfor d stit 

logicdefinedin(Wansing,2006b). Inthepresentpaperweintroducea 

soundandcompleteaxiomatizationofbdi–stitlogicandprovedecidability 

byestablishingthefinitemodelproperty. 

2 Syntaxandsemantics 

Thesyntaxofbdi–stitlogic 

Thelanguageofbdi–stitlogiccomprisesadenumerablesetofsentential 

variables (p1,p2,p3,... ),theconstants ⊥, ⊤,theconnectivesofclassical 

propositionallogic(¬, ∧, ∨, ⊃, ≡),andthemodalnecessityandpossibility 

operators ✷and ✸.Weassumethat ✸isdefinedas ¬✷¬.Thisvocabularyis 

supplementedbyactionmodalitiesandoperatorsusedtoexpressthebeliefs, 

desiresandintensionsofarbitrary(rational)agents.Additionally,thereis 

apossibilityoperator�takenoverfrom(Semmling&Wansing,2008).We 

alsoassumeadenumberablesetofagentvariables (α1,α2,...,αn,...).

202 CarolineSemmling&HeinrichWansing 

Definition14(bdi–stitsyntax). 1.Everysententialvariable p1,p2,... 

andeachconstant ⊥, ⊤isaformula. 

2.If α1, α2areagentvariables,then (α1 = α2)isaformula. 

3.If ϕ, ψareformulasand αisanagentvariable,then ¬ϕ, (ϕ ∧ψ), ✷ϕ, 

�ϕ, αcstit : ϕ, α bel : ϕ, αdes : ϕand α int : ϕareformulas. 

4.Nothingelseisaformula. 

Aformulaconsistingofonlyonesententialvariableoroneconstantis 

calledanatomicformula.Thereadingofaformula α cstit : ϕis“agent α 

seestoitthat ϕ”. In(Semmling&Wansing,2008),insteadofthe cstitoperator,anoperatorofdeliberativelyseeingtoitthat, 

d stit:,isused.We 

introducethe d stit-operatorwiththefollowingequivalence 

α d stit : ϕ ≡ (α cstit : ϕ ∧ ¬✷ϕ). 

Thisisdonebecauseitmakesthepresentationofthecompletenessproof 

easier. Butneverthelessitisalsopossibletouse d stit: asaprimitive 

operatorandtochoosetheaxiomsappropriately,cf.(Belnapetal.,2001). 

Aformula αbel : ϕisreadas“agent αbelievesthat ϕ”or“agent α 

hasthebeliefthat ϕ”.Thereadingsofthedesireoperators α des :andthe 

intentionoperators α int :areconceivedinthisvein,too. 

Thesemanticsofbdi–stitlogic 

Abdi–stitmodelconsistsofaframe F = (Tree, ≤, A,N,C,B,D,I)and 

avaluationfunction v. Theframe F isbasedonabranchingtemporal 

structureasinStit-Theory,(Belnap&Perloff,1988).Theset Treeisanonemptyset(ofmomentsoftime)and 

≤isapartialorder,whichisreflexive, 

transitivebutacyclic,suchthateverymoment m ∈ Treehasaunique 

predecessor.Thus,thesetofhistories H,definedasthesetofallmaximal 

linearlyorderedsubsetsof Tree,andthesetofsituations S = {(m,h)|m ∈ 

Tree,h ∈ H}oftheframeresultfromtheorderedset (Tree, ≤).Thesetof 

historiespassingthroughmoment m ∈ Tree({h | m ∈ h,h ∈ H})isdenoted 

by Hm.Thedenumerable,non-emptyset Aisthesetofagents,and Cis 

afunctionthatmapseverypairof A × Treetoasetofdisjointsubsets 

ofhistoriespassingthrough m,suchthattheunionofallsubsetsis Hm. 

Thus, C(α,m) = C α m 1 definesanequivalencerelationon Hm. Histories 

hand h ′ aresaidtobechoice-equivalent foragent αatmoment m,if 

theybelongtothesamesetin C α m. Theequivalenceclassofanarbitrary 

1 Onthisaccountwedonotexplicitlydistinguishbetweenthevariablesandtheagents 

anddenotebothby α, α1, . . . , αn, . . ..Thecontextwilldisambiguate.

ASoundandCompleteAxiomaticSystemofbdi–stitLogic 203 

history hinmoment misdenotedby Cα m (h) = Cα � 

(m,h) 

Cα � 

(m,h) |h ∈ Hm representsalldistinguishablechoicecellsofagent αat 

situation (m,h). 

Thefunction N : S → P(P(S))assignsaset Nsofnon-emptysubsetsof 

situationstoeverysituation s.Theset Nsiscalledaneighbourhoodsystem 

of s.Itselementsarecalledneighbourhoodsof s.Wealsodenoteby Nthe 

unionofallneighbourhoodsystems, N = {U|U ∈ Ns,s ∈ S}.Thecontext 

willdisambiguate. 

Thefunctions Band Daremappingsfrom A × Sto P(N).Aset U ∈ 

B(α,s) = Bα scanberegardedasaneighbourhoodendorsingcertainbeliefs ofagent αatsituation s.Inthesameway,everyset U ∈ D(α,s) = Dα s is 

aneighbourhoodendorsingcertaindesires. 

Tointerprettheascriptionofintentionstoanagentinasituation s,we 

usethefunction I,whichmapsapair (α,s) ∈ A × Stoaneighbourhood 

I(α,s) = Iα s ∈ Nrepresentingallsituationscompatiblewithwhat αintends 

at s. 

Let F = (Tree, ≤, A,N,C,B,D,I)besuchaframeandlet Selectmbe 

thesetofallfunctions σfrom Aintosubsetsof Hm,suchthat σ(α) ∈ Cα m . 

Fsatisfiestheindependenceofagentsconditionoftheiractions,ifandonly 

ifforevery m ∈ Tree, � 

α∈Agent 

σ(α) �= ∅ 

. Thesetofclasses 

forevery σ ∈ Selectm. 

Apair M = (F,v)isthensaidtobeabdi–stitmodelbasedontheframe 

F,where visavaluationfunctionon F,whichmapstheagentvariables 

intotheset Aof Fandthesetofatomicformulasintothepowersetof 

situations P(S)of Fwiththeconstraintsthat v(⊥) = ∅and v(⊤) = S. 

Satisfiabilityofaformulainabdi–stitmodel Misthendefinedasfollows, 

where,forabbreviation,wedenoteforanarbitraryformula ϕby �ϕ�the 

setofsituationswhichcontaineverysituationof Msatisfyingformula ϕ; 

�ϕ� = {s|M,s |= ϕ}. 

Definition15(bdi–stitsemantics).Let s = (m,h)beasituationinmodel 

M = (F,v),let α, α1, α2beagentvariables,andlet ϕ, ψbeformulas 

accordingtoDefinition14.Then: 

M,s |= ϕ iff s ∈ v(ϕ),if ϕisanatomicformula. 

M,s |= (α1 = α2) iff v(α1) = v(α2). 

M,s |= ¬ϕ iff M,s �|= ϕ. 

M,s |= (ϕ ∧ ψ) iff M,s |= ϕandM,s |= ψ. 

M,s |= ✷ϕ iff M,(m,h ′ ) |= ϕforall h ′ ∈ Hm.


M,s |=�ϕ iff thereexists U ∈ Nswith U ⊆ �ϕ�. 

M,s |= α cstit : ϕ iff {(m,h ′ )|h ′ ∈ C v(α) 

s } ⊆ 

M,s |= α int : ϕ iff I v(α) 

s 

⊆ �ϕ�. 

M,s |= α des : ϕ iff thereexists U ∈ D v(α) 

s 

M,s |= α bel : ϕ iff thereexists U ∈ B v(α) 

s 

{(m,h ′ )|M,(m,h ′ ) |= ϕ,h ′ ∈ Hm}. 

with U ⊆ �ϕ�. 

with U ⊆ �ϕ�. 

Obviously,theoperators�, α des :,and αbel :arenotdefinedbyarelationalsemantics,butbyamonotonicneighbourhood(aliasScott-Montague, 

aliasminimalmodels)semantics,cp.(Chellas,1980;Montague,1970;Scott, 

1970).Forsuchanoperator op,aformula op ϕ ∧op ¬ϕissatisfiableforany 

contingentformula ϕ.Notethatitisnotpossibletosatisfy op ϕforaninconsistent 

ϕinabdi–stitmodel,becauseeveryneighbourhoodisnon-empty. 

Aneighbourhoodsemanticsofanoperator nopisinuse,if M,s |= nopϕ 

iff �ϕ� ∈ Ns. Usually,inthissemantics nopisinterpretedasakindof 

necessity-operator. Sinceforsuchanoperatoritisalsopossibletosatisfyformulas 

nop ϕ ∧ nop ¬ϕ,we,however,prefertoreadtheoperator� 

asakindofpossibility-operator,andcallitneighbourhoodpossibility. A 

formula�ϕistrueatasituation,if ϕiscognitivelypossibleatsituation s. 

Onemaywonderaboutthemeaningofthedualoperator ¬ op ¬ϕ.The 

semanticstellsusthataformula ¬ op ¬ϕissatisfiedatasituation s,if 

eachneighbourhoodof snecessarilycontainsasituation s ′ satisfying ϕ. 

Butformulassuchas ¬ op ¬ϕ ∧ ¬ opϕarealsosatisfiable,forexample, 

ifeveryneighbourhoodcontainsatleasttwosituations,onesatisfying ϕ 

andanothersatisfying ¬ϕ,sothatthedualoperatordoesnotexpressa 

kindofneighbourhoodnecessity,too. Howcanweexpressnecessityina 

neighbourhoodsemantics? Ourproposalis: s |= ⊡ϕiffforevery U ∈ Ns, 

U ⊆ �ϕ�(iff ϕisacognitivenecessityatsituation s). Thenobviouslyit 

holdsthattheimplications�ϕ ⊃ ¬ ⊡ ¬ϕand ⊡ϕ ⊃ ¬�¬ϕarevalid,but 

theimplicationsintheotherdirectionfail. Thus,itispossibletosatisfy 

formulasoftheform ¬ ⊡ ¬ϕ ∧ ¬�ϕ. 

Inadditiontoneighbourhoodnecessityandpossibility,therearemodal 

operatorsnotrelatedtothecognitivepropositionalattitudesofagents: ✷ 

and ✸. Theseoperatorscanbereadasoperatorsofhistoricalnecessity 

andpossibility,respectively,wherehistoricalpossibilityandnecessityare 

definedasdualoperators: ✸ϕ ≡ ¬✷¬ϕ.Theyareadoptedfrom(Belnap 

&Perloff,1988;Belnapetal.,2001).


3 Axiomatization 

Sincethebdi–stitlogicisconstructedon d stitframes,cf.(Belnap&Perloff, 

1988;Belnapetal.,2001),withtheadditionofsomefunctionsasinScott– 

Montaguemodels,cf. (Chellas,1980;Montague,1970;Scott,1970),the 

axiomatizationisnottoodifficult. Weassumeacompleteaxiomatization 

ofthenon-modalpropositionallogicandaddthefollowingaxioms: 

(A1) ✷ϕ ⊃ ϕ, ¬✷ϕ ⊃ ✷¬✷ϕ, ✷(ϕ ⊃ ψ) ⊃ (✷ϕ ⊃ ✷ψ). 

(A2) αcstit : ϕ ⊃ ϕ, ¬α cstit : ϕ ⊃ α cstit : ¬αcstit : ϕ, 

αcstit : (ϕ ⊃ ψ) ⊃ (α cstit : ϕ ⊃ αcstit : ψ). 

(A3) ✷ϕ ⊃ α cstit : ϕ. 

(A4) α = α, (α = β) ⊃ (β = α), ((α = β) ∧ (β = γ)) ⊃ (α = γ). 

(A5) (α = β) ⊃ (ϕ ⊃ ϕ[α/β]). 2 

(AIAk) (∆(β0,... ,βk) ∧ ✸β0 cstit : ψ0 ∧ ... ∧ ✸βk cstit : ψk) ⊃ 

⊃ ✸(β0 cstit : ψ0 ∧ ... ∧ βk cstit : ψk). 

Theaxioms(AIAk)representtheindependenceofagentsconditionfor k ∈ 

Nagents. Theformula ∆(β0,...,βk)statesthat β0,...,βkarepairwise 

distinct.Wealsohaveseveralderivationrules,cf.(Belnapetal.,2001): 

(RN) ϕ/✷ϕ, 

(MP) ϕ,ϕ ⊃ ψ/ψ,, 

(APCn) [✸αcstit : ϕ1 ∧ ✸(α cstit : ϕ2 ∧ ¬ϕ1) ∧ ...∧ 

✸(α cstit : ϕn ∧ ¬ϕ1 ∧ ... ∧ ¬ϕn−1)] ⊃ (ϕ1 ∨ ... ∨ ϕn). 

Bytheaxiomof npossiblechoices(APCn),itisassuredthateveryagent 

hasatmost ndifferentalternativestoact.Iftheaxiom(APCn)isaccepted, 

theresultinglogicisdenotedby Ln.Evidently,itholdsthat Ln+1 ⊆ Ln. 

Thenewbdioperatorsareaxiomatizedbythefollowingaxiomsand 

derivationrulestakenoverfrom(Chellas,1980). 

(Di) α int : ϕ ⊃ ¬αint : ¬ϕ, 

(F,Fb,Fd,Fi) ¬�⊥, ¬αbel : ⊥, ¬αdes : ⊥, ¬αint : ⊥, 

(RM) (ϕ ⊃ ψ)/(�ϕ ⊃�ψ), 

2 Note,thatthesubstitutiondoesnothavetobeuniform.Itispossible,toreplacesome 

oralloccurrencesof αwith β.


(RMi) (ϕ ⊃ ψ)/(α int : ϕ ⊃ αint : ψ), 

(RMb) (ϕ ⊃ ψ)/(α bel : ϕ ⊃ α bel : ψ), 

(RMd) (ϕ ⊃ ψ)/(α des : ϕ ⊃ α des : ψ). 

Fromtheseaxioms,whichareproventobecompleteincombinationwiththe 

axiomsofthe d stitlogicinSection4,wecanderivethefollowingtheorems, 

whichstatethemonotonyoftheneighbourhoodoperators,andformthe 

typicalaxiomsoftherelationallydefinedones. 

(Ni) α int : ⊤, 

(Tc) αcstit : (ϕ ∧ ψ) ≡ (α cstit : ϕ ∧ α cstit : ψ), 

(Ti) αint : (ϕ ∧ ψ) ≡ (α int : ϕ ∧ α int : ψ), α int : ϕ ∨ α int : ψ) ⊃ α int : 

(ϕ ∨ ψ), 

(T)�(ϕ ∧ ψ) ⊃ (�ϕ ∧�ψ), 

(Tb) α bel : (ϕ ∧ ψ) ⊃ (α bel : ϕ ∧ αbel : ψ), 

(Td) α des : (ϕ ∧ ψ) ⊃ (α des : ϕ ∧ α des : ψ) 

4 Completenessanddecidability 

Sincebdi–stitlogicisbasedon d stitlogic,whichisdecidable,andsinceit 

issupplementedwithsomeoperators,whichareinterpretedasindecidable 

classicalmodallogicswithaneighbourhoodsemantics,itisnotsurprising 

thatalsobdi–stitlogicisdecidable.Wefirstshowthecompletenessofthe 

axiomatizationpresentedinSection3,byextendingtheconstructionofa 

canonicalBT+AC(agentsandchoicesinbranchingtime)structureof d stit 

logic,presented,forexample,in(Belnapetal.,2001),totheconstructionof 

aframeofacanonicalbdi–stitmodel.Subsequently,weshowthatbdi–stit 

logichasthefinitemodelproperty,i.e.,eachnon-theoremof Lnisfalsifiable 

inafinitebdi–stitmodel,bydoingthesameasin(Belnapetal.,2001)for 

d stitlogic.Sincethenumberofaxiomschemesandderivationrulesisalso 

finite,thedecidabilityofbdi–stitlogicensues. 

Completeness 

ThesoundnessofthesystemofaxiomsandderivationrulesofSection3is 

straightforwardandfor(APCn)and(AIAk)asadducedin(Belnapetal., 

2001). Thus,thissectiondealsonlywiththecompletenessoftheaxiomatization. 

Therefore,weintendtocombinetheconstructionofacanonical


modelfor d stitlogic,representedin(Belnapetal.,2001),andtheconstructionofacanonicalmodelofaclassicalmonotonicmodallogic,cf.(Chellas, 

1980).Sincetheconstructionofthecanonical d stitmodelhasamorecomplicatedstructure,thisconstructionconstitutesthebasisandweexpand 

itappropriatelytocomprisetheinterpretationofthebelief,desire,and 

intentionoperators. 

WewillpresenttheconstructionofthecanonicalBT+ACstructure 

whichwasdefinedbyMingXu,cf.(Belnapetal.,2001;Xu,1994,1998).But 

first,propertiesoftheset WLnofmaximal Ln-consistentsetsofformulas, 

relationsonthisset WLnaswellasonsubsets X, Wofitandonsetsof 

agentvariablesarestatedinalmostthesamemannerasforthe d stitlogic 

Ldmn.Thesubscript nindicatesthattheaxiom(APCn)isincluded. 

Foragivensubset W ⊆ WLnwedefinearelation ∼ =Wonasetofagent 

variables,bystipulatingthat α1 ∼ =W α2,ifonlyif, α1 = α2 ∈ wforall 

w ∈ W. 

Lemma3.Therelation ∼ =Wisanequivalencerelationforany W ⊆ WLn. 

Proof.Cf. (Belnapetal.,2001). Thepropertyofbeinganequivalence 

relationresultsfromtheaxioms(A4),whichcorrespondtoreflexivity,symmetryandtransitivity. 

Theotherwayaround,wedefinearelationon X ⊆ WLnbyasetof 

agentvariables A: w ∼ =A w ′ ,ifandonlyif, α1 = α2 ∈ wiff α1 = α2 ∈ w ′ 

forall α1,α2 ∈ A. 

Lemma4.Therelation ∼ =Aisanequivalencerelationonanarbitraryset 

W ⊆ WLnforanyset Aofagentvariables. 

Proof.Itisself-evident. 

Forthenexttwolemmaswefixanarbitrarysubset W ⊆ WLn. Then, 

therelation R ⊆ W × Wisdefinedby wRw ′ iff {φ|✷φ ∈ w} ⊆ w ′ . 

Lemma5.Therelation R ⊆ W × Wisanequivalencerelation. 

Proof.Cf. (Belnapetal.,2001). Thepropertyofbeinganequivalence 

relationresultsfromaxioms(A1),since ✷φ ⊃ φcorrespondstoreflexivity 

and ¬✷φ ⊃ ✷¬✷φtoeuclidity. 

Therefore,wecanpartition Wintoequivalenceclasses {Xi}i∈Iwithrespectto 

R. Let Xbeanarbitraryelementof {Xi}i∈I. Forsuchasubset 

X ⊆ Witholds,thatif (α = β) ∈ wforsome w ∈ X,itfollowsbyRule(RN) 

andAxiom(A5)that ✷(α = β) ∈ w,suchthat (α = β) ∈ w ′ forall w ′ ∈ X. 

Thatwarrantstheuseoftheequivalenceclasses {βj}j∈J = {[α]X}of ≡ X


insteadofagentvariablestodefinethefollowingrelations Rβj ⊆ X × Xfor 

all j ∈ J; 

wRβjw′ iff {φ|βj cstit : φ ∈ w} ⊆ w ′ . 3 

Lemma6.For X ∈ {Xi}i∈Itherelation Rβj ⊆ X × Xisanequivalence 

relationforall j ∈ J. 

Proof.Cf. (Belnapetal.,2001). Thefirstandthesecondaxiomof(A2) 

expressreflexivityandeuclidity,respectively. 

Thisdefinitionoftherelations Rβjdependsontheset X.Inthefollowing,this 

Xwillbeanarbitrarybutfixedequivalenceclassofrelation R. 

Wedenoteby Eβjthesetofallequivalenceclassesofrelation Rβjon X. 

Lemma7.Let X, {βj}j∈J, R, Rβj , Eβj forall j ∈ Nbegivenbythe 

definitionsabove.Thenitholdsforall w ∈ X, φ: 

(i) ✷φ ∈ w iff φ ∈ w ′ forall w ′ ∈ X iff ✷φ ∈ w ′ forall w ′ ∈ X. 

(ii) βj cstit : φ ∈ wiff φ ∈ w ′ forall w ′ with wRβj w′ iff βj cstit : φ ∈ w ′ 

forall w ′ with wRβj w′ . 

(iii) βj d stit φ ∈ wiff φ ∈ w ′ forall w ′ with wRβj w′ and ¬φ ∈ w ′′ for 

some w ′′ ∈ X. 

(iv)Assume ftobeanarbitraryfunctionfrom {βj}j∈Jintotheunionof 

Eβj forall j ∈ Jsuchthat f(βj) ∈ Eβj .Thisentails 

� 

f(βj) �= ∅. 

j∈J 

(v)Let Lnwith n ≥ 1andlet X, {βj}j∈J, {Rβj }j∈J, {Eβj }j∈Jbedefined 

withrespectto Ln. Thenthereareatmost ndifferentequivalence 

classes Rβjforevery j ∈ J,i.e. 

� � 

� ≤ n. 

� Eβj 

Proof.Cf.(Belnapetal.,2001).For(i),theclaimfollowsbyAxiom(A1) 

andRule(RN).For(ii)Axioms(A2)and(A3)andRule(RN)areneeded. 

Assertion(iii)resultsfromthedefinitionof d stitand(i),(ii).Clearly,(iv) 

isbackedupby(AIAk)and(v)by(APCn)forappropriate k, n. 

3 Theabbreviation βj c stit : φmeansthatforsome α ∈ βj, αcstit : φ ∈ w,becauseof 

Axiom(A5)itfollowsforall ˜α ∈ βj, ˜αcstit : φ ∈ w.


Theorem8(completeness).Each Ln-consistentset Φofbdi–stitformulas 

issatisfiablebyabdi–stitmodel. 

Proof.Let WLnbethesetofallmaximal Ln-consistentsets,let 

A = {α|theagentvariable αoccursin Φ}. 

Then ∼ =Aisanequivalencerelationon WLn.Wedenoteby Wtheequivalenceclass,suchthatforallagentvariables 

α, ˜α ∈ A, β, ˜ β /∈ Aitholdsthat 

α = ˜α /∈ Φiff α = ˜α /∈ wand β = ˜ β ∈ wand α = β /∈ wforall w ∈ W.Let 

{Xi}i∈Ibethesetofallequivalenceclassesofrelation Ron W. 

Thebasisofourbdi–stitmodelisaframe F = (Tree, ≤, A,N,C,B,D,I) 

definedonaBranching-timestructure (Tree, ≤).Wedefineitasfollows: 

• Tree≔{w|w ∈ W } ∪ {Xi|i ∈ I} ∪ {W }; 

• ≤≔trcl({(w,w)|w ∈ W }∪{(W,Xi),(Xi,Xi),(Xi,w)|w ∈ Xi,i ∈ I}∪ 

{(W,W)}); 4 

• A≔{α|αbelongstoanarbitrarybutfixedsetofclassrepresentatives 

of ∼ =Wonallagentvariables.}; 5 

•forall i ∈ Iwedefinethechoiceequivalenceclassesofanyagent α ∈ A 

atanymomentin Tree: 

C(α,w)≔{{hw}},where hwistheuniquehistorypassingthrough 

moment wwith hw = {w,Xi,W },where Xiistheequivalence 

classcontaining w. 

C(α,W)≔{{hw|w ∈ W }}, 

AccordingtoLemma6,thereisanequivalenceclass βj with 

α ∈ βjandanequivalencerelation R i βj on Xi. Wedenotethe 

classesof R i βj on Xiby E i βj andthenwecandefine: 

C(α,Xi)≔{H |∃e : e ∈ E i βj and H = {hw|w ∈ e}}. 

Sincethereisaone-to-onecorrespondencebetweenall w ∈ W andall 

historiesof (Tree, ≤),thefollowingconceptsarewell-defined. Forall m ∈ 

Tree, w ∈ Wand α ∈ A,wehave: 

• |ϕ|≔ � 

{(Xi,hw ′)|ϕ ∈ w′ ,Xi ∈ hw ′} ∈ N (m,hw)iff�ϕ ∈ w; 

i∈I 

4 Here trclstandsforthetransitiveclosureofabinaryrelation. 

5 Recallthatweusethesame αforagentvariablesandagents. Sincewenowinterpret 

theagentvariablebytheagentvariableitself,thisnamingwasjustakindofforestalling. 

Butnotethat A �= Aingeneral.


• |ϕ| ∈ Bα (m,hw) iffthereis α bel : ϕ ∈ w;6 

• |ϕ| ∈ Dα (m,hw) iffthereis αdes : ϕ ∈ w; 

•Wedefinearelation Sα ⊆ W × Wforall α ∈ A,bystipulatingthat 

wSαw ′ iff {ϕ|α int : ϕ ∈ w} ⊆ w ′ . Thenwechoosethesets I α s for 

everysituation s = (m,hw)in M: 

I α (m,hw) = {(w′ ,hw ′)|wSαw ′ }. 

Wehavetoshowthatforall w ∈ Wthereisaw ′ ∈ Wwith wSαw ′ , 

whichmeansthat Iα (m,hw) �= ∅forany m ∈ Tree .Fromaxiom(Di), 

derivationrule(RMi)andtheorems(Ti),itisevidentthatforany 

w ∈ Wtheset S = {ϕ|α int : ϕ ∈ w}isconsistent,thusthereisa 

maximalconsistentset w ′ with S ⊆ w ′ . 

Inanalogyto(Belnapetal.,2001),weclaimthattheframe Fsatisfies 

theindependenceofagentscondition. Foragivenmoment m ∈ Treelet 

Selectmbethesetofallfunctionsfrom Aintosubsetsof Hm,thesetof 

historiespassingthroughmoment m,whereforall σ ∈ Selectmitholdsthat 

σ(α) ∈ Cα m .Then Fsatisfiesthisconditionifandonlyifforeverymoment 

mandany σ ∈ Selectm � 

σ(α) �= ∅. 

α∈A 

Let m = wforanarbitrary w ∈ W or m = W,thentheconditionis 

evidentlysatisfied. Now,letforanarbitrary i ∈ I, σXibeanyfunction from Ainto P(HXi ),suchthat σXi (α) ∈ Cα forall α ∈ A.Bytheabove 

Xi 

with σXi (α) = 

definitionof C α Xi ,thereisanequivalenceclass ej ∈ E i βj 

{hw|w ∈ ej}.Defineafunction fiby fi(α) = ej ∈ Ei,where α ∈ βj 

βj 

for j ∈ J. Thenforall α, ˜α ∈ βj, fi(α) = fi(˜α),thereisawelldefined 

correspondingfunction ˜ fi: {βj}j∈J → � 

Eβj .Aswell, w ∈ fi(α)iff hw ∈ 

j∈J 

σXi (α)andbyLemma7(iv),itholdsthat 

� 

α∈A 

fi(α) = � 

j∈J 

˜fi(βj) �= ∅,suchthat � 

α∈A 

σXi (α) �= ∅. 

Since |Cα w | = |Cα W | = 1forall w ∈ Wand α ∈ A,andforall i ∈ Iitholds 

that |Cα Xi | = |Ei |,itobviouslyfollowsbyLemma7(v)thatforany α ∈ A 

βj 

and m ∈ Tree, Cα m ≤ n,cf. (Belnapetal.,2001). Soanyagent αhasat 

most npossiblechoicesintheframe F. 

6 BecauseofAxiom(A5)forall β ∈ [α]W itfollows: β op : φ ∈ wiff αop : φ ∈ wfor 

op ∈ {c stit,dstit,bel,des,int}andforall w ∈ W.


Now,wedefineacanonicalmodelonthatframe M = (F,v),where vis 

aninterpretationfunction,whichmapseachagentvariable βon v(β) = α ∈ 

Awith β ∈ [α]Wandeveryatomicformula ponasubsetof Scontaining 

allsituations s = (m,hw)with p ∈ wforall m ∈ Tree.Evidently, v(⊤) = S 

and v(⊥) = ∅.Foranyagentvariable α, h ∈ H, w ∈ Witholdsthat h ∈ 

Cα Xi (hw)iffthereis e ∈ Ei and w, w [α]Xi ′ ∈ e,where h = hw ′.Furthermore, 

w, w ′ ∈ e ∈ E i [α]X i 

iff wRi w [α]Xi ′ ,suchthat h ∈ Cα Xi (hw)iff wRi w [α]Xi ′ . 

Weshowbyinductionthat M,(Xi,hw) |= ϕiff ϕ ∈ wforeverybdi–stit 

formula ϕand w ∈ Xi,forall i ∈ I. 

M,(Xi,hw) |= p ⇔ (Xi,hw) ∈ v(p) bydefinition 

⇔ p ∈ w. 

M,(Xi,hw) |= (α = β) ⇔ v(α) = v(β) ⇔ α ∼ =W β ⇔ α = β ∈ w. 

M,(Xi,hw) |= ¬ϕ ⇔ (Xi,hw) �|= ϕ byinduction 

⇔ ϕ /∈ w ⇔ ¬ϕ ∈ w. 

M,(Xi,hw) |= ϕ ∧ ψ ⇔ (Xi,hw) |= ϕand (Xi,hw) |= ψ byinduction 

⇔ 

ϕ ∈ wand ψ ∈ w ⇔ (ϕ ∧ ψ) ∈ w. 

M,(Xi,hw) |= ✷ϕ ⇔forall h ∈ HXi itholdsthat (Xi,h) |= ϕ 

byinduction 

⇔ forall h ∈ HXi thereis w′ ∈ Xi 

with h = hw ′and ϕ ∈ w′ 

⇔forall w ′ ∈ Xi,ϕ ∈ w ′byLemma7(i) 

⇔ ✷ϕ ∈ w. 

M,(Xi,hw) |= β cstit : ϕ ⇔forall h ∈ C α Xi (hw)with β ∈ [α]W 

itholdsthat (Xi,h) |= ϕ 

⇔forall h ∈ C α Xi (hw)thereis w ′ with h = hw ′ 

anditholdsthat (Xi,hw ′) |= ϕbyinduction ⇔ 

forall w ′ ∈ Xi,if wR i w [α]Xi ′ then ϕ ∈ w ′ 

byLemma7(ii) 

⇔ αcstit : ϕ ∈ w 

byAxiom(A5) 

⇔ β cstit : ϕ ∈ w. 

M,(Xi,hw) |=�ϕ ⇔thereis U ∈ N (Xi,hw)∅ �= U ⊆ �ϕ� 

⇔thereis ψwith ∅ �= |ψ| ⊆ �ϕ�and�ψ ∈ w 

(∗) 

⇔�ϕ ∈ w. 

Wewanttoshowtheequivalence (∗) 

⇔. 

⇐:If�ϕ ∈ w,then |ϕ| ∈ N (Xi,hw)with w ∈ Xi.Thatmeansthereis 

ψ = ϕwith |ϕ| ⊆ �ϕ�byinduction.


⇒:Thereexists ψwith ∅ �= |ψ| ⊆ �ϕ�and�ψ ∈ w.Since 

|ψ| = � 

{(Xi,hw ′)|ψ ∈ w′ � 

,Xi ∈ hw ′}, W = {w|(Xi,hw)isasituation}, 

i∈I 

and |ψ| ⊆ �ϕ�itfollowsforall w ′ ∈ Wand i ∈ I:if M,(Xi,hw ′) |= ψthen 

M,(Xi,hw ′) |= ϕ. Byinductionwehaveforall w′ ∈ W: if ψ ∈ w ′ then 

ϕ ∈ w ′ .Thusitholdsforall w ′ ∈ Wthat (ψ ⊃ ϕ) ∈ w ′ .Byrule(RM)it 

holdsthat (�ψ ⊃�φ) ∈ w ′ forall w ′ ∈ W.Since�ψ ∈ w,wehave�ϕ ∈ w. 

i∈I 

M,(Xi,hw) |= β bel : ϕ ⇔thereis U ∈ B α (Xi,hw) 

∅ �= U ⊆ �ϕ� 

with β ∈ [α]Wand 

⇔thereis ψwith ∅ �= |ψ| ⊆ �ϕ�and αbel : ψ ∈ w 

(∗∗) 

⇔ β bel : ϕ ∈ w. 

M,(Xi,hw) |= β des : ϕ ⇔thereis U ∈ D α (Xi,hw) 

∅ �= U ⊆ �ϕ� 

with β ∈ [α]Wand 

⇔thereis ψwith ∅ �= |ψ| ⊆ �ϕ�and αdes : ψ ∈ w 

⇔ β des : ϕ ∈ w. 

Weonlyshow (∗∗),whichissimilartotheargumentforthe�-operator. 

Thecorrespondingequivalenceforthedesireoperatorisshownanalogously. 

⇐:If β bel : ϕ ∈ w,then,byAxiom(A5), α bel : ϕ ∈ w,suchthat 

|ϕ| ∈ B α (Xi,hw) with w ∈ Xi.Thatmeansthereis ψ = ϕwith |ϕ| ⊆ �ϕ�by 

induction. 

⇒:Thereis ψwith ∅ �= |ψ| ⊆ �ϕ�and αbel : ψ ∈ w.Since 

|ψ| = � 

{(Xi,hw ′)|ψ ∈ w′ � 

,Xi ∈ hw ′}, W = {w|(Xi,hw)isasituation}, 

i∈I 

and |ψ| ⊆ �ϕ�itfollowsforall w ′ ∈ Wand i ∈ I:if M,(Xi,hw ′) |= ψthen 

M,(Xi,hw ′) |= ϕ. Byinductionwehaveforall w′ ∈ W: if ψ ∈ w ′ then 

ϕ ∈ w ′ .Thusitholdsforall w ′ ∈ Wthat (ψ ⊃ ϕ) ∈ w ′ .Byrule(RMb)it 

holdsthat (α bel : ψ ⊃ α bel : φ) ∈ w ′ forall w ′ ∈ W.Since α bel : ψ ∈ w, 

wehave α bel : ϕ ∈ w.Againbecauseof(A5),itfollows β bel : ϕ ∈ w. 

M,(Xi,hw) |= β int : ϕ ⇔ I α (Xi,hw) 

i∈I 

⊆ �ϕ�with β ∈ [α]W 

⇔forall s = (w ′ ,h ′ w) : if wSαw ′ ,then M,s |= ϕ 

byinduction 

⇔ forall s = (w ′ ,h ′ w) :if wSαw ′ ,then ϕ ∈ w ′ 

(∗∗∗) 

⇔ β int : ϕ ∈ w. 

Atlastwehavetoshowtheequivalence (∗ ∗ ∗).


⇐:If β int : ϕ ∈ w,then α int : ϕ ∈ wandforall w ′ ∈ Wwith wSαw ′ it 

followsthat ϕ ∈ w ′ . 

⇒:Forall s = (w ′ ,h ′ w ):if wSαw ′ ,then ϕ ∈ w ′ .Becauseofaxiom(Di) 

andmaximalityforany ϕand w ∈ Witholdsthateither α int : ϕ ∈ wor 

αint : ¬ϕ ∈ worbothisnotthecase. Theassumption α int : ¬ϕ ∈ wis 

contradictory,since ¬ϕ ∈ {ψ|α int : ψ ∈ w} ⊆ w ′ forany w ′ with wSαw ′ . 

Assume ¬αint : ϕ ∈ wand ¬αint : ¬ϕ ∈ w. Then ϕ, ¬ϕ /∈ {ψ|α int : 

ψ ∈ w}. Since wismaximal,theset {ψ|α int : ψ ∈ w}isclosedunder 

implicationby(RMi)and(Ti),suchthatthesets {ϕ} ∪ {ψ|α int : ψ ∈ w} 

and {¬ϕ} ∪ {ψ|α int : ψ ∈ w}arebothconsistent. Butthenthereisa 

maximalworld w ′′ with {¬ϕ}∪{ψ|α int : ψ ∈ w} ⊆ w ′′ and wSαw ′′ .Butthat 

conflictswith ϕ ∈ w ′ forall wSαw ′ .Thus, α int : ϕ ∈ w,resp. β int : ϕ ∈ w. 

Atanyrate,thereisone(maybemorethanone,thenchooseone)maximalconsistentset 

w0 ∈ Wwith Φ ⊆ w0.This w0belongstoanequivalence 

class Xi0of R.Then, M,(Xi0 ,hw0 ) |= ϕforany ϕ ∈ Φ.Thus,anyconsistentset 

Φissatisfiable. 

Finitemodelproperty 

Weconstructtoagivensentence ϕaccordingtoDefinition14afiniteframe 

Ffinandaddaspecialinterpretation v,suchthatforeverysubsentence 

of ϕitisdecidablewhetherthesubsentenceissatisfiableby (Ffin,v). We 

adoptthefiltrationmethodasusedin(Belnapetal.,2001). Wetakethe 

canonicalframe F = (Tree, ≤, A,N,C,B,D,I)oftheprevioussectionand 

defineafiltrationfirstoverallworldsbythesetofallsubformulasofthe 

givenformula ϕincludingallformulasderivedbyAxioms(AIAk)and(Ai) 

forall 1 ≤ i ≤ 5,cf. (Belnapetal.,2001). Thenagain,wefiltratethe 

equivalenceclassesofrelation Rbyasetofformulasimpliedbysubformulas 

of ϕprefixedby d stitor cstitoperators,suchthatwecandefinechoiceequivalenthistories.Tobeginwith,wedefinethesetsofsubformulas: 

Σϕ = {ψ|ψisasubsentenceof ϕ} , 

Σi = Σϕ ∪ {¬β int : ¬ψ|β int : ψ ∈ Σϕ} , 

Σd = Σϕ ∪ {β cstit : ψ, ¬✷ψ|β d stit : ψ ∈ Σϕ} ∪ {¬✷¬ψ|✸ψ ∈ Σϕ} , 

Σe = {ψ|ψisasubsentenceofaformulaof Σdorof 

{β cstit : ¬β cstit : ψ|β cstit : ψ ∈ Σd}}, 

Σp = {✸(β0 cstit : φ0 ∧ · · · ∧ βn cstit : φn)|n ≥ 0,β0,... βndiffer 

pairwisely,occurin ϕ,andforall 0 ≤ i ≤ n,φi = ψ0 ∧ · · · ∧ ψmi , 

0 ≤ j ≤ mi,thereis βi cstit : ψjin Σe}, 

Σa = {ψ|ψisasubsentenceofaformulaof Σp ∪ Σe ∪ Σi}.


Forall w,w ′ ∈ W,wedefinetheequivalencerelation ≡ Σabysetting w ≡ Σa 

w ′ iffforall ψ ∈ Σa: (m,hw) |= ψiff (m,hw ′) |= ψ.By ˜ Wwedenoteachosen 

setofrepresentativesofallequivalenceclasses.By ˜ Xiwedenotethesubset 

of ˜ Wconsistingofallrepresentatives,whichbelongtotheequivalenceclass 

Xiforall i ∈ I. Notethatiftherelation Risfirstappliedtotheset W 

andthenrelation Σaisimplementedontheequivalenceclasses,onegetsan 

isomorphicBranchingTimeStructureintheend.Since Σaisfinite,soitis 

˜Wandtherewithevery ˜ Xi.Thus,wedefineafiniteframe Ffin: 

• Tree≔{w|w ∈ ˜ W } ∪ { ˜ Xi|i ∈ I} ∪ { ˜ W }isfinite. 

• ≤≔trcl({(w,w)|w ∈ ˜ W } ∪ {( ˜ W, ˜ Xi),( ˜ Xi, ˜ Xi),( ˜ Xi,w)|w ∈ Xi,i ∈ I} 

∪ {( ˜ W, ˜ W)}),suchthatthereisagainanone-to-onecorresponding 

relationbetweenthehistories ˜ Handtheset ˜ W, ˜ H = {hw|w ∈ ˜ W }. 

•thesetofagentsischosenas Ã = {α|thereis αoccurringin ϕ},thus 

Ãisalsofinite. 7 

•forall α ∈ Ãwedefinerelations ≡ α Σeoneveryset ˜ Xi,by w ≡ α Σe w′ iff 

forall α cstit : ψ ∈ Σeitholdsthat αcstit : ψ ∈ wiff αcstit : ψ ∈ w ′ . 

wedenotethesetofallequivalenceclasseson ˜ Xi.Withthis 

By Ũi 

[α] ˜ Xi 

definitionitispossibletodefinethechoiceequivalentfunction ˜ Cin 

thefiniteframe: 

˜C(α,w)≔{{hw}},where hwistheuniquehistorypassingthrough 

moment wwith hw = {w, ˜ Xi, ˜ W }, 

˜C(α, ˜ W)≔{{hw|w ∈ ˜ W }}, 

˜C(α, ˜ � 

Xi)≔ H|∃e : e ∈ U i 

[α] Xi ˜ 

� 

and H = {hw|w ∈ e} . 

Sincethereisaone-to-onecorrespondencebetweenall w ∈ ˜ W and 

allhistoriesof (Tree, ≤),thefollowingnotionsarewell-defined. Forall 

�φ,αbel : φ,αdes : φ ∈ Σϕ, m ∈ Tree, w ∈ ˜ Wand α ∈ Ã,wehave: 

• |φ|≔ � 

{(w ′ ,hw ′)|φ ∈ w′ } ∈ N (m,hw)iff�φ ∈ w, 

i∈I 

• |φ| ∈ ˜ Bα (m,hw) iff α bel : φ ∈ w, 

• |φ| ∈ ˜ Dα (m,hw) iff α des : φ ∈ w, 

7 Weneglecttheproblemofidentitystatementsofagents,sinceitcanbehandledas 

above.


•Forall w ∈ ˜ Wandeach α ∈ Ãset tα w 

= {φ|α int : φ ∈ w}. Then, 

thereisatleastone ˜w α ∈ ˜ Wwith t α w ∩Σa ⊆ ˜w α ,since t α w isconsistent 

by(Ti),(Di).Wedefineforall m ∈ Tree: 

Ĩ α (m,hw) = {( ˜wα ,h˜w α)| ˜wα ∈ ˜ W,t α w ∩ Σa ⊆ ˜w α }. 

Byconstruction,thesesetsarenotempty. 

Lemma8.Let F = (Tree, ≤, A,N,C,B,D,I)bethecanonicalframe,let 

ibefixed, Xithecorrespondingequivalenceclass Xi ∈ Tree, ˜Xithecorre- 

spondingclassinthefiniteframe Ffin = (Tree, ≤, Ã,Ñ, ˜ C, ˜ B, ˜ D, Ĩ)filtrated 

thesetsofequivalenceclassesof 

bythesetsofsubformulasof ϕand U i 

[α] ˜ Xi 

≡ Σeon ˜ Xiforall α ∈ Ã. 

(i)If ✷ψ ∈ Σa, w ∈ ˜ Xi,then 

(ii)If α cstit : ψ ∈ Σe, w ∈ ˜ Xi,then 

✷ψ ∈ w iff ψ ∈ w ′ forall w ′ ∈ ˜ Xi. 

α cstit : ψ ∈ w iff ψ ∈ w ′ forall w ′ ∈ ˜ Xiwith w ≡ α Σe w′ . 

(iii)Forallequivalenceclasses eα ∈ Ũi 

[α] ˜ Xi 

� 

α∈ Ã 

eα �= ∅. 

itholdsthat 

(iv)Let ϕ ∈ Lnwith n ≥ 1,thenforany j ∈ [0, | Ã|]itholdsthat 

Proof.Cf.(Belnapetal.,2001). 

� 

� 

�U i 

[α] ˜ Xi 

� 

� 

� ≤ n. 

Thisframesatisfiestheindependenceofagentscondition. Forallmoments 

m ∈ { ˜ W,w|w ∈ ˜ W }itisevidentthatforanarbitraryfunction 

σm : Ã → ˜ Cmtheintersection � {σm(α)|α ∈ Ã}isnotempty.If m = ˜ Xi, 

thenforall α ∈ Ãthereis eα ∈ U i 

[α] ˜ Xi 

with σm(α) = eα,suchthatwiththe 

frameproperty8(iii)theset � {σXi ˜ (α)|α ∈ Ã}isalsonotempty.Assuming 

Ln,thecorrespondingaxiomofpossiblechoicesisalsofulfilled,sinceforall 

α ∈ Ã, 

� 

� 

� ˜ Cα � � 

� � 

W˜ 

� = � ˜ Cα � � 

� � 

w� 

= 1and � ˜ Cα � � 

� � 

Xi 

˜ � = � 

�Ui � 

� 

� 

� ≤ nbyLemma8(iv). 

[α] ˜ Xi 

Forany ψ ∈ Σϕwecanshowthatforanyequivalenceclass ˜ Xiitholds 

that 

Mfin( ˜ Xi,hw) |= ψ iff ψ ∈ w,


where Mfin = (Ffin,v)and visthevaluationfunctiondefinedasforthe 

canonicalmodel,butrestrictedto ˜ W.Theproofisbyinduction. 

Mfin,( ˜ Xi,hw) |= p ⇔ ˜ Xi,hw) ∈ v(p) ⇔ p ∈ w. 

Mfin,( ˜ Xi,hw) |= ¬ψ ⇔ ( ˜ Xi,hw) �|= ψ ⇔ ψ /∈ w ⇔ ¬ψ ∈ w. 

Mfin,( ˜ Xi,hw) |= φ ∧ ψ ⇔ ( ˜ Xi,hw) |= φand ( ˜ Xi,hw) |= ψ 

⇔ φ ∈ wand ψ ∈ w ⇔ (φ ∧ ψ) ∈ w. 

Mfin,( ˜ Xi,hw) |= ✷ψ ⇔forall h ∈ H ˜ Xi itholdsthat ( ˜ Xi,h) |= ψ 

⇔forall h ∈ H ˜ Xi thereis w′ ∈ ˜ Xiwith 

h = hw ′and ψ ∈ w′ 

⇔forall w ′ ∈ ˜ Xi,ψ ∈ w ′byLemma8(i) 

⇔ ✷ψ ∈ w. 

Mfin,( ˜ Xi,hw) |= α cstit : ψ ⇔ forall h ∈ ˜ C α ˜ Xi (hw)itholdsthat 

( ˜ Xi,h) |= ψ 

⇔ eα ∈ U i 

[α] ˜ Xi 

with w ∈ eα : 

⇔forall w ′ ∈ ˜ Xi,if w, w ′ ∈ ejthen ψ ∈ w ′ 

byLemma8(ii) 

⇔ αcstit : ψ ∈ w. 

Mfin,( ˜ Xi,hw) |=�ψ ⇔ thereis U ∈ Ñ ( ˜ Xi,hw) ∅ �= U ⊆ �ψ� 

⇔ thereis φ ∈ Σϕwith ∅ �= |φ| ⊆ �ψ�and 

�φ ∈ w 

⇔�ψ ∈ w. 

If |φ| ⊆ �ψ�,then |φ| ⊆ |ψ|,i.e.forall w ′ ∈ ˜ W : (ψ ⊃ φ) ∈ w ′ .Assumethere 

is w ∈ Wwith (ψ ⊃ φ) /∈ w.Since ˜ Wisacompletesetofrepresentatives 

ofall ≡ Σa-equivalenceclasses,thereis ˜w ∈ ˜ Wwith w ≡ Σa ˜w. Forthe 

canonicalmodel Mitholdsthat (ψ ⊃ φ) /∈ w.Then M,(Xi,hw) |= ¬(ψ ⊃ 

φ)and M,(Xi,h˜w) |= (ψ ⊃ φ).Thus, M,(Xi,hw) �|= ψor M,(Xi,hw) |= φ 

and M,(Xi,h˜w) |= ψand M,(Xi,h˜w) �|= φ,butthisconflictswith w ≡ Σa 

˜w,as φ,ψ ∈ Σa.Therefore,forall w ∈ W : φ ⊃ ψ ∈ w,andsoby(RM) 

�φ ⊃�ψ ∈ w.Consequently,�ψ ∈ w.Theotherdirectionisobviouswith 

ψ = φ.Similarconsiderationsgive: 

Mfin,(Xi,hw) |= αbel : ψ ⇔thereis U ∈ ˜ B α (Xi,hw) ∅ �= U ⊆ �ψ� 

⇔thereis φ ∈ Σϕwith ∅ �= |φ| ⊆ �ψ�and 

α bel : φ ∈ w ⇔ αbel : ψ ∈ w.


Mfin,(Xi,hw) |= α des : ψ ⇔thereis U ∈ ˜ D α (Xi,hw) ∅ �= U ⊆ �ψ� 

⇔thereis φ ∈ Σϕwith ∅ �= |φ| ⊆ �ψ�and 

α des : φ ∈ w ⇔ αdes : ψ ∈ w. 

Mfin,(Xi,hw) |= αint : ψ ⇔ Ĩα (Xi,hw) ⊆ �ψ� 

⇔forall w ′ ∈ ˜ W :if t α w ∩ Σa ⊆ w ′ ,then ψ ∈ w ′ . 

⇔ α int : ψ ∈ w. 

Assume αint : ψ /∈ w,then ¬α int : ψ ∈ w.Becauseof(Di)therearetwo 

differentcasespossible,(i) αint : ¬ψ ∈ wor(ii) ¬αint : ¬ψ ∈ w.If(i),then 

¬ψ ∈ t α wand,since Σi ⊆ Σa, ¬ψ ∈ t α w ∩ Σa ⊆ w ′ .Or(ii) ¬αint : ψ ∈ wand 

¬αint : ¬ψ ∈ w;then ψ, ¬ψ /∈ t α w.Butthenthereisaw ′′ with t α w ∩Σa ⊆ w ′′ 

and ¬ψ ∈ w ′′ .Thesecontradictionsimply α int : ψ ∈ w. 

Tosumup,like d stitlogic,bdi–stitlogicisfinitelyaxiomatizableand 

hasthefinitemodelproperty.Therefore,itisdecidable. 

CarolineSemmling 

InstituteofPhilosophy,DresdenUniversityofTechnology 

Dresden,Germany 

Caroline.Semmling@gmx.de 

HeinrichWansing 

InstituteofPhilosophy,DresdenUniversityofTechnology 

Dresden,Germany 

Heinrich.Wansing@tu-dresden.de 

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27,505–552.

A Procedural Interpretation of Split Negation 


Sebastian Sequoiah-Grayson ∗ 

Takingtheprocedural/dynamicturninthestudyofinformationseriously 

meansthatweneedtomakethetransitionfromthestudyofbodiesofinformation,tothestudyofthemanipulationsofsuchbodiesofinformation. 

Inthiscase,wewillnotbeabletocarryoutthestudyofinformational 

dynamicsbyrestrictingourattentiontobodiesofinformation,orevento 

thestructureofthebodiesofinformation,althoughthisisanimportant 

component.Wewillalsoneedtopayattentiontotheproceduresviawhich 

suchbodiesofinformationarecombinedanddeveloped,andprocessed. 

Onerestrictionthatwemightplaceonaparticularstudyofinformational 

dynamicsisthatweexamineonlypositiveinformation.Thatis,wemight 

restrictourattentiontothepositivefragmentsofvariouslogicsusedto 

underpinlogicsofinformationflow. Restrictingourselvestothestudyof 

positiveinformationisjustifiableonseveralcounts,nottheleastofwhich 

isthatitmakesperfectsensetorestrictourselvestosimplercases,aseven 

thesemayturnouttobesurprisinglycomplicated.However,todojustice 

tothephenomenaofinformationflow,anyadequatetheoryofinformation 

processingwillhavetoallowfortherepresentationofbothpositive,and 

negativeinformation. Inthiscase,attentionwillnotberestrictedtothe 

positivefragmentofthevariouslogicsusedtounderpinlogicsofinformation 

flow. 

∗ ManythankstoVladimírSvoboda,MichalPeliˇs,andallbehindLogica2008! This 

researchwasmadepossiblebythegeneroussupportoftheHaroldHyamWingateFoundation.ThisresearchwascarriedoutwhilstundertakingaVisitingResearchFellowshipattheTilburgInstituteofLogicandPhilosophyofScience(TiLPS),atTilburgUniversity,TheNetherlands. 

IamextremelygratefultoStephanHartmannandeveryone 

atTiLPSforthevibrant,research-griendlyatmosphereprovided. Iamalsoextremely 

gratefultoEdgarAndrade,JohanvanBenthem,FrancescoBerto,CararinaDutilh,Volker 

Halbach,ChristianKissig,GregRestall,HeinrichWansing,andTimWilliamsonformany 

invaluablesuggestions.Anyerrorsthatremainaremyown.

220 SebastianSequoiah-Grayson 

Thisessayisanargumentforaparticularproceduralinterpretationof 

negativeinformation.Inparticular,itisanargumentforaproceduralinterpretationofsplitnegation.Asplitnegationpair 

〈∼, ¬〉isdefinableinany 

non-commutativelogic.Assuch,aproceduralinterpretationofsplitnegationshouldadoptableinprincipleforanynon-commutativelogic,bethisanon-commutativelinearlogic,oravariantoftheLambekcalculusorwhatever.Accordingly,wewillbeabstractingacrossnon-commutativelogicsin 

generalasopposedtolookingindetailatanyonenon-commutativelogicin 

particular.However,aninformation–processingapplicationwillbethegeneralmotivation.Fromaphilosophicalstandpoint,theclosestanaloguesare 

thenon-commutativelinearlogics,albeitunderproceduralinterpretations. 

Linearlogicsweredevelopedinordertotrackresource-use:formulaareunderstoodasresources,andinthiscasenumberoftimestheyoccurbecomes 

relevant. Assuch,themarkoflinearlogicsingeneralistherejectionof 

contraction. However,iftheformulaaretakentobeconcretedata,then 

theaccessibilityoftheseresourcesalsobecomesrelevant.Itisoftenthecase 

thatdatahavespatiotemporallocations,suchasinthememoriesofagents 

orcomputers,andremotedatawillbelesseasytoaccessthanadjacentdata. 

Inamoresensitivelogicofresourcesthen,itisnotonlythemultiplicityof 

data,butalsotheirorderthatisimportant.Spatiotemporalobstaclesoften 

needtobecircumventedsothatdatamaybeaccessed,hencecommutation 

isinappropriatebyvirtueofitsdestroyingtheveryorderingthatwewould 

liketopreserve.Insituationswhereactualinformationprocessingisbeing 

carriedout,thearrangementofthedataiscrucial(Paoli,2002,28-9).For 

recentworkonnon-commutativelinearlogics,see(Abrusci&Ruet,2000), 

andforanexplicitlyproceduralexaminationofcommutationinthecontext 

ofagent-basedinformationprocessing,see(Sequoiah-Grayson,2009). 

Interpretingsplitnegationisaknowndifficulty(Dosen,1993,20). For 

anynegationtypetherewillbemorethanonewayofdefiningit.Givena 

definition,wethenneedtoprovideaninterpretationoftheresultingnegationintermscommensuratewiththeintendedapplication.Inourcase,the 

intendedapplicationistheareaofdynamicinformationprocessing.Given 

theproceduralaspect,wewilldefinethenegationof Aintermsof Aimplyingbottom(0).Thisiscommensuratewithinformationprocessingdue 

totheimplicationdoingtheworkbeinganalysedinproceduralterms.Sans 

theproceduralaspect,aninterpretationofthenegationof Aintermsof A 

implying 0goesbackatleastto(Kripke,1965). 

Thisessaydevelopsandproposesaparticularinterpretationofthenegationof 

Aintermsof Aimplying 0ininformationprocessingterms: In 

section2,informationframesandinformationmodelsareintroduced. We 

alsointroducethedefinitionofsplitnegation. Theinformationalreading 

oftheternaryrelation Rofframesemanticsisintroduced. Insection3,


aproceduralinterpretationofsplitnegationunderthedefinitiongivenin 

section2isproposed.Uptothispointourexplorationwillhavebeenconductedinpurelymodel-theoreticterms.Itisinsection3thatwetouchontoproof-theoreticalmatters.Thisisessentiallytochecktheproceduralinterpretationagainstaseriesofuniversallyvalidproof-theoreticalproperties 

ofsplitnegation.Putsimply,theproposalisthatweinterpretthenegation 

of Aintermsoftherulingoutofparticularprocedures,withtheseproceduresbeinganyprocedurethatinvolvescombiningthenegationof 

Awith 

Aitself. 

Thefirststepistointroducethenotionofaninformationframeand 

model,sothatwemayspecifyourdefinitionofsplitnegation. 

2 InformationFramesandModels 

Takeanon-commutativeinformationframe F 〈S, ⊑, •, ⊗, →,←,0〉where S 

isasetofinformationstates x,y,...thatmaybeinconsistent,incomplete, 

orboth,thebinaryrelation ⊑isapartialorderon Sofinformational 

development/inclusion, •isthebinarycompositionoperatoroninformation 

statessuchthatduetocommutationfailurewehaveitthat x • y �= y • x, 

⊗is(non-commutative)fusion, → and ←arerightandleftimplication 

respectively,and 0isbottom. 1 Makingallofthiscleariseasieroncewe 

haveamodel. 

Amodel M≔〈F,�〉isanorderedpair F 〈S, ⊑, •, ⊗, →,←,0〉and � 

suchthat �isanevaluationrelationthatholdsbetweenmembersof Sand 

formula.Where Aisapropositionalformula,and x,y,z ∈ F, �obeysthe 

hereditycondition: 

Forall A,if x � Aand x ⊑ y,then y � A, (1) 

Andalsoobeysthefollowingconditionsforeachofourconnectives: 

x � A ⊗ Biffforsome y,z, ∈ Fs.t. y • z ⊑ x,y � Aand z � B. (2) 

x � A → Biffforall y,z ∈ Fs.t. x • y ⊑ z,if y � Athen z � B. (3) 

x � A ← Biffforall y,z ∈ Fs.t. y • x ⊑ z,if y � Athen z � B. (4) 

x � 0forno x ∈ F. (5) 

1 Anotationalnote: 0iscommonlywrittenas ⊥.Thedifferenceinnotationistoensure 

thatnoconfusionismadebetweenbottom,andtheperprelationofincompatibility(Dunn, 

1993),(Dunn,1994),(Dunn,1996),writtenas ⊥.Intherecentliteratureonnegation, ⊥ 

issooftenusedfortheperprelationthatusingitforbottomcreatestoogreatariskfor 

misunderstanding.Hence,wefollow(Girard,1987)intheuseof 0forbottom. 

Manynon-commutativelogicsarealsonon-associative, suchasthenon-associative 

Lambekcalculusamongothers. However,sincenothingthatfollowsdependsoneither 

thepresenceorabsenceofassociativity,weshouldbeabletosafelyignorethisissuefor 

ourpurposes.


Theevaluationrelation �maybeunderstoodindifferentways,dependingonthecontextofapplication. 

Forexample,ifweweretobeworking 

withlanguageframesandsyntacticallycategorisingparticularalphabetical 

strings,wewouldunderstand x � Atomeanstring xisoftype A. We 

mightinsteadconsiderascientificresearchprojectwithitsvariousdevelopmentalphases.Inthiscasethedevelopmentrelation 

⊑willorderdifferent 

statesofaresearchprojectovertime(withtheidealisationthatthereis 

noinformation-loss). Herewewouldunderstand x � Atomeanthatthe 

proposition Aisknownatstate x,andthatthisparticularstateofdevelopmentintheprojectsupports 

A.Wewillinfactreturntothisveryideain 

5below.Fornowhowever,weneedsomethingalittlemoregeneral.Along 

with(Mares,2009)wewillunderstand x � Atomeanthattheinformation 

instate xcarriestheinformationthat A. Hence,wemayalsosaythat x 

supportstheinformationthat A. Thisisverysimilartothefamiliarsemanticentailmentrelation 

�. Thedifferenceisthatwewanttoallowfor 

theinformationatxbeingincompleteand/orinconsistent.Therearemany 

applicationswherewemightwanttodothis.Takinginconsistencyasthe 

runningexample,considervariousstatesofanagentastheagentreasons 

deductively. Inthiscase, xmaysupport Awhere Ais‘pandnot p’,but 

thisisdifferentfrom xmaking Atrue,atleastintheusualsenseof“making 

true”,asthereisnopossiblewaythattheworldcanbesuchthat xcouldbe 

trueofit.Onemightwishtounderstand’supports’as‘makestrue’ifone 

holdstoadialethicparaconsistentismwherebyatleastsomecontradictions 

aretakentobetrue.However,wewillsidestepthisparticulardebateand 

staywiththeinterpretationof’supports’thattakesittobethesubtler 

relativeof’makestrue’inthemannerstipulatedabove. 

Thereaderfamiliarwiththeternaryrelation Rofframesemanticswill 

recognise(2)–(4)astheternaryconditionsfor ⊗, →,and ←respectively, 

underanexplicitlyinformationalreading. Rmaybeparsedintermsofthe 

twobinaryrelations •and ⊑andsuchthat Rxyzcomesoutas x•y ⊑ z.How 

shouldwereadformulascontainingthebinarycompositionrelation •? A 

commonandtraditionalwayofunderstandingbinarycompositionissimply 

totake x•yas xtogetherwith y.Inthiscase •willbehavemuchthesameas 

setunionsuchthat xtogetherwith yisnodifferentto ytogetherwith x,and 

xtogetherwithitselfisnodifferentfrom xandsoforth.However,wearenot 

restrictedtosuchareadingof •.Thereisingeneralnocanonicalreading 

of Rxyz. Thatistosaythatthereisnocanonicalinterpretationofthe 

modeltheory.Althoughthisisfrustratingwhenoneencounterstheternary 

relationforthefirsttime,itisakeypointwithrespecttotheflexibilityof 

ternarysemantics. Inourcase,wehaveitthat •isnon-commutative,so 

x•y �= y•x.Internaryterms,non-commutationcomesoutas Rxyz �= Ryxz. 

Hence,simplyinterpreting x • yas xtogetherwith ywillblurthevery


orderingthatnon-commutationistryingtopreserve. Wecouldstipulate 

that“xcomposedwith y”differsfrom“ycomposedwith x”,howeverthis 

isslightlystrainedanddoesnotreadstraightoffacasualuseof‘composed’. 

Abetterwaytokeepthisdistinctionrobustistoread x•yas xappliedto y. 

“Applying”isanorder-sensitivenotion,andonethatfitscomfortablywith 

dynamic/proceduraloperations.Sowecanthinkof x•yasthecomposition 

of xwith y,wherethiscompositionisorder-sensitive,andwewillmarkthis 

order-sensitivitybyspeakingof“application”insteadof“composition”. 

Ofcoursewearenotmerelyconcernedwithsyntacticconstructions,as 

x,y,z ∈ S,and Sisasetofinformationstates.Weareconcernedwiththe 

applicationoftheinformationinonestatetotheinformationinanother. 

Onewaythen,ofreading Rxyz,isthat Rxyzholdsifftheresultofapplying 

theinformationin xtotheinformationin yiscontainedintheinformation 

in z,andthisispreciselywhat x • y ⊑ ztellsus.Anotherwayofputting 

thisistosaythattheinformationin zisadevelopmentoftheinformation 

resultingfromtheapplicationoftheinformationin xtotheinformationin 

y. 

Therolethatinformationapplicationplayshereisnotredundant,and 

neitherisitmerelytomarkorder-sensitivity.Wearenotsimplyconcerned 

withorderedsequencesofinformationstates—somethinglikeanordersensitiveconjunctionwherewewouldhaveonepieceofinformation,then 

anotherandthenanotheretc. Weareconcernedwithsomethingmuch 

moresubtle. Weareconcernedwiththeinteractionbetweeninformation 

states.Thisconcernwithinteraction,orprocess,ispreciselywhyitisthat 

weareconcernedwithorder-sensitivityinthefirstplace.Order-sensitivity 

isinthissenseameanstoanend,withthisendbeingtheindividuation 

ofproceduresofdynamicinformationprocessing. Thissenseof“applied” 

carriesoverinanaturalwayfromtheinformationstatesthemselves,tothe 

propositionssupportedbytheinformationstates. Itiseasiesttoseethis 

withanexample. 

Takefusion,anditsframeconditionsgivenin(2).(2)canbeinterpreted 

tostatethataninformationstate xcarriestheinformationresultingfrom 

theapplicationoftheinformationthat Atotheinformationthat Bifand 

onlyif xisitselfadevelopmentoftheapplicationoftheinformationin 

state ytotheinformationinstate z,where ycarriestheinformationthat A 

and zcarriestheinformationthat B.Thisisalittlelongwinded,andgoing 

intherighttolefthanddirectionisalittlemorestraightforward:fortwo 

states yand zthatcarrytheinformationthat Aandthat Brespectively, 

theapplicationof yto zwillresultinanewinformationstate, x,suchthat x 

carriestheinformationthatresultsfromtheapplicationoftheinformation 

that Atotheinformationthat B. Theanalogousinterpretationsofthe 

frameconditionsforrightandleftimplication((3),and(4),respectively)


unpackinasimilarmanner. 

Thefusionconnectiveandtheimplicationconnectivesarenotindependent;theyformafamilyofsorts. 

Ourfusionandimplicationconnectives 

interrelateinthefollowingmanner: 

A ⊗ B ⊢ Ciff B ⊢ A → C. (6) 

A ⊗ B ⊢ Ciff A ⊢ C ← B. (7) 

Indeductiveinformationprocessing,weunderstandthepremisesasdatabasesandtheconsequencerelation‘⊢’astheinformationprocessingmechanism,amorebrutallysyntacticoperationthattheinformationcarrying/supportingof 

�. Ininformationalterms,wemayread A ⊢ Basinformationoftype 

Bfollowsfrominformationoftype A,ortheinformation 

in Bfollowsfromtheinformationin Aetc.Wecanthinkoftypingasencoding,inwhichcasewemightalsoread 

A ⊢ Bastheinformationencoded 

by Bfollowsfromtheinformationencodedby A.(6)and(7)makesense. 

Take(6),startingwiththeleft-to-right-handdirection:Iftheinformation 

in C followsfromtheinformationresultingfromtheapplicationofthe 

informationin Atotheinformationin B,thenfromtheinformationin 

Baloneitfollowsthatwehavetheinformationin Cconditionalonthe 

informationin A. Theright-to-left-handdirectionworksoutsimilarly: If 

fromtheinformationin Balonewecangettheinformationin Cconditional 

ontheinformationin A,thenwecangettheinformationin Cviathe 

applicationoftheinformation Atotheinformationin B. Nowtakethe 

left-to-right-handdirectionof(7): If,again,theinformationin Cfollows 

fromtheinformationresultingfromtheapplicationoftheinformationin A 

totheinformationin B,thenfromtheinformationin Aaloneitfollowsthat 

wehavetheinformationin C,thistimeconditionalontheinformationin 

B.Theright-to-left-handdirectionworksonsimilarlyheretoo:Iffromthe 

informationin Aalonewecangettheinformationin Cconditionalonthe 

informationin B,thenwecangettheinformationin Cviatheapplicationof 

theinformationin Atotheinformationin B.(6)and(7)areinformational 

processingversionsofthedeductiontheorem. 

Withregardsto(5),noinformation,inanycontextwhatsoever,isof 

type 0.Thereisnothingthatwecandotoget 0,and 0isnotsupported 

byanyinformationstateinourframe F. 

Nowwehavethelogicaltoolsthatweneedinordertobeginlookingat 

negativeinformation.Wecandefineasplitnegationpairintermsofdouble 

implication: 

∼A≔A → 0, (8) 

¬A≔0 ← A. (9)


Inthiscase,theframeconditionsfor ∼Aand ¬Aarecashedoutinexplicit 

informationaltermsasfollows: 

x � ∼A[A → 0]iffforeach y,zs.t. x • y ⊑ z,if y � Athen z � 0, (10) 

x � ¬A[0 ← A]iffforeach y,zs.t. y • x ⊑ z,if y � Athen z � 0. (11) 

Themajorpointssofarhavebeentheinformationaltranslationofthe 

ternaryrelation R,suchthat Rxyzcomesoutasas x • y ⊑ z,andthe 

definitionofsplitnegationintermsofdoubleimplication,suchthat ∼A≔ 

A → 0and ¬A≔0 ← A. Thedefinitionalcomponenthereisimportant. 

Ourdoubleimplicationconnectives →and ←havetheirconditionsgiven 

by R,albeitunderaninformationalreading,in(3)and(4)respectively. 

Thismeansthatoursplitnegationconnectives ∼and ¬ultimatelyhave 

theirdefinitionsintermsoftheternaryrelationalso. 

3 AProceduralInterpretationofSplitNegation 

Howshouldweinterpret ∼Aand ¬Agiventheirrespectivedefinitions, 

A → 0and 0 ← A? Thetypeofanswerwegiveherewilldependon 

thedomain.Forexample,ifwewereworkingwithactions,thenwecould 

interpret ∼Aasthetypeofactionthatcannotbefollowedbyanaction 

oftype A,andwecouldinterpret ¬Aasthetypeofactionthatcannot 

followanactionoftype Aetc.Foranyinterpretationthatwegivetosplit 

negation,theinterpretationhastobecompatiblewithcertainproperties 

thatholduniversallyforanysplitnegation.Thepurposeofthissectionis 

tocheckthepresentlyproposedproceduralinterpretationofsplitnegation 

againsttheseproperties,whicharelistedas(12)–(18)below. 

Weareworkingwithinformation.Thisisstillfairlygeneralthough,and 

variousinformationalapplicationswilllikelyinfluenceourchoiceofinterpretation.Bytakingtheprocedural/dynamicturnandworkingwithinformationflow,theapplicationaspectatworkinbothfusionandthebinary 

combinationoperatorgettakenveryseriously.Inthiscase,thesuggestion 

isthatweinterpret ∼Aasthebodyofinformationthatcannotbeapplied 

tobodiesofinformationoftype A,andthatweinterpret ¬Aasthebody 

ofinformationthatcannothavebodiesofinformationoftype Aappliedto 

it.Theinterpretationissupportedbythemodeltheory;bytheinformation 

statessupporting ∼A, ¬A,and A.If xsupports ∼Aand ysupports A, x 

cannotbeappliedto y.Similarly,if xsupports ¬Aand ysupports A,then 

ycannotbeappliedto x. Thisisnotbecausesuchanapplicationwould 

causeanexplosionofinformation,butbecauseitcouldnevergenerateany 

information. Theinterpretationofsplitnegationintermsofrulingout 

particularinformationalapplicationsisnotgerrymandered. Itisdirectly 

supportedbytheframeconditionsfor ∼Aand ¬A.


Toseethis,notethattheframeconditionsininformationaltermsfor 

∼Alaidoutin(10)abovetellusthatthereisnoinformationresultingfrom 

theapplicationof xto ywhere x � ∼Aand y � A,since x • y ⊑ zand 

z � 0(and z � 0nowhere). Supposethoughthatweweretoattemptto 

apply ∼Ato A,inotherwordstoattempt ∼A ⊗A.Ininformationalterms, 

theframeconditionsforfusion(2)tellusthataninformationstate xwill 

support ∼A ⊗ Aiffforsomeinformationstates yand zsuchthat xisan 

informationaldevelopmentoftheapplicationoftheinformationin ytothe 

informationin z, ysupports ∼Aand zsupports A.However,weknowfrom 

ourdefinitionof ∼Aintermsof A → 0,thatthereisnostate xsuchthat 

itsupportstheapplicationof ∼Ato A,thisissimplywhat(10)tellsus. 

Supportfortherulingoutconditionson ¬Afromtheframeconditions 

for ¬Aworkssimilarly,andinvolvesonlyadirectionalchange.Theframe 

conditionsfor ¬Alaidoutin(11)abovetellusthatthereisnoinformation 

resultingfromtheapplicationoftheinformationstate ytotheinformation 

state xwhere y � Aand x � ¬Asince y • x ⊑ zand z � 0(and z � 

0nowhere). Ifweweretoattempttoapply Ato ¬A,inotherwords 

attempt A ⊗ ¬A,thentherewouldneedtobeaninformationstate xthat 

supported A⊗¬A,andthiswouldbethecaseiffthereweresomeinformation 

states yand zsuchthat ysupported Aand zsupported ¬Aand xwasan 

informationaldevelopmentoftheapplicationof yto z.Fromourdefinition 

of ¬Aintermsof 0 ← Ahowever,weknowthatthereisnostate xsuch 

thatitsupportstheapplicationof Ato ¬A,thisismarkedoutby(11). 

Giventhenon-gerrymanderednatureoftheinterpretationofsplitnegationintermsofproceduralprohibition,weshouldbeabletogiveanaturalinterpretationofgeneralproof-theoretic,henceinformation–processingpropertiesofsplitnegationinsuchterms.Foranysplitnegation,independentlyofwhichstructuralrulesarepresent,thefollowing(12)–(18)hold: 

A ⊢ B 

∼B ⊢ ∼A 

(12) 

(12)makessenseintermsoftherulingoutofinformationprocessingprocedures.Givenasplitnegation,andgivenalsothatinformationoftype 

B 

followsfrominformationoftype A,thenrulingouttheprocedure ∼A ⊗ A 

followsfromrulingouttheprocedure ∼B ⊗ B. Thisisjusttosaythat 

giventhatwecangetinformationoftype Bfrominformationoftype A, 

thenfromthebodyofinformationthatcanneverbeappliedtobodiesof 

type B,wecangetthebodyofinformationthatcanneverbeappliedto 

bodiesofinformationoftype A. 

A ⊢ B 

¬B ⊢ ¬A 

(13) 

Thereasoningwithregardsto(13)isdirectlyanalogoustothatsurrounding 

(12):Againgivenasplitnegation,andagaingiventhatinformationoftype


Bfollowsfromtheinformationoftype A,thenrulingouttheprocedure 

A ⊗ ¬Afollowsfromrulingouttheprocedure B ⊗ ¬B.Thisisjusttosay 

thatgiventhatwecangetinformationoftype Bfrominformationoftype 

A,thenfromthebodyofinformationthatcanneverhavebodiesoftype 

Bappliedtoit,wecangetthebodyofinformationthatcanneverhave 

bodiesofinformationoftype Aappliedtoit. Thereasoningsurrounding 

(14)and(15)isslightlymoreinvolvedthanin(12)and(13). 

A ⊢ ∼B 

B ⊢ ¬A 

B ⊢ ¬A 

A ⊢ ∼B 

Wecantake(14)and(15)together,gettingthesplitnegationproperty: 

(14) 

(15) 

A ⊢ ∼Biff B ⊢ ¬A. (16) 

Startingwiththeleft-to-right-handdirection: Ifwecan,onthebasisof 

informationoftype Aalone,getthebodyofinformationthatcanneverbe 

appliedtobodiesofinformationoftype B,thenonthebasisofinformation 

oftype Balone,wecangetthebodyofinformationthatcanneverhave 

bodiesofinformationoftype Aappliedtoit.Theintermediatestepisthis: 

Ifweweretoapply Ato B(i.e. A ⊗ B)thenwewouldgetnothing,viz. 

0,since A ⊗ B ⊢ 0,sinceif A ⊢ ∼Bthen A ⊗ B ⊢ 0. Assuch,from 

informationoftype Balonewecangetthebodyofinformationthatcan 

neverhavebodiesofinformationoftype Aappliedtoit. Theright-toto-left-handdirectionissimilar: 

Ifwecan,onthebasisofinformationof 

type Balone,getthebodyofinformationthatcanneverhavebodiesof 

informationoftype Aappliedtoit,thenweretoapply Bto A(i.e. B ⊗ A) 

thenwewouldgetnothing,viz. 0,since B ⊗ A ⊢ 0,sinceif B ⊢ ¬Athen 

B ⊗ A ⊢ 0.Assuch,thenfrominformationoftype Aalonewecangetthe 

bodyofinformationthatcanneverbeappliedtobodiesofinformationof 

type B.(17)and(18)aremorestraightforward: 

A ⊢ ¬∼A, (17) 

A ⊢ ∼¬A. (18) 

Onthebasisofinformationoftype Aalone,wecanruleoutthebodyof 

informationthatcanneverbecomposedwithbodiesofinformationoftype 

A.Thisisjusttosaythatwecanruleout A → 0.Thisisjustwhat(17) 

statesunderaprocedurallyfocusedinformationalinterpretation.Similarly, 

onthebasisofinformationoftype Aalone,wecanalsoruleoutthebody 

ofinformationthatcanneverhavebodiesofinformationoftype Aapplied 

toit.Inthiscasewearerulingout 0 ← A.Inthiscontext,“rulingout”is 

aformofproceduralprohibition.


Theproposalforaproceduralinterpretationofsplitnegationhasbeen 

thatweinterpret ∼A(thatis A → 0)asthebodyofinformationthat 

cannotbeappliedtobodiesofinformationoftype A,andthatweinterpret 

¬A(thatis 0 ← A)asthebodyofinformationthatcannothavebodiesof 

informationoftype Aappliedtoit.Wehaveseenthatthisinterpretationof 

splitnegationisentirelynaturaloncewetranslatetheternaryrelation Rinto 

itsinformationalform,inwhichcasetheinterpretationisdirectlysupported 

bytheframeconditionsfor ∼Aand ¬A,(10)and(11)respectively.Wehave 

alsoseenthattheinterpretationiscompatiblewiththeuniversallyvalidsplit 

negationproperties(12)–(18). 

4 Conclusion 

Wehaveseenhowitisthatwemayreconstructtheternaryrelationofframe 

semantics, Rxyzinexplicitlydynamicinformationalterms,as x • y ⊑ z. 

Thisdynamicinformationalreconstructioncarriesovertoanyconnective 

definedintermsoftheternaryrelation,allowingustogiveexplicitlyproceduralaccountsofdoubleimplicationandfusion.Sincewehaveuseddouble 

implicationtodefineasplitnegation, ∼A≔A → 0,and ¬A≔0 ← A,we 

haveaproceduraldefinitionofsplitnegation. 

Giventhedefinitionofsplitnegationinthesedynamicinformational 

terms,wehavebeenableto“readoff”anaturalproceduralinterpretation 

ofsplitnegation.Thisinterpretationhasbeenshowntobecompatiblewith 

theuniversallyvalidpropertiesofasplitnegation. 

SebastianSequoiah-Grayson 

FormalEpistemologyProject 

IEG–ComputingLaboratory–UniversityofOxford 

sebsequoiahgrayson@hiw.kuleuven.be 

http://users.ox.ac.uk/∼ball1834/index.shtml 

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1 Theproblem 

Reference to Indiscernible Objects 

Stewart Shapiro ∗ 

Somecriticsofmyanteremstructuralism(Shapiro,1997)arguethatI 

haveanissuewithstructuresthathaveindiscernibleplaces. 1 Astructure 

issaidtobe“rigid”ifitsonlyautomorphismisthetrivialonebasedonthe 

identitymapping.Themainexemplarsoftheallegedproblemarenon-rigid 

structures.Itisaneasytheoremthatisomorphicstructuresareequivalent: 

Let fbeanautomorphismonagivenstructure Mandlet Φ(x1,...,xn) 

beanyformulainthelanguageofthestructure. Thenforanyobjects 

a1,... ,aninthedomainof M, Msatisfies Φ(a1,... ,an)ifandonlyif M 

satisfies Φ(fa1,...,fan).If fisanon-trivialautomorphism,thenthereis 

anobject asuchthat fa �= a. Inthiscase, aand faareindiscernible,at 

leastconcerningthelanguageofthestructure:anythingtrueofoneofthem 

willbetrueoftheother.Sonon-rigidstructureshaveindiscernibleobjects. 

Themost-citedexampleisthatofcomplexanalysis.Startwiththelanguageoffields,andconsiderthealgebraicclosureofthereals,whichis 

uniqueuptoisomorphism(initssecond-orderformulation).Thecomplex 

numbersaretheintendedmodel.Thefunctionthattakesacomplexnumber 

a + bitoitsconjugate a − biisanautomorphism. Itfollowsthatfor 

anyformula Φ(x),withonly xfree, Φ(a + bi)ifandonlyif Φ(a − bi). In 

particular, Φ(i)ifandonlyif Φ(−i). So iand −iareindiscernible;they 

∗ Igaveearlyversionsofpartsofthispaperatthephilosophyofmathematicsworkshop 

atOxford,theArchéResearchCentreattheUniversityofSt. Andrews,OhioState 

UniversityandtheUniversityofMinnesota. Thankstoalloftheaudiencesthere. I 

amindebtedtoCraigeRoberts,GabrielUzquiano,OfraMagidor,CathyMüller,Dan 

Isaacson,GrahamPriest,KevinScharp,RobertKraut,andJasonStanley. 

1 Theearlycriticsinclude(Burgess,1999,pp.287–288),(Hellman,2001,pp.192–196), 

and(Keränen,2001,2006). Morerecentparticipantsinthedebateinclude(Ladyman, 

2005),(Button,2006),(Ketland,2006),(MacBride,2005, §3),(MacBride,2006b),and 

(Leitgeb&Ladyman,2008). Myowncontributionsinclude(Shapiro,2006b),(Shapiro, 

2006a),and(Shapiro,2008).

232 StewartShapiro 

havethesameproperties,atleastamongthosethatcanbeexpressedin 

thelanguage. Anotheroft-citedexampleisEuclideanspace,wherethings 

areevenworse.AnytwopointsinEuclideanspacecanbeconnectedwith 

alineartranslation,whichisanautomorphism. So,itseems,allofthe 

pointsinEuclideanspaceareindiscernible,atleastwithrespecttopropertiesthatcanbeexpressedinthelanguageofgeometry.HannesLeitgeband 

JamesLadyman(2008)pointoutthatsincesome(simple)graphshaveno 

relations,anybijectiononthemisanisomorphism.Sowiththosegraphs, 

everypointisindiscerniblefromeveryother.Thesimplestofthesesimple 

graphsareisomorphictothefinitecardinalstructuresintroducedinmy 

chapteronepistemologyin(Shapiro,1997,Ch.4). 

Whyisthisaproblemforanteremstructuralism? Someill-chosenremarksinmybookatleastsuggestaprincipleoftheidentityofindiscernibles, 

which,inlightofexampleslikethese,wouldreducetheviewtoabsurdity. 

I’dbecommittedtosayingthat i = −i,andthatthereisonlyonepointin 

Euclideanspace. Butthereislittlepointintryingtofigureoutwhatmy 

viewwas.Irejecttheidentityofindiscerniblesnow. 

Muchofthediscussionofthisissueismetaphysical.JohnBurgess(1999, 

p.288)pointsoutthatalthoughthetwocomplexrootsof −1aredistinct, 

onmyview“thereseemstobenothingtodistinguishthem.”Thisseemsto 

invokesomethingintheneighborhoodofthePrincipleofSufficientReason. 

Ifsomethingisso,thentheremustbesomethingthatmakesitso,oratleast 

somethingthatexplainswhyitisso. JukkaKeranänen(2001)articulates 

ageneralmetaphysicalthesisthatanyonewhoputsforwardatheoryof 

atypeofobjectmustprovideanaccountofhowthoseobjectsaretobe 

“individuated”.AccordingtoKeränen,foreachobject ainthepurviewof 

aproposedtheory,wehavetobetold“thefactofthematterthatmakes a 

theobjectitis,distinctfromanyotherobject”ofthetheory,by“providing 

auniquecharacterizationthereof.” 

Someauthorsenteredthediscussion,onmybehalf,bysuggestingmetaphysicalprinciplesthatareweakerthanKeränen’sindividuationrequirementbutstillmeetBurgess’sdemandthatthetheoristfindsomethingthat 

distinguishesdistinctobjects.Theidea,itseems,isthatonecandistinguish 

objectswithoutindividuatingthem.Theweakestoftheserequirementsis 

athesisthatforany a, b,if a �= bthenthereisanirreflexiverelation R 

suchthat Rab(Ladyman,2005).ComplexanalysisandEuclideangeometry 

easilypassthistest. Forexample, iand −isatisfytheirreflexiverelation 

ofbeingadditiveinversetoeachotheranddistinctfrom 0,andanypairof 

distinctpointsinEuclideanspacesatisfytheirreflexiverelationofdeterminingexactlyonestraightline.Nevertheless,thefinitecardinalstructuresand 

somegraphsstillfailthetest,unlessnon-identitycountsasanirreflexive 

relation(inwhichcase,ofcourse,wedonothaveasubstantialtest).


Iwishtoputasidethesemetaphysicalmattershere,atleastasfaras 

possible.Therearesomerelatedand,Ithink,moreinterestingissuesconcerningthesemanticsandpragmaticsofmathematicallanguages,andperhapslanguagesgenerally.Theseissuesalsobearonlogic,andtheygowellbeyondlocaldisputesconcerninganteremstructuralism.Howdowemanagetotalkabout,andthus,insomesense,refertoindiscernibleobjects? 

Idonotintendtoofferadetailedsolutiontothissemanticproblemhere, 

justtohighlightitandindicateitsgenerality.Asolution,Ibelieve,would 

involveanextendedforayintolinguistics,thephilosophyoflanguage,and 

thephilosophyoflogic. 

Tobesure,thewholeprojectpresupposesthatthereareindiscernible 

objects,andthispresupposesthatthemetaphysicalprinciplesadoptedby 

someofmyopponentsarefalse. Ialsodonotintendtoargueforthatin 

detailhere(butsee(Shapiro,2008)andrelatedworkcitedthere). The 

informallanguageofcomplexanalysishasaterm iwhichissupposedto 

denoteoneofthesquarerootsof −1.Atleastgrammatically, iisaconstant, 

apropername.Andtheroleofaconstantistodenoteasingleobject—at 

leastinasufficientlyregimentedlanguage. Butwhichofthesquareroots 

does idenote?Isitnotasifthemathematicalcommunityhasmanagedto 

singleoutoneoftheroots,inordertobaptizeitwiththename“i”.They 

cannotdoso,asthetworootsareindiscernible. 

GottlobFrege(1884)seemstohavenotedourproblem: 

Wespeakof‘thenumber 1’,wherethedefinitearticleservestoclass 

itasanobject(§57). If,however,wewishedtouse[a]conceptfor 

defininganobjectfallingunderit[byadefinitedescription],itwould, 

ofcourse,benecessaryfirsttoshowtwodistinctthings: 

thatsomeobjectfallsundertheconcept; 

thatonlyoneobjectfallsunderit(§74n). 

Nothingpreventsusfromusingtheconcept‘squarerootof −1’;but 

wearenotentitledtoputthedefinitearticleinfrontofitwithout 

moreadoandtaketheexpression‘thesquarerootof −1’ashavinga 

sense.(§97) 

Complexanalysisisperhapsthemostsalientexampleofthelogicalsemanticphenomenoninquestion,butthereareasomeothers,atleast 

ifwegowithastraightforwardreadingofvariousmathematicallanguages 

(see(Brandom,1996)). Consider,forexample,theintegers,withaddition 

astheonlyoperation. Itis,ofcourse,anAbeliangroup,whoseelements 

are: 

... , −3, −2, −1,0,1,2,3,...


Intherelevantlanguage,theoperationthattakesanyinteger ato −ais 

anautomorphism. Soanythingintherelevantlanguagethatholdsofan 

integer aalsoholdsof −a. Inthisstructure,then, 1isindiscerniblefrom 

−1,but,ofcourse, 1isdistinctfrom −1. AnotherexampleistheKlein 

group. Ithasfourelements,whichareusuallycalled e, a, b,and c,and 

thereisoneoperation,givenbythefollowingtable: 

e a b c 

a e c b 

b c e a 

c b a e 

Itiseasytoverifythatanyfunction fthatisapermutationonthefour 

elementssuchthat fe = eisanautomorphism. Thethreenon-identity 

elementsarethusindiscernible,inthelanguageofgroups,andyetthereare 

threesuchelementsandnotjustone.Butwhichoneis a? 

Categorytheoryisrampantwithexamplesofthephenomenonunder 

studyhere. Themainreasonforthis,itseems,isthateverycategorical 

notionispreservedunderisomorphism.Totakeoneinstance,anobject O 

inacategoryiscalledterminal,ifforanyobject Ainthecategory,there 

isexactlyonemapfrom Ato O.Incategorieswithaterminalobject,itis 

commontointroduceaterm“1”forsuchanobject. Itistrivialtoshow 

thatanytwoterminalobjectsareisomorphic.Indeed,if 1and 1 ′ areboth 

terminal,thenthereisexactlyonemapfrom 1to 1 ′ andexactlyonemap 

from 1 ′ to 1.Thesetwomapsmustcomposetoanidentitymap—either 

theuniquemapfrom 1to 1,ortheuniquemapfrom 1 ′ to 1 ′ ,dependingon 

theorderofcomposition. Moreover,anyobjectisomorphictoaterminal 

objectisitselfterminal. Soifacategoryhasaterminalobject,itusually 

hasmany. Inthecategoryofsets,forexample,anysingletonisterminal. 

Whichofthemistheterminalobjectofthecategory,theonepickedoutby 

theterm“1”?Theanswer,ofcourse,isthatitdoesnotmatter—justasit 

doesnotmatterwhichsquarerootof −1is i.And,heretoo,“1”functions 

asasingularterm,atleastasfarassyntaxgoes. 

Inacategorywithaterminalobject,itiscommontodefineanelement 

ofanobject Atobeamapfrom 1to A.So,itwouldseem,toknowwhich 

mapsaretheelementsof A,wehavetoknowwhichobjectis 1.Inasense, 

wecan’tknowthis,but,again,itdoesnotmatter.Inlikemanner,aproduct 

oftwoobjects A, Bisanobject,usuallywritten A ×B,andapairofmaps, 

onefrom A × Bto Aandonefrom A × Bto B,thatsatisfiesacertain 

universalproperty. Again,productsarenotusuallyunique: anyobject


isomorphictoaproductof Aand Bcanitselfbeshowntobeaproduct 

of Aand B.Nevertheless,the“×”symbolseemstobeafunctionsymbol, 

anditiscommontotalkabout A × Bastheproductof Aand B—just 

asitiscommontotalkabout iasthesquarerootof −1. 

Thereisarelatedphenomenonconcerningtheuseparametersorfreevariablesinmathematicaldiscourse.Thoseactlikesingulartermsincontext, 

butoftenfailuniqueness.Supposethatinthecourseofademonstration,a 

geometersays“let ABCDbeanyparallelogram,withtheline ABcongruentandparalleltotheline 

CD.” Itfollowsthatthepairofpoints A, B, 

(andthelinesegment AB)areindiscerniblefromthepair C, D(andthe 

segment CD).Anythingonecansayaboutoneofthepairs(andoneofthe 

segments)willbetrueoftheotherpair(andtheothersegment).Sowhich 

oneis ABandwhichoneis CD? 

2 Indiscernibility,semantics,andexpressiveresources 

Tosaythattwoobjectsareindiscernibleistosaythattheycannotbetold 

apart. Thisbriefcharacterizationsuggeststhatindiscernibilityisrelative. 

Twoballsmaybevisuallyindiscernibletosomeonewhoiscolorblindwhile 

beingdiscernibletosomeonewithmorenormalvision.Inthepresentcontext,indiscernibilityisrelativetoexpressiveresources.Twomathematical 

objectsmaybeindiscerniblewithsomebatchofresources,butdiscernible 

onceexpressiveresourcesareadded. Inthecaseoftheintegers,forexample, 

1and −1arediscernibleifonecaninvokemultiplication: 1isthe 

multiplicativeidentityand −1isnot. 

Attheoutset,Iformulatedtheissueintermsofrigidstructures,with 

“rigidity”definedintermsofautomorphisms.Thisregisterstherelativity 

toexpressiveresourcesaswell,since“automorphism”isitselfdefinedin 

termsofthebackgroundlanguage: alloftheprimitiverelationsmustbe 

preserved. 

Suppose,then,thatwejustaddaconstant itotheofficiallanguageof 

complexanalysis,withtheobviousaxiom i 2 = −1. Then,technically,the 

structurebecomesrigid:therearenonon-trivialautomorphisms.Thereasonisthatisomorphismsmustpreserveallofthestructureinthelanguage, 

and,inparticular,itmustpreservethedenotationsoftheconstants. If f 

isanisomorphismbetween M1and M2,inthelanguageofarithmetic,for 

example,theniftheconstant“0”denotes ain M1,then“0”mustdenote fa 

in M2.Similarly,let Nbeanymodelofcomplexanalysis,intheenvisioned 

languagewithaconstant“i”,andlet fbeanautomorphism.If“i”denotes 

ain N,then“i”denotes fain N.Thatis, a = fa.Itfollowsthatforeach 

element binthedomain, fb = b.So fistrivial.


Nevertheless,itseemstomethat,intherelevantintuitivesense,thetwo 

squarerootsof −1arestillindiscernible,eveninthislanguage.Let N ′ be 

amodelofthetheorythatisjustlike N,exceptthatin N ′ ,“i”denotes −a 

(andthus“−i”denotes a).Technically, Nisnotisomorphicto N ′ ,forthe 

abovereason.However,itseemstomethatthetwomodelsareequivalent, 

intheintuitivesense. Bothhavethesamedomainandtheyagreeonthe 

operations.Inparticular,ineachmodel,thesametwoobjectsaretheroots 

of −1. Theonlydifferencebetweenthemisthat Ncallsoneofthem“i”: 

and N ′ callstheotherone“i”.Itseemstomethatthisisnotasignificant 

difference—notunlessweaddsomestructuretothenamingrelation. 

Todevelopthispointabit,letusgoupalevel,sotospeak,andthink 

ofthesemanticrelationsthemselvesinformal,orstructuralterms. Begin 

withasimplegraphthathastwonodesandnoedges. Asnotedabove, 

thisstructureiscompletelyhomogeneous.Nowaddtwonewobjects, a, b, 

andarelation Rtothestructure. Thenewitem abears Rtooneofthe 

nodesintheoriginalgraphand bbears Rtotheothernode. Thisisthe 

structureofsomeverysimplesemanticrelationsonthegraph:thinkof a 

and basnames,and Rasthereferencerelation.Thismathematical-cumsemanticstructureisnotrigid.Ifwemodifyitbyswitchingthe“referents” 

of aand b,wegetanautomorphism. And,intuitively,wehavenotreally 

changedtheextendedstructurewiththisswitch.Itisstillthesamesimple 

graphwiththesametwonewobjects,thesamerelation R,andthesame 

structural–semanticrelations. 

Wecandothesamewithourmorestandardmathematicalexample. 

Considerastructure Mthatincludestheplacesandrelationsofourmodel 

N,ofcomplexanalysis.Inaddition, Mhasnodesrepresentingtheprimitive 

terms ofthelanguageofcomplexanalysis(“0”,“i”,“+”,“A”),anda 

relation Rrepresentingreference.So,forexample, Rixwouldbeanatomic 

formulaintheenvisionedobjectlanguage,sayingthattheconstant“i”refers 

to,ordenotes, x.Thetheorywouldincludetheaxiomsofcomplexanalysis 

(over N)andtheTarskiansatisfactionclausesbetweenthe“terms”(“0”, 

“i”,“+”,“A”)andtherelevantitemsconstructedfrom N.So,forexample, 

ourstructurewouldsatisfy ∀x∀y((Rix&Riy) → x = y)and Ria(recalling 

that aisoneofthesquarerootsof −1in N). 

Thismathematical-cum-semanticstructureisnotrigid. Thefunction 

thattakes x+ayto x−ay(within N),takeseach“term”(“0”,“1”,“i”,“+”, 

“A”)toitself,andadjuststherelation Raccordingly,isanautomorphism. 

Westillhaveamodelofcomplexanalysis,asabove,andalloftheTarskian 

satisfactionclausesarestillsatisfied(see(Leitgeb,2007,pp.133–134)).The 

problem,here,istosaysomethingaboutthesemanticsandlogicofthe 

languagesofmathematics,soconstrued.


3 Theidentityofindiscernibles 

TheissuesherearerelatedtothoseinMaxBlack’s(1952)celebrateddiscussionoftheidentityofindiscernibles.Thepaperisintheformofadialogue. 

Onecharacter, A,takestheidentityofindiscerniblestobe“obviouslytrue”, 

whiletheopponent, B,takesittobe“obviouslyfalse”.Thelattergivesa 

thoughtexperimentmeanttorefutetheprincipleinquestion: 

Isn’titlogicallypossiblethattheuniverseshouldhavecontainednothingbuttwoexactlysimilarspheres?Wemightsupposethateachwas 

madeofchemicallypureiron,hadadiameterofonemile,thatthey 

hadthesametemperature,color,andsoon,andthatnothingelse 

existed. Theneveryqualityandrelationalcharacteristicoftheone 

wouldalsobeapropertyoftheother.(Black,1952,p.156) 

Black’stwospheresareanalogoustothetwosquarerootsof −1.Ofcourse, 

Iamnotclaimingthatthereisapossibleworldwhichconsistsofjustthese 

twocomplexnumbers. Buttherestoftheanalogyholds,atleastinthe 

languageofcomplexanalysis. 

Themainthrustof(Black,1952)ismetaphysicaland,asnotedabove, 

suchmattersarebeingputasidehereasmuchaspossible.Alongtheway, 

however,thearticlebroachesthelogico-semanticissuesofpresentconcern. 

Thedefenderoftheindiscernibilityofidenticals, A,firstdeniesthat Bhas 

describedacoherentpossibility,andthencontinues,“Butsupposingthat 

youhavedescribedapossibleworld,Istilldon’tseethatyouhaverefuted 

theprinciple.Consideroneofthespheres, a.”Atthispoint, Binterrupts, 

protesting:“HowcanI,sincethereisnowayoftellingthemapart?Which 

onedoyouwantmetoconsider?”.Thatis, Brefusestolethisopponentuse 

avariable,aparameter,orasingulartermforoneofthespheres.Character 

Aresponds:“Thisisveryfoolish.Imeaneitherofthetwospheres,leaving 

youtodecidewhichoneyouwishedtoconsider.” Inourcase,itstrikes 

measeminentlyreasonabletosay,“let idesignateoneofthesquareroots 

of −1. Idon’tcarewhich.” Thepresentproblemistomakesenseofthis 

locution.Character Bdeniesthatitcanbemadesenseof. 

RobertBrandom(1996,p.298)putsourprobleminsimilarterms: 

Nowifweaskamathematician‘Whichsquarerootof −1is i?,’she 

willsay‘Itdoesn’tmatter:pickone.’Andfromamathematicalpoint 

ofviewthisisexactlyright.Butfromthesemanticpointofview,we 

havetherighttoaskhowthistrickisdone—howisitthatIcan 

‘pickone’ifIcan’ttellthemapart?WhatmustIdoinordertobe 

pickingone,andpickingone? Forwereallycannottellthemapart 

—and... notjustbecauseofsomelamentableincapacityofours.


ThenextexchangeinBlack’sdialogueputssomedetailtothediffering 

semanticpresuppositions.Character A,theproponentofthe(obvioustruth 

of)theidentityofindiscernibles,continues,“IfIweretosaytoyou‘Take 

anybookofftheshelf’itwouldbefoolishonyourparttoreply‘Which?’.” 

Bretorts:“It’sapooranalogy.Iknowhowtotakeabookoffashelf,but 

Idon’tknowhowtoidentifyoneofthetwospheressupposedtobealone 

inspace...” Itseemsthat,forthepurposesofthisargument, Bclaims 

thatonecannotuseasingulartermtodesignateanobjectwithoutfirst 

“identifying”it,oratleastknowinghowtoidentifyoneoftheobjectsin 

question.That,Itakeit,isthematterathandhere,whetheritispossible 

tointroducealexicalitemtorefertoanindiscernibleobject.Thecharacter 

Atakesthebait:“Can’tyouimaginethatonespherehasbeendesignated 

as‘a’?”Thedialoguecontinues: 

B. Icanimagineonlywhatislogicallypossible. Nowitislogically 

possiblethatsomebodyshouldentertheuniverseIhavedescribed, 

seeoneofthespheresonhislefthandandproceedtocallit‘a’... 

A.Verywell,nowletmetrytofinishwhatIbegantosayabout a... 

[ellipsisinoriginal] 

B. Istillcan’tletyou,becauseyou,inyourpresentsituation,have 

norighttotalkabout a.AllIhaveconcededisthatifsomethingwere 

tohappentointroduceachangeinmyuniverse,sothatanobserver 

enteredandcouldseethetwospheres,oneofthemcouldthenhave 

aname. ButthiswouldbeadifferentsuppositionfromtheoneI 

wantedtoconsider.Myspheresdon’tyethavenames... Youmight 

justaswellaskmetoconsiderthefirstdaisyinmylawnthatwould 

bepickedbyachild,ifachildweretocomealonganddothepicking. 

Thisdoesn’tnowdistinguishanydaisyfromtheothers.Youarejust 

pretendingtouseaname. 

A.AndIthinkyouarejustpretendingnottounderstandme. 

4 Isthisaproblem? 

Whatseemstomatterhere,ataminimum,isone’sphilosophyofmathematics,andone’saccountofreference. 

Ifsomeonehasaphilosophyof 

mathematicsthatacceptsaprincipleoftheidentityofindiscernibles,and 

alsoacceptsacertainnaiveaccountofwhatindiscernibilitycomesto,then 

hewillnotallowtheforegoing,implicitcharacterizationofthecomplex 

numbersasthealgebraicclosureofthereals(orasthestructurecharacterizedbythestandardaxiomatization). 

Thatverycharacterizationviolates 

theidentityofindiscernibles,sinceitintroducestwodistinctobjectsthat 

cannotbetoldapart. Thesamegoesfortheotherexamples,theintegers


underaddition,theKleingroup,andallofcategorytheory.Thephilosopher 

whoacceptsthesestrictureswillnotfacetheforegoingproblemofreference 

—sincethereisnosuchproblemonthatview—butshewillneedsome 

otheraccountofthevariousstructuresandtheories. 

Onewaytoavoidtheproblemistobreakthesymmetry.Whetherone 

acceptstheidentityofindiscerniblesornot,onecanthinkofcomplexnumbersaspairsofreals,followinganowcommonmathematicaltechnique.In 

thatcase, iisthepair 〈0,1〉and −iis 〈0, −1〉.Sinceonecandistinguish 1 

from −1inthereals,andthusinthesecondcoordinate,thereisnoproblem 

distinguishingthosetwopairs. 

Theproblemwithcomplexanalysis,however,atleastappearstobe 

robust—liabletoreappear. Thecustomnowadaysistousepolarcoordinates,inwhichcase 

iis � 1, π 

� � � 

3π 

2 and −iis 1, 2 . Soitnowbecomesa 

matterofdistinguishingthosepairsfromeachother.This,inturn,becomes 

amatterofdistinguishingtheangle π 

3π 

2fromtheangle 2 . Andhowdoes 

onedothat? Well,wecansaythatthefirstispositiveandthesecondis 

negative;orthatthefirstgoescounterclockwiseandthesecondclockwise; 

orthatthefirstisabovethe x-axisandthesecondbelowit. Butwhen 

itcomestoangles,“positive–negative”,“counterclockwise–clockwise”,and 

“above–below”allpointtosymmetries—automorphismsoftheplane.So 

itseemsthattheindiscernibilityhasreturned.Tobreakthesymmetryin 

thecomplexnumbers,weneedtobreakthesymmetryintheplane. 

Asnotedabove,onecanbreakthesymmetryintheintegersunderadditionbythinkingofthatasasubstructureoftheintegersunderaddition 

andmultiplication. Thereisnoproblemdistinguishing 1from −1inthat 

structure.AndperhapsonecandenythatthereissuchathingastheKlein 

group.Instead,thereareanumberofKleingroups.Ineachsuchgroup,the 

fourelementsareproperlyindividuated.Onewouldhavetogiveasimilar 

interpretationofthelanguagesinvokedincategorytheory.Onemightjust 

breakthesymmetryglobally—onceandforall—byinsistingthatthe 

ontologyofallofmathematicsistheiterativehierarchy,orsomeotherrigid 

structure. 

Allthiscouldbedone,ofcourse. Thetechnicalresourcesrequiredare 

well-known.Notice,however,thattheneedtobreakthesymmetryinvolves 

reinterpretingthelanguagesofmathematics. Onequestionwouldconcern 

hownaturalthereinterpretationsare,asreadingsoftheoriginallanguages 

ofmathematics. 

IsuspectthattherewouldnotbeaproblemforFregeconcerningour 

issue. Asnotedabove,hedemandedtwothingsbeforeonecouldusethe 

definitearticleinaproperlyrigorousmathematicaltreatment.Onehasto 

show“thatsomeobjectfallsunder”theconceptinquestion,andtheotheris 

“thatonlyoneobjectfallsunderit”(Frege,1884, §74n).Itwouldnotdo,for


Frege,tosimplydeclarethatthecomplexnumbersarethealgebraicclosure 

ofthereals,oreventosaythatweareinterestedinanalgebraicclosureof 

thereals.Indoingthis,wewouldfailthefirstrequirement,ofshowingthat 

someobjectfallsundertheconcept“squarerootof −1”.Howdoweknow 

thatthereareanyalgebraicclosuresoftherealnumbers?Presumably,Frege 

wouldhavegivenanexplicitdefinitionofthecomplexnumbers,perhapsas 

pairsofreals(which,inturn,wouldbedefinedintermsofcertaincoursesof-values). 

2 Thisexplicitdefinitionwouldpresumablybreakthesymmetry 

between iand −i. Ifitdidn’t,thentheaccountwouldfailFrege’ssecond 

requirement,“thatonlyoneobjectfallsunder”theconcept. Inthatcase, 

Fregewouldnotallowtheuseofthedefinitearticle. 

Consider,next,anominalist,aphilosopherwhodeniestheexistenceof 

mathematicalobjects.Onsuchviews,mathematicshasnodistinctontology. 

Thenthedevilisinthedetailsoftheview. Idonotseeanissueherefor 

afictionalist,onewholikensmathematicstomake-believe.Onecansurely 

tellacoherentstoryaboutobjectsthatareindiscernibleasfarasthedetails 

providedbythestorygo(Black’scharacter Bnotwithstanding).Consider, 

forexample,thefollowingshortstory: “Oneday,twopeoplemet,fellin 

love,andlivedtogether,happilyeverafter.”Whateveritsliterarymerits, 

thisissurelyacoherentpieceoffiction.Inbothcases,thereisnothinginthe 

storytodistinguishthecharacters. Anythinginthelanguageofthestory 

thatholdsofonealsoholdsoftheother.And,ofcourse,wehavenothing 

togoonbesidesthedetailsofthestory,pluscommonknowledgeofhuman 

psychology,namingconventions,andthelike. Considerthisvariationon 

ourstory:“Oneday,ChrismetKelly.Theyfellinlove,andlivedtogether, 

happilyeverafter.” Onemightclaimthat,here,Chrisisdistinguished 

fromKellybecauseheorsheistheonewhoiscalled“Chris”inthatstory. 

But,aswithcomplexanalysis,thisdoesnotseemlikeadistinctionthat 

matters. Tobesure,thereareinterestingissuesconcerningthesemantics 

and,perhaps,theontologyandmetaphysicsoffiction,butIdonotpropose 

toenterthatrealmhere. 

Reconstructivenominalistsprovidetranslationsofmathematicallanguagesintovocabularythatdoesnotcommitthemathematiciantothe 

existenceofmathematicalobjects. Typically,singulartermsandbound 

variablesinmathematicallanguagesarerenderedasboundvariableswithin 

thescopeofmodaloperators. Insteadofspeakingofwhatexists,thereconstructivenominalistspeaksofwhatmightexist(asinGeoffreyHellman(1989)),whatcanbeconstructed(alaCharlesChihara(1990)),orwhatfollowsfromaxioms,orwhatever.Whetheranissueanalogoustothepresent 

2 ThankstoMichaelDummettforsomekeyinsights.Onemustbespeculativehere,since 

Fregeonlygavethebaresthintsathowrealanalysisistofitintohislogicistprogram 

(see,forexample,(Simons,1987)).


onearisesdependsonthedetailsofthetranslation,andIproposetoavoid 

thataswell. 3 

Suppose,finally,thatsomeonedoesaccepttheexistenceofmathematical 

objects—contranominalism—andagreesthatinsomecases,distinct 

objectsareindiscernible—contratheprogramofsymmetry-breaking-viareinterpretation.Forpresentpurposes,itdoesnotseemtomatterwhatthe 

metaphysicalnatureofthesemathematicalobjectsmaybe.Ourphilosopher 

maybeatraditionalplatonist,orshemayholdthatmathematicalobjects 

aresomehowmentalconstructions,orthattheyaresocialconstructs,or 

whateverelsethephilosopherdreamsup. Ourphilosophermayevenbe 

aquietistaboutmathematicalontology,insistingthattheonlythingsto 

sayaboutthemarewhatfollowsfromthemathematicaltheories.Allthat 

matters,fornow,isthatthelanguagesbeunderstoodliterally,andthat 

somenumericallydistinctobjectsareindistinguishable. 

Withoutmuchlossofgenerality,wemightaswellkeeponwithour 

standardexample:ourphilosopherholdsthatthecomplexnumbersexist, 

thatthesquarerootsof −1areindiscernible,andthatthereisaroleplayed 

bytheterm“i”.Thenourproblemarises.Hemusteithercomeupsomehow 

withareferentfor‘i’,whichwouldbetobreakthesymmetry,orelsehe 

mustdescribethelogico-semanticroleofthatterm. 

Thisisnottheplacetoattempttosolvethepresentproblem.Thatisa 

matterforfutureworkwhich,Ibelieve,involvessubstantialthemesinsemantics,pragmatics,andlogic,bothforthelanguagesofmathematicsand 

fornaturallanguagesgenerally. Whatistheroleandfunctionofsingular 

terms(orlinguisticitemsthatlookandfunctionlikesingularterms),and 

howdosuchtermsgetintroducedintothelanguage? Thepurposeofthe 

presentarticleistoarticulatetheissue,andtodelimittherangeofphilosophersofmathematicsforwhomitisasubstantialissue.Attheveryleast, 

Ihopetohaveconvincedthegentlereaderthatitisnotaproblemlocalto 

anteremstructuralism. 

StewartShapiro 

DepartmentofPhilosophy,TheOhioStateUniversity 

350UniversityHall,230NorthOvalMall,Columbus,Ohio43210,USA 

shapiro.4@osu.edu 

3 Chihara’s(1990)modalconstructivismisaninterestingcasehere. Accordingly,asingularterm,suchasanumeral,representsthepossibilityofconstructinganopensentence 

withcertainsemanticfeatures. Soonecanwonderwhichopensentencewouldcorrespondto“i,asopposedtotheopensentencethatcorrespondswith“−i”.Ipresumethat 

Chiharawouldlikencomplexnumberstopairsofreals,asabove.Thiswouldavoidthe 

(analogueof)thepresentproblem,bybreakingthesymmetry.


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Sequent Calculi and 

Bidirectional Natural Deduction: 

On the Proper Basis of Proof-theoretic Semantics 

Peter Schroeder-Heister ∗ 

Philosophicaltheoriesoflogicalreasoningareintrinsicallyrelatedtoformal 

models.ThisholdsinparticularofDummett–Prawitz-styleproof-theoretic 

semanticsandcalculiofnaturaldeduction.Basicphilosophicalideasofthis 

semanticapproachhaveacounterpartinthetheoryofnaturaldeduction. 

Forexample,the“fundamentalassumption”inDummett’stheoryofmeaning(Dummett,1991,p.254andCh.12)correspondstoPrawitz’sformal 

resultthateveryclosedderivationcanbetransformedintointroduction 

form(Prawitz,1965,p.53). Examplesfromotherareasinthephilosophy 

oflogicsupportthisclaim. 

Ifconceptualconsiderationsaregeneticallydependentonformalones, 

wemayaskwhethertheformalmodelchosenisappropriatetotheintended 

conceptualapplication,and,ifthisisnotthecase,whetheraninappropriate 

choiceofaformalmodelmotivatedthewrongconceptualconclusions.We 

willposethisquestionwithrespecttotheparadigmofnaturaldeduction 

andproof-theoreticsemantics,andpleadforGentzen’ssequentcalculusasa 

moreadequateformalmodelofhypotheticalreasoning.Ourmainargument 

isthatthesequentcalculus,whenphilosophicallyre-interpreted,doesmore 

justicetothenotionofassumptionthandoesnaturaldeduction. Thisis 

particularlyimportantwhenitisextendedtoawiderfieldofreasoningthan 

justthatbasedonlogicalconstants. 

Toavoidconfusion,aterminologicalcaveatmustbeputinplace:When 

wetalkofthesequentcalculusandthereasoningparadigmitrepresents, 

wemean,asitscharacteristicfeature,itssymmetryorbidirectionality,i.e., 

∗ ThisworkhasbeensupportedbytheESFEUROCORESprogramme“LogiCCC—ModellingIntelligentInteraction”(DFGgrantSchr275/15–1)andbythejointGerman-French 

DFG/ANRproject“HypotheticalReasoning:LogicalandSemanticalPerspectives”(DFG 

grantSchr275/16–1).IwouldliketothankLucaTranchiniandBartoszWięckowskifor 

helpfulcommentsandsuggestions.

246 PeterSchroeder-Heister 

thefactthatitusesintroductionrulesforformulasoccurringindifferent 

positions. Wedonotassumethatthesepositionsaresyntacticallyrepresentedbytheleftandrightsidesofasequent,i.e.,wedonotsticktothe 

sequentformatwhichgavethecalculusitsname.Inparticular,wepropose 

anatural-deductionvariantofthesequentcalculuscalledbidirectionalnaturaldeduction,whichembodiesthebasicconceptualfeaturesofthesequent 

calculus. 1 Conversely,thenatural-deductionparadigmtobecriticizedis 

thereasoningbasedon(conventional)introductionandeliminationinferences,eventhoughitcanbegivenasequent-calculusformatasinso-called 

“sequent-stylenaturaldeduction”. 2 Theconceptualmeaningofnaturaldeductionvs. 

sequentcalculus,whichwetrytocapturebythenotionsof 

unidirectionalityvs. bidirectionality,istobedistinguishedfromtheparticularsyntaxofthesesystems. 

Wehopeitwillalwaysbeclearfromthe 

contextwhetheraconceptualmodeloraspecificsyntacticformatismeant. 

Wedonotclaimoriginalityforthetranslationofthesequentcalculus 

intobidirectionalnaturaldeduction.Thistranslationisspelledoutindetail 

in(vonPlato,2001).Thesystemitselfhasbeenknownmuchlonger. 3 Here 

wewanttomakeaphilosophicalpointconcerningtheproperconceptof 

hypotheticalreasoningthatalsopertainstoapplicationsbeyondlogicand 

logicalconstants. Theterm“bidirectionalnaturaldeduction”seemstous 

tobeaveryappropriatecharacterizationofthesystemconsidered.Toour 

knowledge,ithasnotbeenusedbefore. 4 

1 Assumptionsinnaturaldeduction 

Inanatural-deductionframework,thereareessentiallytwothingsthatcan 

bedonewithassumptions:introducinganddischarging.Ifweintroducean 

assumption 

A. 

1 OthervariantswouldbeSchütte-stylesystemswithmetalinguisticallyspecifiedrightand 

leftpartsofformulas(Schütte,1960)orevenFrege-stylesystems,see(Schroeder-Heister, 

1999). 

2 FirstsuggestedbyGentzenin(Gentzen,1935),thoughnotunderthatname. 

3 See,forexample,(Tennant,1992),(Tennant,2002).Forabriefhistorysee(Schroeder- 

Heister, 2004, p.33) (footnote). Unfortunately, theearliest proposal ofthissystem 

(Dyckhoff,1988)wasaccidentallyomittedthere. 

4 VonPlato(2001)simplyspeaksof“naturaldeductionwithgeneraleliminationrules”, 

whichcanalsobeunderstoodintheunidirectionalway(dependingonthetreatment 

ofmajorpremissesofeliminationinferences). —Theterm“bidirectional”cameupin 

personaldiscussionswithLucaTranchinionthepropertreatmentofnegationinprooftheoreticsemantics,atopicwhichiscloselyrelatedtobidirectionalreasoning. 

Seehis 

contributiontothisvolume(Tranchini,2009).

BidirectionalNaturalDeduction 247 

thenwemakethederivationbelow Adependentonthatassumption,and 

ifwedischargeitatanapplicationofaninference 

(n) 

A. 

B (n) 

C 

weretractthisdependency,sothattheconclusionofthatinferenceisnot 

longerdependenton A. Asproposedin(Prawitz,1965),thenumeral n 

indicatesthelinkbetweenassumptionsandinferencesatwhichtheyare 

discharged. OthernotationsareFitch’s(Fitch,1952)explicitnotationof 

subproofs,whichgoesbackto(Jaśkowski,1934),wheretheideaofdischargingassumptionswasdevelopedevenbefore(Gentzen,1934/35). 

Introducinganddischargingassumptionsisnotverymuchonecando. 

Especially,therearenooperationsthatchangetheformofanassumption 

andthereforehavetodowithitsmeaning. Inthissense,theyarepurely 

structuraloperations. However,itisdefinitelymorethancanbedonein 

Hilbert-typecalculi,wherewehaveatbesttheintroductionofassumptions 

butnevertheirdischarging.InHilbert-typesystemsassumptionscannever 

disappearbymeansofaformalstep. However,wecanmetalinguistically 

provethatwecanworkwithoutassumptionsbyusingthemastheleftside 

ofaconditionalstatement.Thisisthecontentofthedeductiontheorem:If, 

inaHilbert-typesystem,wehavederived Bfrom A,wecaninsteadderive 

A →Bbyanappropriatetransformationofthederivationof Bfrom A. 

Sinceinnaturaldeductionwehavethedischargingofassumptionsas 

aformaloperationattheobjectlevel,wecanexpressthecontentofthe 

deductiontheoremasaformalruleofimplicationintroduction: 

(n) 

A. 

B (n) 

A →B 

AlthoughthisisanimportantstepbeyondHilbert-typecalculi,itisnot 

allthatcanpossiblybedoneinextendingtheexpressivepowerofformal 

systems.Ourclaimisthatagenuinelysemantictreatmentofassumptions 

ismoreappropriatethanapurelystructuraloneasinnaturaldeduction. 

Innaturaldeduction,assumptionshaveacloseaffinitytofreevariables: 

Assumptionswhicharenotdischargedarecalledopen,whereasdischarged 

assumptionsarecalledclosed. Thisterminologyisjustifiedsinceundischargedassumptionsareopenforthesubstitutionofderivationswhoseend 

formulaistheassumptioninquestion,whereasclosedassumptionsarenot.


Givenaderivation 

A,then 

A 

Dwiththeopenassumption 

Aandaderivation 

B 

D1 

A of 

D1 

A 

D 

B 

isaderivationof Bwhichmaybeconsideredasubstitutioninstanceofthe 

originalderivation.Inthissenseanopenderivationcorrespondstoanopen 

term,andaclosedderivation,i.e.aderivationwithoutopenassumptions, 

correspondstoaclosedterm. Thisrelationshipbetweenopenandclosed 

proofsandopenandclosedtermscanbemadeformallyexplicitbyaCurry– 

Howard-styleassociationbetweentermsandproofs,wherethedischarging 

ofassumptionsbecomesaformalbindingoperation. 

Inourexample,thederivation D1 

A canitselfbeopen,justlikeavariable 

whichissubstitutedwithanopenterm.Sointheformalconceptofnatural 

deductionandthecompositionofderivationsthereisnoprimacyofclosed 

derivationsoveropenones. However,thisprimacyenterswiththephilosophicalinterpretationofnaturaldeductioninthetraditionofDummett 

andPrawitz. Thereopenassumptionsareinterpretedasplaceholdersfor 

closedproofs. 5 

2 AssumptionsinDummett-Prawitz-style 

proof-theoreticsemantics 

Proof-theoretic semantics as advanced by Dummett and Prawitz 6 was 

framedbyPrawitzintheformofadefinitionofvalidityofproofs,where 

aproofcorrespondstoaderivationinnatural-deductionform. According 

tothisdefinition,closedproofsinintroductionformareprimaryasbased 

on“self-justifying”steps,whereasthevalidityofclosedproofsnotinintroductionformaswellasthevalidityofopenproofsisreducedtothat 

ofclosedproofsusingcertaintransformationproceduresonproofs,called 

“justifications”. Givenanotionofvalidityforatomicproofs(i.e.proofs 

ofatomicsentences),thedefinitionofvalidityforthecaseofconjunction 

andimplicationformulas(totaketwoelementarycases)canbesketchedas 

follows: 

5 Hereweswitchterminologyfrom“derivation”to“proof”,asinthesemanticalinterpretationwearenolongerdealingwithpurelyformalobjects,forwhichwereservetheterm“derivation”.Prawitzhimselfoftenspeaksof“arguments”toavoidformalisticconnotationsstillpresentwith“proof”. 

6 Foranoverviewofthissortofsemanticssee(Schroeder-Heister,2006)andthereferences 

therein.


•Aclosedproofofanatomicformula Aisvalidifthereisavalidatomic 

proofof A. 

•Aclosedproofof A∧Bintheintroductionform 

D1 

D2 

A B 

A∧B 

isvalidifthesubproofs D1and D2arevalidclosedproofsof Aand 

B,respectively. 

•Aclosedproofof A →Bintheintroductionform 

(n) 

A 

D 

B (n) 

A →B 

isvalidifforeveryclosedproof D1 

A 

of Bisvalid. 

D1 

A 

D 

B 

of A,theclosedproof 

•Aclosedproofof Anotinanintroductionformisvalidifitreduces, 

bymeansofthegivenjustifications,toavalidclosedproofof Ainan 

introductionform. 

Ifweareonlyinterestedinclosedproofs,thisdefinitionissufficient. In 

viewofthelastclause,itisageneralizedinductivedefinitionproceedingon 

thecomplexityofendformulasandthereductionsequencesgeneratedby 

justifications. Ifwealsowanttoconsideropenproofs,wewouldhaveto 

define: 

•Anopenproof 

D1 

A1 

,..., Dn 

,theproof 

An 

A1,...,An 

D 

B 

D1 

isvalidifforallclosedvalidproofs 

Dn 

A1,... ,An 

D 

B 

isavalidclosedproof. 

Giventhisclauseforopenproofs,thedefiningclauseforthevalidityofa 

closedproofof A →Binintroductionformmightbereplacedwith


•Aclosedproofof A →Bintheintroductionform 

(n) 

A 

D 

B (n) 

A → B 

isvalidifitsimmediateopensubproof 

isvalid, 

yieldingauniformclauseforallclosedproofsinintroductionform.However, 

thiswayofproceedingmakesthedefinitionsofvalidityofopenandclosed 

proofsintertwined,whichobscuresthefactthatthereisanindependent 

definitionofvalidityforclosedproofs. 

Accordingtothisdefinition,closedproofsareconceptuallypriortoopen 

proofs.Furthermore,assumptionsinopenproofsareconsideredtobeplaceholdersforclosedproofs,asthevalidityofopenproofsisdefinedbythevalidityoftheirclosedinstancesobtainedbysubstitutingafreeassumptionwithaclosedproofofit.Sowehaveidentifiedtwocentralfeaturesof 

standardproof-theoreticsemantics: 

A 

D 

B 

Theprimacyofclosedoveropenproofs (α) 

Theplaceholderviewofassumptions (β) 

Thedefinitionofvalidityshowsafurtherfeaturewhichisconnectedto(α) 

A 

and(β). Thefactthatinanopenproof Dtheopenassumption 

Aisa 

B 

placeholderforclosedproofs D1 

of A,yieldingaclosedproof 

A 

D1 

A 

D 

B 

A 

meansthatthevalidityof Disexpressedasthetransmissionofvalidity 

B 

from[theclosedproof] D1to[theclosedproof] 

D1 

A 

D 

B


A 

Ifoneconsidersanopenproof Dtobeaproofoftheconsequencestatement 

B 

that Bholdsunderthehypothesis A,thisexpresses 

Thetransmissionviewofconsequence (γ) 

i.e.,theideathatthevalidityofaconsequencestatementisbasedonthe 

transmissionofthevalidityofclosedproofsfromthepremissestotheconclusion. 

Thisideaiscloselyrelatedtotheclassicalapproachaccordingto 

whichhypotheticalconsequenceisdefinedasthetransmissionofcategorical 

truth(inamodel)fromthepremissestotheconclusion. Inthatrespect, 

Dummett–Prawitz-styleproof-theoreticsemanticsdoesnotdepartfromthe 

classicalviewpresentintruth-conditionsemantics(see(Schroeder-Heister, 

2008b)).Ofcourse,therearefundamentaldifferencesbetweentheclassical 

andconstructiveapproaches,whichmustnotbeblurredbythissimilarity, 

inparticularwithrespecttoepistemologicalissues(see(Prawitz,2009)). 7 

Afurtherpointshowingupinthedefinitionofvalidityistheassumption 

ofglobalreductionproceduresforproofs(called“justifications”). Thisis 

whatmakesthe(generalized)inductiononthereductionsequenceforproofs 

possible.Itisassumedthatitisnotindividualvalidproofstepsthatgenerateavalidproof,buttheoverallproofwhichmayreducetoaproofofa 

particularform(viz.,aproofinintroductionform).Wecallthis 

Theglobalviewofproofs (δ) 

Thesefourfeaturesareintimatelyconnectedtothemodelofnaturaldeductionasitsformalbackground. 

Thisholdsespeciallyfor(β)and(δ), 

whichspecify(α)and(γ),respectively.Naturaldeductionpermitstoplace 

aderivationontopofanotherone,anditisnaturaldeductionwherewe 

havethenotionofproofreduction. Inthesequentcalculus,thissortof 

connectionisnotpresent. 

Inthesequentcalculus,logicalinferencesnotonlyconcerntherightside 

ofasequent(correspondingtotheendformulainnaturaldeduction)butthe 

rightandleftsidesofsequentslikewise.Inthissensethesequentcalculusis 

7 

Itmightbementionedthatthedefinitionofvalidityforaclosedproofof A → Bisclosely 

relatedtoLorenzen’sadmissibilityinterpretationofimplication.Accordingto(Lorenzen, 

1955), A →Bexpressestheadmissibilityoftherule A 

. Theclaimthateveryclosed 

B 

proofof Acanbetransformedintoaclosedproofof Bcanbevieweduponasexpressing 

admissibility.Atfirstglance,thiscontradictsthefactthatinnaturaldeductionanopen 

A 

... 

proof isaproofof Bfrom Aandshouldassuchbedistinguishedfromanadmissibility 

B 

statement. However,evenif,intheformalsystem,wearedealingwithproofsfrom 

assumptionsratherthanadmissibilitystatements,thesemanticinterpretationintermsof 

validitycomesveryclosetotheadmissibilityview.See(Schroeder-Heister,2008a).


inherentlybidirectionalcomparedtotheunidirectionalformalismofnatural 

deductionthatunderliesDummett–Prawitz-styleproof-theoreticsemantics. 

Inthefollowingwewillmakeacaseforthebidirectionalframework. 

3 Thesequentcalculusandbidirectionalnaturaldeduction 

Accordingtothetraditional,i.e.pre-natural-deductionreasoningmodel,we 

startwithtruesentencesandproceedbyinferenceswhichleadfromtrue 

sentencestotruesentences. Thisguaranteesthatwealwaysstayinthe 

realmoftruth. 8 Alternatively,wecouldstartwithassumptionsandassert 

sentencesunderhypotheses. Thisisthebackgroundofnaturaldeduction. 

Naturaldeductionaddsthefeatureofdischargingassumptions,i.e.,the 

dependencyonassumptionsmaydisappearinthecourseofanargument. 

Inthiswaythedynamicsofreasoningnotonlyaffectsassertionsbutat 

thesametimethehypothesesassumed. However,thisdynamicsisvery 

limitedastheonlyoptionsareintroducinganddischarging,sothereisno 

morethanayes/noattributiontohypotheses. Wecannotintroduceand 

eliminateassumptionsaccordingtotheirspecificmeaning,whichwouldbe 

amoresophisticateddynamics.Inthissensereasoninginstandardnatural 

deductionisassertioncentredandunidirectional.Thisisevenmoreso,as 

thehypothesesassumedareplaceholdersforclosedproofs. 9 

Agenuinelydifferentmodelisgivenbythesequentcalculus. Theparticularfeatureofthissystem,i.e.introductionrulesontheleftsideofthesequentsign,canbephilosophicallyunderstoodasthemeaning-specificintroductionofassumptions.Considerconjunctionwithleftsequentrules 

Γ,A⊢C 

Γ,A∧B ⊢C 

Γ,B ⊢ C 

Γ,A∧B ⊢ C 

Theserulescanbeinterpretedasfollows:Supposewehaveasserted Cunder 

thehypotheses Γand A. Thenwemayclaim Cbyassuming A∧Basan 

assumptionanddischarging Aasanassumption,andsimilarlyfor B.Writteninnatural-deductionstylethiscorrespondstothegeneralelimination 

rulesforconjunction 

A∧B 

C 

(n) 

A. 

C (n) 

A∧B 

C 

(n) 

B. 

C (n) 

8 Thiswas,forexample,thepicturedrawnbyBolzanoandFrege. 

9 Thisisnotessentiallychangedifwereplaceassertionwithdenialandinthissense 

dualizenaturaldeduction.Unidirectionalitywouldjustpointintotheoppositedirection. 

See(Tranchini,2009).


butwiththecrucialmodificationthatthemajorpremissmustnowbean 

assumption,i.e.,mustoccurintopposition 10 (thisishereindicatedbythe 

lineoverthemajorpremiss).Similarly,theleftimplicationrule 

Γ ⊢ A Γ,B ⊢ C 

Γ,A→B ⊢ C 

isinterpretedasfollows: Supposewehaveassertedboth Aunderthehypotheses 

Γ,and Cunderthehypotheses Γand B. Thenwemayclaim 

Cundertheassumption A → Binsteadof B,i.e.,discharge Bandassert 

A →Binstead. Writteninnatural-deductionstyle,thisyieldsthegeneral 

→-eliminationrule 

(n) 

B. 

A → B A C 

(n) 

C 

againwiththecrucialdifferencetothestandardgeneraleliminationrule 

thatthemajorpremissoccursintopposition. 11 

Bypresentingthesequent-calculusrulesinanatural-deductionframeworkwearenolongerworkingin“standard”or“genuine”naturaldeductionbutinthereasoningmodelsuggestedbythesequentcalculus,asthe 

restrictiononmajorpremissesofeliminationrulesrunscountertotheway 

premissesaretreatedinstandardnaturaldeduction.Wecallthismodified 

systembidirectionalnaturaldeductionasitactsonboththeassertionand 

theassumptionside,withrulesthatdependontheformsoftheformulasassumedorasserted.Sothepossibleoperationsonassumptionsarenolonger 

merelystructural. 12 

Inproposingbidirectionalnaturaldeduction,asanatural-deduction-style 

variantofthesequentcalculus,asourmodelofreasoning,weestablisha 

symmetrybetweenassertionsandassumptions. Likeassertions,assumptionscanbeintroducedaccordingtotheirmeaning,namelyasmajorpremissesofeliminationinferences. 

Byimposingtherestrictionthatmajor 

premissesmustalwaysbeassumptions,eliminationinferencesreceivean 

10 InTennant’s(Tennant,1992)terminology,themajorpremiss“standsproud”. 

11 Atranslationbetweensequentcalculusandnaturaldeductionwithgeneralelimination 

rulesiscarriedoutinfulldetailin(vonPlato,2001). Notethatforimplication,weare 

hereconsideringthegeneraleliminationruleusedbyvonPlato,astheycorrespondto 

theleftsequentcalculusrule,ratherthanthemorepowerfuloneproposedin(Schroeder- 

Heister,1984),whichextendsthestandardframeworkofnaturaldeductionwithrulesas 

assumptions. 

12 Wealsocallit“natural-deduction-stylesequentcalculus”,asitisconceptuallyasequent 

calculuswhichispresentedintheformofanaturaldeductionsystem(Schroeder-Heister, 

2004). In(Negri&vonPlato,2001),thistermisusedinadifferentsense,meaninga 

specificformofthesequentcalculus.


entirelydifferentreading. Theyarenolongerjustifiedbyreferencetothe 

waythemajorpremisscanbe(canonically)derived.Theyareratherviewed 

aswaysofintroducingcomplexassumptions,giventhederivationsofthe 

minorpremisses.Eliminationinferencesinbidirectionalnaturaldeduction 

combinetheintroductionofanassumptionwithaneliminationstepand 

canthusbeviewedasaspecialformofassumptionintroduction. Thereforewealsocallthem“upwardintroductions”,asopposedto“downward 

introductions”whicharethecommonintroductionrules. 

Assumptionswhicharemajorpremissesofeliminationinferencesareno 

longerplaceholdersforclosedproofsastheycannotbeinferredbymeans 

ofaninference. Theyarealwaysstartingpointsofeliminationinferences. 

Ofcourse,itmightbepossibletoshowthatgivenaproof D1 

of Aand 

A 

A 

aproof D2of 

Bfrom A,wecanobtainaproof 

B 

D 

of B. However,this 

B 

wouldhavetobeestablishedasatheoremcorrespondingtocutelimination 

forthesequentcalculus. Itisnolongeratrivialmatterasinstandard 

(unidirectional)naturaldeduction,since 

D1 

A 

D2 

B 

isnolongerawell-formedproofif Aisamajorpremissofanelimination 

inference.Thereforebidirectionalityovercomestheplaceholderviewofassumptions(β).Withthisitalsoovercomestheprimacyofclosedoveropen 

proofs(α)asclosedproofsarenolongerusedtointerpretassumptions. 

Onlyapremissofanintroductionrulecanbeviewedasaplaceholderfora 

closedproof,whichmeansthattheuniforminterpretationofassumptions 

byreferencetoclosedproofsisgivenup. 

Thetransmissionviewofconsequence(γ)disappearsaswell.Asassumptionscanbeintroducedinthecourseofaproof(inthesequentcalculusby 

leftintroduction,inbidirectionalnaturaldeductionasthemajorpremiss 

ofaneliminationinference),itisnolongeradefiningfeatureofthemthat 

theytransformclosedproofsintoclosedproofs. Ifthishappenstobethe 

case,thenitis“accidental”andhastobeproved. Theintroductionof 

anassumptionisjustasprimitiveastheintroductionofanassertion. In 

theterminologyofDummett–Prawitz-styleproof-theoreticsemantics,both 

theintroductionofanassertionandtheintroductionofanassumptionis 

acanonical,i.e.definitionalstep. Moreprecisely,thedistinctionbetween 

canonicalandnon-canonicalstepsdisappears.Inthissensetheconceptof 

validityismuchmorerule-orientedthanproof-oriented:Wenowconsidera 

prooftobevalidifitconsistsofproperapplicationsofrightandleftrules


(inthesequentcalculus)ordownwardandupwardintroductionrules(in 

bidirectionalnaturaldeduction)ratherthanifitreducestoaproofinintroductionformforallitsclosedinstances.Inthisway,theglobalviewof 

proofs(δ)alsodisappears,asitisbasedonthefundamentalassumption 

thatproofsareprimarytorulesandthatthevalidityofrulesisbasedon 

proofsandproofreduction.Theideaofbidirectionalreasoningisverymuch 

localratherthanglobal. 13 

Thisdoesnotmeanthatrightandleft(sequentcalculus),ordownward 

andupward(bidirectionalnaturaldeduction)introductionsareunrelatedto 

eachother.Wewillstillrequiresomenotionofharmonybetweenthetwo 

sortsofinferencesasanadequacycondition.However,thisharmonywillbe 

localratherthanglobal,andnotbasedonproofreduction. Onecriterion 

wouldbeuniquenessinthesenseof(Belnap,1961/62),whichmeansthat 

ifweduplicaterulesforaconstant ∗,yieldingaconstant ∗ ′ withthesame 

right(ordownward)andleft(orupward)rules,wecanprove A[∗] ⊣⊢ A[∗ ′ ] 

inthecombinedsystem. There A[∗]isanyexpressioncontaining ∗,and 

A[∗ ′ ]isobtainedfrom A[∗]byreplacing ∗with ∗ ′ .However,unlikeBelnap, 

wewouldnotrelyonconservativeness,asthisisaglobalconcept,but 

ratheronlocalinversioninthesensethatthedefiningconditionsfora 

constant ∗canbeobtainedbackfromthisconstant. Ourmaincriticism 

ofBelnap’sproposalofconservativenessanduniquenessinhisdiscussionof 

theconnective“tonk”isthathemixesalocalcondition(uniqueness)with 

aglobalone(conservativeness). 14 

4 Whygoinglocal? 

Whyshouldweswitchtoaconceptofhypotheticalreasoningwhichisdifferentfromthestandardonecharacterizedby(α)–(δ),andwhichisprevailing 

bothinclassicalandconstructivesemantics? Thelackofanintuitivejustificationoftheprinciples(α)–(δ)isnoreasonforabandoningthem,ifwe 

cannotalsotellwhythebidirectionalalternativehasgreaterexplanatory 

power.Infact,wegainaccesstoamuchwiderrangeofphenomena,ifwe 

sticktothebidirectionalparadigm.Wejustmentiontwopoints. 

Atomicreasoningandinductivedefinitions 

Thediscussioninproof-theoreticsemanticshastraditionallyfocusedonlogicalconstants.Logicalconstantsareaparticularlywell-behavedcasewherewecanapplytheglobalconsiderationscharacteristicofthestandardapproach.Natural-deduction-basedproof-theoreticsemanticshasbeendevel- 

13 Thelocalapproachtohypotheticalreasoningputforwardherewasoriginallyproposed 

byHallnäs(Hallnäs,1991,2006). 

14 Thispointwillbeworkedoutelsewhere.


opedasasemanticsoflogicalconstants. However,thisfocusismuchtoo 

narrow.Proof-theoreticdefinitionsoflogicalconstantsjustfeatureasparticularcasesofinductivedefinitions. 

Lookingatinductivedefinitionsas 

basicstructuralentitiesthatconfermeaningtoobjects,thedistinctionbetweenatomicandnon-atomic(i.e.logicallycompound)objectsdisappears. 

Mostgenerally,wewoulddealwithdefinitionalclausesoftheform 

a ⇐ C 

where aisanobjecttobedefinedand Cisadefiningcondition.Starting 

withadefinitionofthiskind,right(downward)andleft(upward)introductionrulescanbegeneratedfromthisinductivedefinitioninacanonicalway,representingawayofputtinginductivedefinitionsintoaction,andresultinginpowerfulclosureandreflectionprinciples. 

Theformofdefinitional 

clauseslooklikeclausesinlogicprogramming,andlogicprogramscanbe 

viewedasparticularcasesofinductivedefinitions.Wewouldevengeneralize 

theframeworksetupbylogicprogrammingbyconsideringclauseswhere 

thebody Cofaclausemaycontainhypotheticalstatementsandtherefore 

negativeoccurrencesofdefinedobjects. ThisgoesbeyondstandarddefiniteHornclauseprogrammingandeventranscendstheclassicalfieldof 

logicprogrammingwithnegation(Hallnäs&Schroeder-Heister,1990/91). 

Itdiffersfromsystemsinvestigatedin(Martin-Löf,1971)inthatitisnot 

mainlydirectedatinductionprinciplesbutratherthelocalinversionof 

rules. SystemsofthiskindhaverecentlybeenconsideredbyBrotherston 

andSimpson(Brotherston&Simpson,2007),wherealsotherelationship 

betweeninversion-basedreasoningandinductionprinciplesforiteratedinductivedefinitionsisdiscussed.Consideringinductivedefinitionsingeneral 

opensupawiderperspectiveathypotheticalreasoningwhichisnolonger 

basedonlogicalconstants. Itcanalsointegratesubatomicreasoningin 

thesenseof(Więckowski,2008),wherethevalidityofatomicsentencesis 

reducedtocertainassumptionsconcerningpredicatesandterms. 

Non-wellfoundedphenomena 

Theglobalreductionistperspectiveunderlyingunidirectionalnaturaldeductionexcludesnon-wellfoundedcasessuchastheparadoxes. 

Theinductive 

definitionofvalidityexpectsthatthereisnolooporinfinitedescentinthe 

reasoningchain.However,inthecaseoftheparadoxes,wehaveexactlythis 

situation. Ourlocalframeworkcaneasilyaccommodatesuchphenomena. 

Forexample,ifwedefine pby ¬qand qinturnby p,thenboth pand q 

arelocallydefined.Thegloballoopisirrelevantforthelocaldefinition.In 

suchasituationwecannolongerproveglobalpropertiesofproofssuchas 

cutelimination,butthiswedonotrequire.


Asitisnowamatterof(mathematical)factratherthanadefinitional 

requirementwhethercertainglobalpropertieshold,wedonotruleoutnonwellfoundedphenomenabydefinition.Thisisagreatadvantage,asitgives 

usabetterchancetounderstandthem.Following(Hallnäs,1991),wemight 

callthisapproachapartialapproachtomeaning.AccordingtoHallnäs,this 

wouldbeincloseanalogytorecursivefunctiontheory,whereitisapotential 

mathematicalresultthatagivenpartialrecursivefunctionistotal,rather 

thansomethingthathastobeestablishedforthefunctiondefinitionto 

makesense. 

Thereareotherapplicationsofthelocalapproachthatwecannotmentionheresuchasthepropertreatmentofsubstructuralissues,generalized 

inversionprinciples,evaluationstrategiesinextendedlogicprograms,etc. 

5 FinalDigression:Dialogues 

Wehavepleadedforabidirectionalviewofreasoningasitisincorporated 

inGentzen’ssequentcalculusandcanbegiventheformofbidirectional 

naturaldeduction.Astherearecertainadequacyconditionsgoverningsuch 

asystemthatrelateright/downwardsandleft/upwardsruleswithoneanother,sothattheyarelinkedtogetherinacertainway,wemightaskof 

whetheritwouldbepossibletoobtainthemfromasingleprinciple. One 

possibleanswermightbethedialogicalapproachproposedbyLorenzen 

(Lorenzen,1960)andhisfollowers.Ifonecarriesitsideasovertothecase 

ofinductiveclauses 

a ⇐ C1 

. 

a ⇐ Cn 

onewouldbeleadtoanapproachwhereanattackonthedefinedobject 

awouldhavetobedefendedbyachoiceamongthedefiningconditions 

Ci,whicharethemselvesattackedbychoosingoneofitscomponents.The 

distinctionbetweenrightandleftruleswouldthenbeobtainedbystrategy 

considerationsforandagainstcertainatoms. Inthiswayamoreunified 

approachcouldbeachieved. Thedialogicalmotivation,asbasedonlocal 

attackanddefencerules,wouldnotinvolveglobalreductivefeaturescomparedtovaliditynotionsinstandardproof-theoreticsemantics.Therefore,itappearstobemorefaithfultoourlocalapproach,astheglobalperspectiveisonlyintroducedatalaterstageintermsofstrategiesandtheir 

transformations.Inthiswaythedialogicalresearchprogrammepromisesa 

novelperspectiveatthelocal/globaldistinction.


PeterSchroeder-Heister 

Wilhelm–Schickard–InstitutfürInformatik,UniversitätTübingen 

Sand13,72076Tübingen,Germany 

psh@informatik.uni-tuebingen.de 

http://www-ls.informatik.uni-tuebingen.de/psh 

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Relatives of Robinson Arithmetic 

Vítězslav ˇ Svejdar ∗ 

1 Introduction:numbers,orstrings? 

Robinsonarithmetic Qwasintroducedin(Tarski,Mostowski,&Robinson, 

1953)asabaseaxiomatictheoryforinvestigatingincompletenessandundecidability.Itisveryweak,butallitsrecursivelyaxiomatizableconsistent 

extensionsarebothincompleteandundecidable.Inlogictextbooks,itoften 

playstheroleoftheweakestreasonabletheorywiththisproperty. 

A.Grzegorczykrecentlyproposedtostudythetheoryofconcatenation 

asapossiblealternativetheoryforstudyingincompletenessandundecidability. 

UnlikeRobinson(orPeano)arithmetic,wheretheindividualsare 

numbersthatcanbeaddedormultiplied,inthetheoryofconcatenation 

onehasstrings(ortexts)thatcanbeconcatenated.Sointhelanguageof 

thetheoryofconcatenationthereisabinaryfunctionsymbol ⌢ forlaying 

twostringsendtoendtoformanewstring.Axiomsofthetheoryofconcatenationpostulate,e.g.,associativityoftheoperation 

⌢ ,ortheexistence 

ofirreducible,i.e.single-letter,strings.Particularvariantsofthetheoryof 

concatenationmaydifferinthenumberofirreduciblestrings(withtwoas 

themostobviouschoice),orintheexistenceoftheemptystring. 

BeforeGrzegorczyk,someaspectsofconcatenationwereconsideredand 

someaxiomswereformulatedbyQuine(1946)andTarski. Onevariant 

ofthetheory,calledtheory F,appearsalreadyinthebook(Tarskietal., 

1953),whereitisclaimedbutnotprovedthat Fisessentiallyundecidable. 

Grzegorczyk’smotivationtostudythetheoryofconcatenationisphilosophical.Whenreasoningorwhenperformingacomputation,wedealwith 

texts.Ourhumancapacitytoperformtheseintellectualtasksdependson 

ourabilitytodiscerntexts.Thenitisnaturaltodefinenotionslikeundecidabilitydirectlyintermsoftexts,withoutreferencetonaturalnumbers. 

∗ ThisworkisapartoftheresearchplanMSM0021620839thatisfinancedbytheMinistry 

ofEducationoftheCzechRepublic.

262 Vítězslav ˇ Svejdar 

WhenprovingGödel1 st incompletenesstheorem,choosingthetheoryof 

concatenationasthebasetheorycouldbepreferabletochoosingPeano 

orRobinsonarithmetic,becausethenoneoftheessentialstepsintheincompletenessproof,formalizationoflogicalsyntax,wouldbepractically 

effortless. 

Wewilldiscusspropertiesoftwotheoriesofconcatenation,theory FdefinedinTarskietal.(1953)andtheory 

TCproposedbyGrzegorczyk. It 

appearsthatanappropriatemethodofshowingundecidabilityofallconsistentextensionsisprovingmutualinterpretabilityofthesetheorieswith 

Robinsonarithmetic Q.Wewillconsidermethodsofconstructinginterpretations,oneofthesebeingthewellknownSolovaymethodofshorteningofcuts.WewillalsodiscusstheGrzegorczyk’sprojectofreplacingRobinson’s 

Qbysomeversionoftheoryofconcatenationinmoredetails. The 

prosoftheprojectareobvious,buttherearealsosomecons. 

2 Somepreliminaries 

Foranaxiomatictheory T,let Thm(T)bethesetofallsentencesprovable 

in T,insymbols Thm(T) = {ϕ;T ⊢ ϕ},andlet Ref(T)bethesetofall 

sentencesrefutablein T,insymbols Ref(T) = {ϕ;T ⊢ ¬ϕ}.Atheory Tis 

consistentif Thm(T)∩Ref(T) = ∅,i.e.,ifnosentenceof Tissimultaneously 

provableandrefutablein T.Atheory Tiscompleteifitisconsistentand 

eachsentenceof Tiseitherprovableorrefutablein T.Atheory Tisrecursivelyaxiomatizableifitisequivalenttoatheory 

T ′ withanalgorithmically 

decidablesetofaxioms(i.e.with T ′ algorithmicallydecidable).Atheoryis 

decidableifthereexistsanalgorithmthatdecidesaboutitsprovability,i.e., 

iftheset Thm(T)isalgorithmicallydecidable. 

Atheory Sisanextensionofatheory Tifthelanguageof T(i.e.the 

setofallnon-logicalsymbolsof T)isasubsetofthelanguageof S,and 

eachsentenceof Tprovablein Tisprovablealsoin S.Atheory Tisessentiallyincompleteifnorecursivelyaxiomatizableextensionof 

Tiscomplete; 

Tisessentiallyundecidableifnoconsistentextensionof Tisdecidable.It 

isknownthatatheoryisessentiallyincompleteifandonlyifitisessentiallyundecidable.Thusweusethesenotionsinterchangeablyor,following 

Grzegorczyk,wepreferablyspeakaboutessentialundecidability. 

Aninterpretationofatheory Tinatheory Sisamappingfromformulas 

of Ttoformulasof Sthatwell-behavesw.r.t.logicalsymbolsandmapsall 

axiomsof Ttosentencesprovablein S. Atheory Tisinterpretablein S 

ifthereexistsaninterpretationof Tin S.Thenotionofinterpretation,as 

wellasthenotionofessentialundecidability,firstappearedin(Tarskietal., 

1953). Importantfactsaboutinterpretabilityarethefollowing:(i)if Tis 

interpretablein Sand Sisconsistentthen Tisconsistent,too;(ii)if Tis


interpretablein Sand Tisessentiallyundecidablethenthen Sisessentially 

undecidable,too.Thenotionofinterpretabilitycanbeusedasameansto 

measurestrengthofaxiomatictheories:if Tisinterpretablein Sandvice 

versa,i.e.,if Tand Saremutuallyinterpretable,thenwecanthinkthat 

Tand Srepresentthesameexpressiveanddeductivestrength. 

3 TheimportanceofRobinsonarithmetic 

Robinsonarithmetic Qisanaxiomatictheoryhavingsevensimpleaxioms 

formulatedinthelanguage {+, ·,0,S}withsymbolsforadditionandmultiplication(ofnaturalnumbers),aconstantforthenumberzero,andaunary 

functionsymbol Sforthesuccessorfunction x ↦→ x+1.Peanoarithmetic PA 

isobtainedfrom Qbyaddingtheinductionschema.Thetheory I∆0islike 

Peanoarithmetic,butwiththeinductionschemarestrictedto ∆0-formulas 

(boundedformulas)only. Thetheory I∆0+Ω1is I∆0enhancedbytheaxiomassertingthetotalityofthefunction 

x ↦→ x log x .Foranon-expert,the 

propertiesofnaturalnumbersexpressibleby ∆0-formulasconstituteaclass 

thatisasubclassofallalgorithmicallydecidableproperties. Anexample 

ofa∆0-formulaistheformula ∃v(v · x = y),i.e.theformulathenumber x 

isadivisorofthenumber y.Twootherexamplesarethenumber xisprime 

andthenumber xisdivisiblebysomeprime.Anexampleofaformulathat 

isnot ∆0isthereexistsay> xsuchthat y �= 0and yisdivisiblebyall v 

suchthat v �= 0and v ≤ x;thisformulaspeaksaboutathingsimilarto 

thefactorialof x.Anotherexampleofanon-∆0formulaisthereexistsay 

suchthat y > xand yisprime. Inthetheory I∆0,onecannotprovethat 

afactorialof xexistsforeachnumber x,whileprovabilityofthesentencea 

prime y > xexistsforeach xisadifficultopenproblem.Bothsentencesare 

easilyprovedbyunrestrictedinduction,i.e.inPeanoarithmetic. 

Basicpropertiesofnaturalnumbers,likeassociativityandcommutativityofadditionandmultiplication,areprovablein 

I∆0,butunprovable 

in Q. Generally,universalsentencesareseldomprovablein Q. However, 

I∆0+Ω1isinterpretablein Q.Gödel1 st incompletenesstheorem,orbetter, 

itsRossergeneralization,saysthatanyrecursivelyaxiomatizableextension 

of Qisincomplete.So Qisessentiallyincomplete(essentiallyundecidable). 

ThemeaningofGödel2 nd incompletenesstheoremissomewhatquestionablefor 

Q.However,itsusualproofgoesthroughin I∆0+Ω1withoutany 

changes. 

ThusRobinsonarithmetic Qisaveryweakbutstillessentiallyundecidabletheory. 

Itrepresentsarich“degreeofinterpretability”becausealot 

ofstrongertheories,like I∆0+Ω1,areinterpretableinit.Sinceitisfinitely 

axiomatizable,itcanbeusedinastraightforwardproofofundecidabilityof 

classicalpredicatelogic.


4 ThetheoryTC 

� 

x 

�� 

y 

�� 

w 

w 

� �� 

u 

� �� 

v 

Figure1.Theeditorsaxiom 

Thetheoryofconcatenation TChasthelanguage { ⌢ ,ε,a,b}withabinaryfunctionsymbol 

⌢ ,aconstant εfortheemptystring,andtwoother 

constants aand b.Weusuallyomitthesymbol ⌢ ,i.e.,write xyfortheconcatenation 

x ⌢ yofthestrings xand y.Theaxiomsof TCarethefollowing: 

TC1: ∀x(xε = εx = x), 

TC2: ∀x∀y∀z(x(yz) = (xy)z), 

TC3: ∀x∀y∀u∀v(xy = uv → 

→ ∃w((xw = u & wv = y) ∨ (uw = x & wy = v))), 

TC4: a �= ε & ∀x∀y(xy = a → x = ε ∨ y = ε), 

TC5: b �= ε & ∀x∀y(xy = b → x = ε ∨ y = ε), 

TC6: a �= b. 

TheaxiomsTC1andTC2canbedescribedasaxiomsofsemigroups;byTC2 

wecanomitparenthesesinexpressionswheneverconvenient. Theaxioms 

TC4–TC6postulatethatthestrings aand baredifferent,andeachofthem 

isnon-emptyandirreducible(cannotbenon-triviallydecomposedintotwo 

strings).TheaxiomTC3iscallededitorsaxiomin(Grzegorczyk,2005).It 

describeswhathappensiftwoeditorsofalargeworkindependentlysuggest 

splittingthetextintotwovolumes. Iftheirsuggestionsare x,yand u,v 

respectively,asshownintheFigure1,thenthefirstvolumeofoneofthe 

editorsconsistsoftwoparts:theothereditor’sfirstvolume,andatext w 

(possiblyempty)thatsimultaneouslyoccursasastartingpartoftheother 

editor’ssecondvolume.In(Ganea,2007)thistext wiscalledaninterpolant 

(oftheequation xy = uv). 

Thetheoryofconcatenation TCwasdefinedin(Grzegorczyk,2005). 

However,theeditorsaxiomisattributedtoTarski,andtheideaaboutthe 

importanceofconcatenationinincompletenessproofscanbetracedback 

toQuine,whoin(Quine,1946)citesTarskiandHermesandsays:Gödel’s 

proof[...]dependedonconstructingamodelofconcatenationtheorywithin 

arithmetic. NotethatQuinedoesnotlistanyaxioms,andthuswhenhe


says“concatenationtheory”,heinfactmeansitsstandardmodel(defined 

below). Grzegorczyk(2005)proved(mere)undecidabilityof TC. Later 

GrzegorczykandZdanowski(2008)showedessentialundecidabilityof TC 

andleftopenthequestionwhetherRobinsonarithmeticisinterpretable 

in TC.A.VisserandR.Sterken,see(Visser,2009),M.Ganeain(Ganea, 

2007),andthepresentauthorin( ˇ Svejdar,2009)independentlygaveapositiveanswertothisquestion.Moreaboutinterpretabilityin(andof) 

TCis 

inSection5below. 

Thepapers(Grzegorczyk,2005)and(Grzegorczyk&Zdanowski,2008) 

workwithavariantof TChavingnoemptystring. Then,forexample, 

theaxiomTC4hastheform ∀x∀y(xy �= a). Thepaper( ˇ Svejdar,2009) 

workswithavariantof TChavingthreeinsteadoftwoirreduciblestrings. 

Theexactchoiceofvariantofthetheoryisamatteroftastebecause,as 

shownin(Grzegorczyk&Zdanowski,2008),allvariantsofthetheoryof 

concatenationaremutuallyinterpretable,providedtheirreduciblestrings 

areatleasttwoinnumber. 

Let Abetheset {a,b} ∗ ofallstringsinthetwo-letteralphabet {a,b}, 

andlet Abethestructurewith Aasauniverse,withconcatenationdefined 

“normally”andwithconstants aand brealizedby aand b,respectively. 

Then Aisthestandardmodelof TC. Thestructure Bhavingtheset 

B = {a,b,e} ∗ asitsuniverseandwithallsymbolsalsodefinednormallyis 

anotherexampleofamodelof TC. Let x ⊑ ymean ∃u∃v(uxv = y),and 

let x ymean ∃u(ux = y). Theformulas x ⊑ yand x ycanberead 

thestring xisasubstringof yandthestring yendsby xrespectively. The 

model Baboveshowsthatthesentence ∀x(x �= ε → a ⊑ x ∨b ⊑ x)isnot 

provablein TC. 

Thefollowingtheoremgivessomemoreexamplesofprovableandunprovablesentences. 

Itspurposeistogivethereadersomefeelingabout 

provabilityin TC. 

Theorem9.Thefollowingsentences(a)–(d)areprovablein TC, 

(a) ∀x(xa �= ε), 

(b) ∀x∀y(xy = ε → x = ε & y = ε), 

(c) ∀x∀y(xa = ya → x = y), 

(d) ∀x∀y(a xy → y = ε ∨a y), 

whilethefollowingsentence(e)isnotprovablein TC: 

(e) ∀x∀y∀z(xz = yz → x = y). 

Proof.(a)Assume xa = ε. Then,byTC1andTC2,wehave (bx)a = b. 

Irreducibilityof b,i.e.TC5,yields bx = εor a = ε.Thelatterisexcluded


byTC4. Thenfrom bx = ε, (bx)a = b,andTC1wehave a = b,a 

contradictionwithTC6. 

(b)If xy = εthen x(ya) = ausingTC1andTC2.So x = εor ya = ε 

byTC4.From(a)wehave x = ε.Then xy = εyields y = ε. 

(c)Let xa = ya.BytheeditorsaxiomTC3,thereexistsawsuchthat 

xw = y & wa = aor yw = x & wa = a.Considerthefirstcase,thesecond 

oneissymmetric.From wa = aandirreducibilityof awehave w = ε.From 

thatand xw = yweindeedhave x = y. 

(d)Let a xy,andlet ubesuchthat ua = xy.TheaxiomTC3yields 

a wsatisfying uw = x & wy = a,or xw = u & wa = y.Inthesecondcase 

weobviouslyhave a y.Soconsiderthefirstcase.From wy = awehave 

w = εor y = ε.If y = εthenwearedone.If w = εthen y = a,andthus 

a y. 

(e)Let Dbethesetofallstringsin {a,b,e} ∗ thathavenooccurrences 

of ae.Realize aand bby aand brespectively,anddefine x+yaccordingly: 

x + yresultsfrom xybyrepeatingthesubstitution ae→ewhilepossible. 

Forexample, bab + eb = babeb,but baa + eb = beb.Onecancheck,incase 

ofTC3withalittleeffort,incaseoftheremainingaxiomsrathereasily, 

thatthestructure D = 〈D,+,ε,a,b〉isamodelofthetheory TC. In D 

wehave a + e = ε + e. Sotheformula x ⌢ z = y ⌢ zisnottruein Dif x, 

y, zareevaluatedby a,theemptystring,and erespectively,andthusthe 

sentence(e)isnotvalidin D. 

Anotherusefulsentenceis ∀x∀y(a ⊑ xy → a ⊑ x ∨ a ⊑ y). Weleave 

itsproofin TCasanexercise. Moreaboutthetheory TCandaboutits 

modelsisin(Visser,2009). 

5 ThetheoryF,interpretability 

Theorem10.Robinsonarithmetic Qisinterpretablein TC. 

Proof.Weonlygivethebasicideaoftheproofgivenin( ˇ Svejdar,2009). 

Thefullproofisrathertechnical. 

Whenconstructinganinterpretation,onefirsthastospecifyitsdomain, 

whichinourcasemeanstoworkin TCandselectstringsthatwillplaythe 

roleofnaturalnumbers.Itappearsthatthefollowingdefinitionworks: 

Num(x) ≡ ∀u(u ⊑ x & u �= ε → a u), 

astring xisanumberifeachnon-emptysubstringof xendsby a. Note 

that,inthemodel DintheproofofTheorem9(e),thestring estartsby a 

(since e = a + e). However, eisnotanumberbecauseitisanon-empty 

substringofitselfandcannotbewrittenas e = z +a,i.e.doesnotendby a.


Havingnumbers,additionisinterpretedasconcatenation,zeroisinterpretedastheemptystring 

ε,andthesuccessorfunction Sisinterpretedas 

thefunction x ↦→ xa.Thesedefinitionsworkbecausein TConecanprove 

that εand aarenumbersandthatnumbersareclosedunderconcatenation. 

Allaxiomsof Qabout 0, S,and + translatetosentencesprovablein TC 

underthisinterpretation. 

Tointerpretmultiplication,astraightforwardideaistofirstdefinethe 

notionofawitnessingsequence. Asequenceofpairs [u0,v0],.. ,[uq,vq]is 

awitnessingsequencefor x · yif: u0 = v0 = ε,foreach i < qthepair 

[ui+1,vi+1]equals [uia,viy],and uq = x.Thenonecandefinethat x · y = z 

ifthereexistsawitnessingsequencefor x · ywith [x,z]asthelastmember. 

Theproblemhereisthatin TCitisnotpossibletoprovethatawitnessing 

sequenceexistsforeachchoiceof x,y,anditisalsonotpossibletoprove 

thatifitexists,itisuniquelydetermined. Awayhowtoovercomethis 

problemisinterpretingnotthefullRobinsonarithmetic Q,butratherits 

variant Q − inwhichadditionandmultiplicationarenon-totalfunctions. 

Thentheresultisobtainedbycombiningtheconstructedinterpretation 

of Q − in TCwithafactknownfrom( ˇ Svejdar,2007)that Qisinterpretable 

in Q − . 

Thetheory Q − usedintheproofofTheorem10wasalsointroduced 

byGrzegorczyk. Theinterpretationof Qin Q − in( ˇ Svejdar,2007)isconstructedusingtheSolovaymethodofshorteningofcuts. 

Thismethodis 

nowwidelyknown,butwasneverpublished:itisonlyexplainedinaletter 

toPetrHájek(Solovay,1976).M.Ganeain(Ganea,2007)givesadifferent 

proofofinterpretabilityof Qin TC,buthealsousesthedetourvia Q − . 

SterkenandVissergiveaproofnotusing Q − ,see(Visser,2009). 

Aconsequenceofthefactthat Qisinterpretablein TCisessentialundecidabilityof 

TC.Allproofsofinterpretabilityof Qin TCaresomewhat 

involved,butstillsimplerthanthedirectproofofessentialundecidability 

of TCgivenin(Grzegorczyk&Zdanowski,2008). Theseinterpretability 

proofsmightusesomeideasdevelopedbyGrzegorczykandZdanowski:that 

iscertainlytrueabouttheauthor’sproofin( ˇ Svejdar,2009). 

Since TCisinterpretablein I∆0,thetheories TCand Qaremutually 

interpretable;thustheyrepresentthesameexpressiveanddeductivepower. 

Thisisapieceofinformationmissingin(Grzegorczyk&Zdanowski,2008). 

Aninterestingalternativetheoryofconcatenationisthetheory F.Ithas 

thesamelanguageas TC,anditsaxiomsare: 

F1: ∀x(xε = εx = x), 

F2: ∀x∀y∀z(x(yz) = (xy)z), 

F3: ∀x∀y∀z(yx = zx ∨ xy = xz → y = z),


F4: ∀x∀y(xa �= yb), 

F5: ∀x(x �= ε → ∃u(x = ua ∨ x = ub)). 

AxiomsF1andF2arethesameasaxiomsTC1andTC2of TC.Itiseasy 

toverifythataxiomF4isprovablein TC;axiomsF3andF5,asisevident 

frommodels Dand Bintheprevioussection,arenotprovablein TC.From 

theoppositepointofview,axiomsTC4–TC6andsentences(a)and(b)in 

Theorem9areexamplesofsentencesprovablein F;weleavetheirproofs 

tothereaderasaninterestingexercise. AlbertVisser,see(Visser,2009), 

hasconstructedamodel Mof Fsuchthat M |= / ∀x∀y(a ⊑ xy → a ⊑ 

x ∨a ⊑ y).Thusin F,onecanhavestrings w1and w2suchthat a ⊑ w1w2, 

a �⊑ w1, a �⊑ w2;AlbertVisserdescribesthissituationascreatingaletter 

exnihilo. Aconsequenceoftheseremarksisthat Thm(TC)and Thm(F) 

areincomparablesetsofsentences. 

Itisclaimedin(Tarskietal.,1953)thatW.SmielewandA.Tarski 

provedessentialundecidabilityof Fbyinterpreting Qin F;however,no 

proofisgiven.Ganea(2007)constructedaninterpretationof TCin F.In 

conjunctionwithTheorem10,thisgivesaproofofthetheoremofSmielew 

andTarski. Wegive(aslightsimplificationof)Ganea’sproofbelowin 

Theorem11.Notehowever,thatitisstillaninterestinghistoricalproblem 

whatproofcouldSmielewandTarskihavehadinmind. Ours(Ganea’s) 

proofimplicitlyusestheSolovay’sshorteningtechnique,formulatedlong 

afterthebook(Tarskietal.,1953)waspublished. A.Visserhassome 

possibleexplanationofthishistoricalproblem. 

Theorem11(Ganea). TCisinterpretablein F. 

Proof.Workin Fanddefinetamestringsasfollows: 

Tame(x) ≡ ∀v∀z(z vx → z x ∨ x z), 

where hasthesamemeaningasin TC. 

(i)Wefirstshow(provewithin F)thattamestringsareclosedunder 

concatenation.Soassumethat xand yaretame,andlet vand zbesuch 

that z vxy. Weneedtoshowthat z xyor xy z. Since yistame, 

wehave z yor y z.If z ythen z xyandwearedone.Soassume 

that y zandtake tsuchthat ty = z. From z vxywehaveausuch 

that uz = vxy;thus uty = vxy.FromaxiomF3wehave ut = vx.Since x 

istame,wehave t xor x t.Then ty xyor xy ty.Since ty = z, 

weindeedhave z xyor xy z. 

(ii)Nextweshowthatif wyistame,thenalso wistame.Solet vand z 

besuchthat z vw.Wewanttoshowthat z wor w z.From z vw 

wehave zy vwy.Since ywistame,wehave zy wyor wy zy.Then 

astraightforwarduseofaxiomF3yields z wor w z.


Nowwearereadytoverifythatthedomainoftamestrings,togetherwith 

theidenticalmappingofsymbols(a, b,and εto a, b,and εrespectively, 

concatenationtoconcatenation),definesaninterpretationof TCin F. It 

isnotdifficulttoverifythat a, b,and εaretame;thistogetherwith(i) 

meansthatthedomainoftamestringsisclosedunderalloperations.The 

axiomTC1translatestothesentence ∀x(Tame(x) → xε = εx = x). This 

sentenceisevidentlyprovablein F.Asimilarargumentshowsthataxioms 

TC2andTC4–TC6translatetosentencesprovablein Faswell.Thisisso 

easybecauseTC2andTC4–TC6areuniversalsentences. 

ThusitremainstoprovethethetranslationoftheeditorsaxiomTC3is 

provablein F.NotethatTC3istheonlyaxiomof TCthatisnotauniversal 

sentence;itcontainsanexistentialquantifier.Let x, y, u, v,betamestrings 

suchthat xy = uv.Wehavetoshowthatthereexistsatame wsatisfying 

xw = u & wv = yor uw = x & wy = v.Since yistame,from uv = xy 

wehave v yor y v.Itissufficienttoconsiderthelatter,theformeris 

symmetric.Wehaveawsuchthat wy = v.Then uwy = uvand uwy = xy. 

FromaxiomF3wehave uw = x.So wisaninterpolant.Since vistame, 

from wy = vand(ii)aboveweknowthat wistame. 

Since Fiseasilyinterpretablein I∆0,fromtheotherresultsmentionedin 

thispaperweknowthat Fand TCaredeductivelyincomparable,butfrom 

interpretabilitypointofviewtheyrepresentthesamedegreeofdeductive 

strength.Itmaybeofsomeinteresttodirectlyinterpret Fin TC. 

Theorem12. Fisinterpretablein TC. 

Proof.Nowin TC,workwithradicalstrings,where 

Rad(x) ≡ ∀y∀z(yx = zx → y = z). 

Itisnotdifficulttoshowthatradicalstringsinclude ε, a,and b,andthat 

thedomainofallradicalstringsthatareemptyorendineither aor bis 

closedunderconcatenationanddefinesaninterpretationof F. 

6 OntheGrzegorczyk’sproject 

LetusrepeatfromtheIntroductionthatGrzegorczyk’ssuggestionisto 

considerstringsandconcatenationonbothformalandmetamathematical 

level.Onformallevel,thetheoryofconcatenationcanserveasanalternativetoRobinsonarithmetic;onmetamathematicallevel,dealingwithtextsisphilosophicallybetterjustifiedbecauseintellectualactivitieslikereasoningandcomputinginvolveworkingwithtexts. 

Briefly,themotivationsof 

thatprojectcanbesummarizedasfollows:


•inGödel’sargument,theonlyuseofnumbersiscodingofsyntactical 

objects, 

•thenGödeltheoremsarepresentedasapartofmathematics,buttheir 

significanceisbroader, 

•whenreasoning,communicating,orevencomputing,wedealwith 

texts,notwithnumbers, 

•onmetamathematicallevel,thenotionofcomputabilitycanbedefined 

withoutreferencetonumbers. 

Onecouldremarkthatmathematicsinnotnecessarilyidentifiedbyworking 

withnumbers;Gödeltheoremscouldbepresentedaspartofmathematics 

evenifreformulatedwithoutnumbers,andtheytranscedemathematicsregardlesswhethertheirformulationinvolvesstringsornumbers.Withthis 

littleremarkinmind,onecansaythattheargumentsproGrzegorczyk’s 

projectareclearandeasilyacceptable.Thedefinitionofrecursivenesswithoutusingnumbers,asdonein(Grzegorczyk,2005),isveryinteresting. 

However,itisalsopossibletofindsomeargumentsthatspeakcontrathat 

project,oratleastformodifyingorextendingit. First,whenreasoning 

orcomputing,wenotonlyconcatenate: wealsosubstitute. Creatinga 

grammaticallycorrectsentenceinanaturallanguagecanbedescribedas 

substitutingintopatterns.Inlogic,wehavesubstitutioninformulationof 

predicateaxioms. Soonecanthinkthatthetheoryofconcatenation,if 

enhancedbysomenotionofoccurrenceorsubstitution,couldbetterserve 

itspurpose.Second,whenprovingessentialundecidability,onealsoneeds 

anorder. Knownproofsusually(isitamistaketosayalways?) contain 

somesortofRossertrick,i.e.,speakaboutaneventthatoccursbeforesome 

otherevent. Onecanthinkthatconsideringorderismorenaturalinthe 

environmentofnumbersthanintheenvironmentofstrings.Infact,defining 

anorderofstringsisoneofcrucialandratherdifficultstepsintheessential 

undecidabilityproofof TCcontainedin(Grzegorczyk&Zdanowski,2008). 

Vítězslav ˇ Svejdar 



vitezslav.svejdar@cuni.cz,http://www.cuni.cz/∼svejdar/ 

References 

Ganea,M.(2007).ArithmeticonSemigroups.J.SymbolicLogic,74(1),265–278. 

Grzegorczyk,A.(2005).UndecidabilitywithoutArithmetization.StudiaLogica, 

79(2),163–230.


Grzegorczyk,A.,&Zdanowski,K. (2008). Undecidabilityandconcatenation. 

InA.Ehrenfeucht,V.W.Marek,&M.Srebrny(Eds.),AndrzejMostowskiand 

foundationalstudies(pp.72–91).Amsterdam:IOSPress. 

Quine,W.V.O. (1946). Concatenationasabasisforarithmetic. J.Symbolic 

Logic,11(4),105–114. 

Solovay,R.M.(1976).Interpretabilityinsettheories.(UnpublishedlettertoP. 

Hájek,Aug.17,1976,http://www.cs.cas.cz/hajek/RSolovayZFGB.pdf.) 

ˇSvejdar,V.(2007).AnInterpretationofRobinsonArithmeticinitsGrzegorczyk’s 

WeakerVariant.FundamentaInformaticae,81(1–3),347–354. 

ˇSvejdar,V. (2009). Oninterpretabilityinthetheoryofconcatenation. Notre 

DameJ.ofFormalLogic,50(1),87–95. 

Tarski,A.,Mostowski,A.,&Robinson,R.M. (1953). Undecidabletheories. 


Visser,A.(2009).GrowingCommas:AStudyofSequentialityandConcatenation. 

NotreDameJ.ofFormalLogic,50(1),61–85.

The Role of Negation in Proof-theoretic Semantics: 

a Proposal 

Luca Tranchini ∗ 

Proof-theoreticsemantics,asdevelopedbyauthorssuchasDummettand 

Prawitz,triestoaccountforthemeaningoflogicalconstantsthroughthe 

usemadeoftheminpractice. Thetypicalcontextinwhichtheyfigure 

isdeduction,sotheprogrambecomestheoneofshowinghowtherules 

governingdeductivepracticesfixthemeaningoflogicalconstants. The 

theoreticalrequirementruleshavetosatisfyisharmony,whichisendorsed 

inGentzen’sinversionprinciple. 

Oneofthedistinctivefeaturesofhumanlanguageiscompositionality, 

thatisthepossibilityofproducingsentencesofarbitrarycomplexityby 

meansoflogicaloperators. Henceproof-theoreticsemanticscanaimat 

beingthecoreofafully-fledgedtheoryofmeaning,thatisofanexplication 

ofspeakerslanguagecompetence. 

1 Verificationism:ProofandAssertion 

Theverificationisttheoryofmeaning,isgroundedonthechoiceofassertion 

asthebasiclinguisticact.Assertionistakentobegovernedbythefollowing 

principle 

Theassertionofasentenceiswarrantedonlyifitstruthisrecognized 

wheretheintuitivenotionoftruthrecognitionistobeexplainedbymeans 

ofthenotionofproof. 

Clearly,theproofsthatcountasevidenceforthetruthofsentencesare 

onlyclosedproofs,i.e.proofsinwhichtheconclusiondoesnotdependon 

anyassumption. 

Openproofsareonlymediatelyconnectedwithlinguisticacts.Takean 

openproofof Bfrom A 

∗ ThisworkhasbeensupportedbytheESFEUROCORESprogramme“LogiCCC— 

ModellingIntelligentInteraction”(DFGgrantSchr275/15–1).

274 LucaTranchini 

A. 

B 

thesentence Bcanbeassertedonlyifevidencefor Aisavailable. If A 

isatomic,evidencewillconsistintheopportunecomputation,if Aisa 

mathematicalsentence.Whenever Aisanempiricalsentencewecanthink 

ofevidenceforitasanopportuneempiricalobservation. 

Technically,toaccountforthese“atomicproofs”inastandardnatural 

deductionsystemweextendthevocabularywithaset Pofpropositional 

constants,standingfortheatomicsentencesoflanguage.Asubsetofsuch 

sentences, Twillconsistsofthesentencesforwhichextra-deductiveevidence 

isavailable.Thesetofopenassumptionsinadeductionwillberestricted 

tothosenotbelongingto T. 

If Aisacomplexsentence,onecantrytoobtainaproofofitfromatomic 

assumptionsin T.Still,itisnotalwayspossibletodoso.Nonetheless,even 

incasessuchasthesewecanobtainaclosedprooffromtheopenone,even 

thoughtheconclusionoftheclosedproofisnottheconclusionoftheopen 

one,butamorecomplexsentence. Typically,implicationisthedeviceby 

meansofwhichanopenproofistakenintoaclosedonehavingasconclusion 

theimplicationoftheassumptionandtheconclusionoftheopenproof: 

[A] 

. 

B 

A → B 

Ingeneral,wecansaythatverificationismfocusesontheroleofsentencesasconclusionsofdeductiveprocesses. 

Asaconsequence,therules 

thataretakentofixthemeaningoflogicalconstantsareintroductionrules. 

For,theyspecifytheconditionsunderwhichasentencehavingtherelevant 

constantasprincipaloperatorcanbeintroducedasconclusionofaderivation.Accordingly,thenotionofcanonicalproof(thatisofaproofinwhich 

introductionrulesplayaprominentrole)hasbeentakenastheexplicans 

ofthenotionofmeaning.Thatis,toknowthemeaningofasentenceisto 

knowwhatcountsasacanonicalproofofit. 

2 Negation 

Toaccountfornegationinverificationism,thesymbol ⊥,standingforabsurdity,isintroduced.Thenegationofasentence 

A, ¬A,isdefined,infull 

analogywiththeBHKclause,as A → ⊥.Sowehavetworulesfornegation 

(anintroductionandanelimination),whicharenothingbutspecialcases 

oftheimplicationrules.

NegationinProof-theoreticSemantics 275 

[A] 

. 

⊥ 

¬A I¬ 

. 

A 

. 

¬A 

⊥ 

Obviously,theinversionprincipleholdsfortheserulesaswell(asaconsequenceofitsvalidityforimplication). 

Theproblem 

Suchacharacterizationgraspsallpropertiesofintuitionisticnegationexcept 

thefactthatnoconstructionoffersevidencefortheabsurdity.So,further 

ruleshavetobeadded,tofixtheintendedmeaningof ⊥. 

Itisnosimpletasktoexplicitlyexpresswitharulethefactthatno 

constructionsatisfiestheabsurdity:rulesspecifywaysofobtainingproofs, 

whileouraimistospecifytheabsenceofproofs. 

IntuitionisticNaturalDeductionNJisobtainedbyextendingminimal 

logicwiththefollowingrule,socalledexfalsoquodlibet: 

. 

⊥ 

A ef 

Oncetheabsurdityhasbeenderived,itispossibletoderiveeverything. 

Theruleisintendedtoholdforanysentence A. Still,withoutlossof 

generality,itisusefultorestrictittoatomicsentences.Asaconsequence, 

⊥canbetakenasanabbreviationofaself-contradictoryatomicsentence, 

forinstance‘0 = 1’,deductivelycharacterizedbythefactthatitentailsall 

otheratomicsentences. 

Aswiththeexfalsowecanformallyseizeintuitionisticlogic,itisnatural 

tothinkofitasgraspingtheintendedmeaningoftheabsurdity. Yet,as 

someauthorshavenoticed,itisdubiousthattheexfalsoconveysto ⊥the 

desiredmeaning.Thefactthatitistobereadasabsurdityseemstodepend 

onwhichatomicsentencesareprovable.Indeedifallatomicsentenceswere 

provable,therewouldbenothingwronginasserting ⊥asonlytruesentences 

couldbeinferredfromit.Butif ⊥hastobeconsideredasstandingforthe 

absurdity,thenitshouldnotbeassertibleinanysituation. 

Apossiblewayout(Dummett,1991)istobanthepossibilitythatall 

atomicsentencescanbesimultaneouslyasserted,thatistoassumethatat 

leasttwoatomicsentencesaremutuallyincompatible. Nonetheless,this 

restrictionsoundsdefinitelyadhoc:wehavenoreasontoproposeit,apart 

fromtheneedofwarrantingthattheexfalsoconveystheexpectedmeaning 

to ⊥and,consequently,tonegation. 

E¬


Brouweronabsurdityandnegation 

InBrouwer’searlywritingswefindtheideathatthenegationofasentence 

iswarrantedwhen 

wearrivebyaconstructionatthearrestmentoftheprocesswhich 

wouldleadto[aconstructionforthesentence]. 1 

Toclarifythispoint,onecanimagineamathematician(or,rather,the 

idealizedmathematician)attemptingtoproduceaconstructionforasentence 

A. Unfortunately,itisimpossibletoobtainaconstructionfor A. 

Hence,eachattemptreachesacertainpointafterwhichitisnotpossibleto 

carryouttheconstruction,apointbeyondwhich,inBrouwer’swords,“the 

constructionnolongergoes.” 

AccordingtoBrouwer,whenthemathematicianfindsherself(i.e.,she 

producesaconstructionshowingthatsheis)insuchasituationshecan 

declarethesentencefalseor,equivalently,acceptitsnegationaswarranted. 

Itisonlylaterthattheideaofarrestintheprocessofconstructionis 

substitutedbytheoneofcontradiction,glorifiedintheBHKspecification 

ofthesemanticsfortheintuitionisticlogicalconstants. Thetreatmentof 

contradictionasasentence,implicitintheintuitionisticinformalsemantics 

hasbeenfullyfledged,aswebrieflyshowed,withthemarriagebetween 

intuitionismandnaturaldeduction,throughtheopportunereadingofthe 

exfalsorule. 

Tennanton ⊥ 

Recently,Tennant(1999)triedtochallengetheverificationistaccountof 

whatisaproofforthenegationofasentence. Evenifhedoesn’tmake 

explicitreferencetoBrouwer,itisquitenaturaltoputtheconceptionof ⊥ 

heproposessidebysidewiththeideathatacontradictionisnothingbuta 

deadendintheprocessofconstruction. 

Tennantstartsfromtherefusalofconsideringproofsofthenegationof 

asentenceasmethodstoobtainaproofofafalsesentence, ⊥.Rather,he 

proposesconsidering ⊥asamarkerofadeadendintheprocessofconstruction. 

Clearly, ⊥isdevoidofsententialcontent,i.e.,itisnomorean 

abbreviationfor‘0 = 1’. Hence,weareforcedtowithdrawtheinterpretationof 

¬Aas A → ⊥: as ⊥hasnosententialcontentwecan’tapply 

toitsententialoperators,inparticularimplication.Wecanconcludewith 

Tennant’sthat, 

accordingly,anoccurrenceof ⊥isappropriateonlywithinaproof,as 

akindofknot—theknotofpatentabsurdity,orofself-contradiction. 2 

1 (Brouwer,1908,p.109). 

2 (Tennant,1999,pp.203–204).


However,TennantdoesnotcompletelyembodyBrouwer’ssolution.For, 

accordingtoBrouwer,provingthenegationof Ameansfindingadeadend 

intheroutetowardtheproofof A.Ontheotherhand,Tennantthinksthat 

theroleof Aisnotthatofanunreachablegoal,butratherastartingpoint. 

Weprovethenegationof Awhenwereachadeadendstartingfrom A.It 

isimportanttoobservethatthisisnottosaythatwestartfromahypotheticalconstructionfor 

A,asinHeytinginterpretationofthemeaningof 

negation.Ratherweareperforminganactivitywhichisdifferentfromthe 

productionofproof.Suchanactivityisnotorientedbytheconclusionthat 

wewanttoreach,butratherbythepointfromwhichwestart. Tennant 

introducesanewprimitivenotiontorefertothisalternativeactivity:disproof.Theactivityofconstructionisthensplitintwodifferentsubspecies: 

theproductionofproofsandtheproductionofdisproof.Whenweattempt 

todisproofasentencewedonotstartfromaproofofitthatthenturnsout 

tobeimpossible.Wesimplylookforadisproofofit. 

Asaconsequence,theBHKnegationclauseisreformulatedas: 

•Aproofof ¬Aisadisproofof A 

droppingthe ⊥clause.So,insteadofanalyzingnegationintermsofimplicationandabsurdity,wetrytodothisintermsofdisproofs. 

Tennantpresentshisnotionofdisproofwithoutanyreferencetorelated 

work. Thoughitseemsquitenaturaltocomparetheseideaswithwhat 

DummettandPrawitzsaidaboutthepossibilityofdevelopingatheoryof 

meaningcenteredaroundthenotionofrefutation. 3 

3 Falsificationism:RefutationandDenial 

AccordingtobothDummettandPrawitz, 4 itispossibletothinkoftheories 

ofmeaningalternativetoverificationism,inparticulartooneinwhichthe 

meaningofsentencesisspecifiedbywhatcountsastheirrefutation. The 

primitivecharacterofrefutationscanbeendorsedbyconsideringthelinguisticactparalleltotheoneofassertion,thelinguisticmanifestationofthe 

possessionofarefutationofasentence:denial.Therelationshipofdenial 

torefutationisgovernedbytheprinciple: 

Thedenialofasentenceiswarrantedonlyifitsfalsityisrecognized 

wheretherecognitionofthefalsityofasentenceamountstothepossession 

ofarefutationofit. 

3 WhileTennantspeaksofdisproofs,wepreferrefutation. Aswillbecleared,theintuitionsbehindthetwonotionarecommon,eventhoughthedetailedtreatmentissensibly 

different. 

4 Theideaspresentedinthissectioncomefrom(Dummett,1991,Ch.13)and(Prawitz, 

1987, §6).


ThistheoreticalperspectiveofPopperianflavorisgroundedontheintuitionaccordingtowhich,inacceptingasentence,aspeakermustalso 

bereadytoacceptallitsconsequences.Wheneveroneofitsconsequences 

turnsouttobeunacceptable,sotoomustthesentenceuponwhichitdependsberejected.Hencethecentralnotionofatheoryofmeaninginwhich 

denialplaysabasicrolewillbetheoneofconsequenceofasentence.Thus, 

themeaningofthelogicalconstantsisfixedbyeliminationrules,asthey 

typicallyspecifyhowasentencecanbeusedasassumptioninaderivation. 

AccordingtoDummettandPrawitz,oneneedsnottointroducenew 

technicaltoolstoaccountforrefutations. Ratherthenotionofrefutation 

canbedefinedinastandardnaturaldeductionframeworkprovidingan 

alternativeinterpretationofthedeductivesystem. 

Accordingtotheverificationistreading,onecaneasilyconstructproofsof 

morecomplexsentencesstartingfromproofsofsimpleroneswithintroductionrules.So,accordingtofalsificationism,onecanconstructrefutationsof 

morecomplexsentencesfromrefutationsofsimpleroneswithelimination 

rules.Forexampletakenarefutationof A: 

A. 

onecanobtainarefutationof A ∧Bwiththehelpofthe ∧eliminationrule: 

A ∧ B 

A E∧ 

. 

Thecoreofthedeductiveprocesseswillthenbetheassumption,thatacts 

likeastartingpointofthederivationandthatonetriestorefute. Asa 

consequence,therulesthataretakentofixthemeaningoflogicalconstantsareeliminationrules.For,theyspecifytheconditionsunderwhicha 

sentencehavingtherelevantconstantasprincipaloperatorcanbeusedas 

assumptioninaderivation.Accordingly,thenotionofcanonicalrefutation 

(thatisofadeductioninwhicheliminationrulesplayaprominentrole)has 

beentakenastheexplicansofthenotionofmeaning.Thatis,toknowthe 

meaningofasentenceistoknowwhatcountsasacanonicalrefutationof 

it. 

Asintheverificationistframework,soherenotallderivationsaredirectly 

linkedtothebasiclinguisticact.Again,anopenderivationof Bfrom A 

A. 

B 

receivesahypotheticalreading:ifonecomesintopossessionofarefutation 

of B(i.e.,ifsheisinthepositionofdenying B),thenshewillalsobeinthe 

positionofdenying A.


Justasintheverificationistcase,weintroduceanotionofevidencefor 

atomicsentencestowhichwereferasextra-deductiverefutingevidence. 

Suppose Bisthesentence‘Thecuponthetableisblue’,theempirical 

observationthatthecuponthetableisredcanbetakenasrefutingevidence 

for B. 

Technically,wedefineasubsetofthepropositionalconstants, F,containingtheatomsforwhichanextra-deductiverefutationisavailable.RefutationsendingwithatomsbelongingtoF,havingasopenassumptionsinstancesofonlyonesentence,willallowthedenialofthatsentence. 

Thedisanalogybetweenthetwoperspectivesconsistsinthelackofa 

connectiveactinginfalsificationismasimplicationdoesinverificationism. 

Suchaconnectiveshouldallowthedenialofasentencealsoinsituations 

inwhichonlyanopendeductionisathand. Asimplicationissaidto 

dischargetheopenassumption,sotheconnectiveinquestioncouldbesaid 

to“dischargetheconclusion”ofthededuction. 

Butdoesthestandardlanguagepossessatoolwhichcanbetakenin 

somesensetodischargeconclusions? Negationcanbe(partially)thought 

ofintheseterms.Consideraderivationof A 

. 

A. 

Thenegationeliminationrulecanbeseenasawayofclosingtheconclusion 

ofthederivationbyintroducingamorecomplexassumption: 

. 

A ¬A 

⊥ . 

Thisisactuallyinfullanalogywiththewayinwhichimplicationworks:it 

closesanassumptionandintroducesamorecomplexconclusion. 

4 Towardaunifiedframework 

Thetwotheoriesofmeaning,verificationistandfalsificationism,havebeen 

treatedbybothDummettandPrawitzastwodifferent(concurrent)theoreticalenterprises.Thatis,asemanticsforagivenlanguagecanbedeveloped 

eitheraccordingtotheverificationistorthefalsificationiststandpoint. 5 

Onthecontrary,Tennant’ssuggestionsontheroleof ⊥inadeduction 

areveryneartothefalsificationistperspective.Bylookingathisproposal 

inmoredetail,onerealizesthatitisnothingbutamixtureofthetwoviews 

onmeaning. 

5 Dummettactuallygivesreasonsfordevelopingsimultaneouslybothperspectives. But 

eveninsuchacasethetwotheoriesaredistinct.


ThelimitsofTennant’sapproach 

Aswesaw,Tennant’ssuggeststoread ⊥asamarkerofdeadendsindeductions. 

Inotherwords,adeductionendingwith ⊥istakenasadeduction 

withnoconclusion.Hence,heproposestointerpretnaturaldeductionsystemsasprovidingthemeansforproducingopenproofs,closedproofsand 

disproofs. 

Supposeonehasalogicalsystemforwhichtheexistenceof 

proofsisindicatedbytheusualturnstile ⊢,arelationofexactdeducibilityholdingbetweenpremisesontheleftandaconclusion 

ontheright.Theintuitivemeaningof‘X ⊢ A’isthatthereisa 

proofwhoseconclusionis Aandwhosepremises(undischarged 

assumptions)formtheset X.[...] 

Therearetwoextremecases. 

1. Xisempty. Then‘⊢ A’means Aisatheorem. Thatis, 

thereisaproofof A‘fromnoassumptions’.[...] 

2. Ais‘empty’.Then‘X ⊢’meansthatthereisadisproofof 

X,thatis,adeductionshowingthat Xisinconsistent. 

Accordingly,insteadoftheusualinductivedefinitionofproof,Tennant’s 

givesasimultaneousdefinitionofthenotionsofproofanddisproof. 

Still,inthelightoftheconsiderationsonfalsificationism,Tennant’sapproachcanbecriticizedfortheasymmetryinthetreatmentofthetwonotions.Inparticular,totreatopendeductionsasopenproofsmeanstotreat 

themas“incompleteproofs”:theyaremeansofobtainingclosedproofsof 

theconclusions,providedclosedproofsoftheassumptions.Butwhyshould 

theynotbeconsideredas“incompleterefutations”,thatisasmeansof 

rejectingthepremisesoncerefutationsoftheconclusionsareprovided? 

ThisasymmetrycanalsobeseeninTennant’swayofdealingwithrules, 

ingivingthedefinitionofproofanddisproof.Introductionrulescanbeused 

onlytoproduceproofs. Eliminationrules,ontheotherhandcanbeused 

toproduceeitherproofsordisproofs,dependingonwhetherthedeductions 

oftheminorpremisesareproofsordisproofs. Herearethetwocasesfor 

disjunction: 

. 

A ∨ B 

[A] 

. 

C 

C 

[B] 

. 

C 

. 

A ∨ B 

[A] 

. 

⊥ 

⊥ 

Thepointisthatalsointroductionrulescanbeusedinproducingrefutations.Justlikeintheverificationistperspectiveoneproducesnon-canonical 

[B] 

. 

⊥


proofswitheliminationrules,soinfalsificationismoneproducesnon-canonicalrefutationswithintroductionrules. 

Finally,itisimplicitinTennant’slineofargumentthattheroleof ⊥in 

disproofsisanalogoustotheroleofdischargedassumptionsinproofs.Asa 

closedproofisadeductionwithnoopenassumptions,soarefutationisadeductionwithnoconclusion.Furthermore,evenifTennantdoesnotconsiderit,wesawthatinordertouseanaturaldeductionsystemformeaningtheoreticalpurposesonealsohastoaccountforextra-deductiveevidence 

foratomicsentences.Thereisadeepanalogyoftheroleofextra-deductive 

probativeandrefutingevidenceforatomicsentencesand(respectively)the 

roleofdischargingtheassumptionsandreachingadeadendinadeduction. 

Forallthesearethemeansthroughwhichanopendeductionistakeninto 

aclosedone(eitheraprooforarefutation). 

Alltheseconsiderationssuggestthepossibilityofre-framingthenatural 

deductionsysteminordertoexplicitlyshowthesesymmetries. 

Top-closedandbottom-closedderivations 

Bothperspectivesonmeaningdistinguishbetweenderivationsthatimmediatelyallowalinguisticperformanceandthosethatdonot.Inverificationismwehaveadistinctionbetweenclosedandopenproofs.Itseemsnatural 

toadaptthisterminologytorefutations,sothatwehaveopenandclosed 

refutations. 

Aswesaw,Tennantproposestotreat(whatinthestandardframework 

areconsidered)derivationsofconclusion ⊥asdisproofs.For, ⊥hasnosententialcontentandhencecan’tbetakentobetheconclusionofadeductive 

process.Rather,itregistersthefactthatthedeductivepathleadingtothe 

conclusionofthederivationisadeadend,orinotherwords,itisclosed. 

Canthesetwonotionof“closure”,theoneregisteredby ⊥andtheoneof 

deductiveprocesseslinkedtolinguisticacts,betakenintoone? 

Toexplicitlystatetheanalogyweintroducethesign ⊤tomarkassumptionclosure.Sowheneveranassumptionisclosed,wewillmarkit 

⊤.Inthe 

caseofassumptionsdischargethroughimplicationthissimplyamountstoa 

notationalchange.Insteadofputtingthesentenceinbrackets(oroverlining 

it),weputthesign ⊤overit.So,theintroductionruleforimplicationwill 

appearas: 

⊤ A. 

B 

A → B 

Asweobservedtherearetwodifferentwaysinwhichanopendeduction 

canbetakenintoaclosedone. Onecancloseoneoftheedges(assump-


tionsorconclusion)bylogicalmeans,inverificationismwithimplication,in 

falsificationismwithnegation;alternativelyonecantrytoreachtheatomic 

componentsofthesentencetobeprovedorrefuted,toseeifthereisextradeductiveevidencepurportingorrefutingsuchcomponents. 

Ifweconsiderverificationism,thenotionofclosure(bymeansofwhich 

weusuallyrefertodischargedassumptions)appliesquitewellalsotoatomic 

sentencesforwhichwehaveextra-deductiveprobativeevidence:anassumptionisclosedwhentheconclusionofthedeductiveprocessdoesnotdepend 

onit.Andclearly,notonlytheassumptionsdischargedthroughimplication 

areclosed,butalsotheatomiconesforwhichextra-deductiveevidenceis 

available.Thissuggeststheideaofextendingtheuseof ⊤tomarktheclosureoftheatomicassumptionsaswell.Accordingtothewayinwhichwe 

introducedatomicsentencesinnaturaldeductioninsection1,wecanuse 

⊤toexplicitlymarktheatomicsentencesbelongingtotheset Tofverified 

atoms.Todothis,weaddanewruletothenaturaldeductionsystem: 

if A ∈ Tthen 

isaderivationofconclusion Afromnoassumptions. 

⊤ A 

Forexample,supposetheweatheriswindy:insuchacase,theconclusion 

ofthederivation 

⊤ ⊤ 

Itrains Itiswindy 

Itrainsanditiswindy I∧ 

Ifitrainsthenitrainsanditiswindy 

I → 

canbeasserted,becauseitdoesnotdependonanyassumption,evenifthe 

twoassumptionsareclosedindifferentways:thefirstoneisdischargedby 

theapplicationofthe I →rule;thesecondoneisclosedbytheavailability 

oftheempiricalevidenceforit. 

Inanalogywiththis,infalsificationismwehavetwowaysoftakingan 

opendeductionintoarefutation:eitherrefutingevidenceisprovidedforthe 

conclusion;oralternatively,wecanusesomelanguagedevicesto“discharge” 

theconclusioninthecourseofthederivation. 

Ifwetakeacloselookatthefirstpossibility,Tennant’sideaof ⊥as 

registeringaknotofinconsistencyfitsthissituationquitewell.For, ⊥can 

betakentoregisteranincompatibilitybetweentheoutputofthedeductive 

processandtheavailableevidence.Thissuggeststhepossibilityofextendingtheuseof 

⊥,bymarkingwithittheatomicconclusionsofderivations, 

forwhichweareinpossessionofextra-deductiverefutingevidence.Wecan 

formallyachievethiswitharuleanalogoustotheoneforatomicassumptions:


if A ∈ Fand 

isaderivationofconclusion Afromassumptions Γthen 

Γ. 

A 

Γ. 

A 

⊥ 

isaderivationofnoconclusionfromassumptions Γ. 

Forexample,supposethecuponthetableisred:insuchacasewemark 

theconclusionofthefollowingderivationwith ⊥: 

Thecuponthetableisblueanditisfulloftea 

Thecuponthetableisblue 

⊥ 

Thisextensionoftheuseof ⊥makesitpossibletoschematicallyrepresent 

thecoreprocessesoffalsificationismas 

A. 

⊥ 

Thispatternstandsforarefutation(andhenceallowsthedenial)ofagiven 

sentence A.Wewillalsorefertosuchdeductivepatternsasbottom-closed 

derivations.Onceintroducedthesign ⊤inordertomarkclosedassumptions 

indeductions,itispossibletorepresentthecoreprocessesofverificationism 

withthescheme: 

⊤. A 

standingforaproof(andhenceallowingtheassertion)ofthesentence A. 

Wewillrefertosuchdeductivepatternsastop-closedderivations. 

Newhorizonsforproof-theoreticsemantics 

Aswepreviouslyunderscoredwhatweareproposingisaunifiedframework 

inwhichbothproofsandrefutationscanbeaccountedfor.Todothiswe 

havetoaddasetofpropositionalconstants Ptoastandardnaturaldeductionsystemandbothasubset 

Tofverifiedatomsandasubset Fofrefuted 

atomshavetobespecified.Atthispointwehavethattop-closedderivation 

andbottom-closedderivationscountasclosedproofandclosedrefutations


forsentences,i.e.,theyallowtheassertionanddenialofsentences. Itremainsonlyadisanalogybetweenthetwokindsofderivations,namelythat 

whiletoatop-closedderivationalwayscorrespondstheassertionofthe 

conclusion,toabottom-closedderivationcorrespondsadenialonlyifthe 

assumptionsofthedeductionareoccurrencesofthesamesentence.Thisis 

duetothefactthatthenaturaldeductionframeworkallowsatmostone 

conclusionbutthereisnolimitonthenumberofpossibleassumptions. 

Besidethis,whatlooksreallyproblematicforthefulldevelopmentofthis 

perspectiveisthedefinitionofvalidity.For,inverificationismanopendeductionisvalidif,providedclosedderivationsoftheassumptions,itreduces 

toaclosedproof;infalsificationismanopendeductionisvalidif,provided 

aclosedderivationoftheconclusion,itreducestoaclosedrefutationof 

theassumption(s).Inotherwords,inbothperspectivesthecategoricalnotionofclosedderivationhasprimacyoverthehypotheticalnotionofopen 

derivation.Thepointisthatitisnotclearhowthenotionofvalidityisto 

beshapedinasysteminwhichwehavetwodistinctcategoricalnotions. 

Thedirectioninwhichthesolutioncanbefoundistherejectionofthe 

proof-theoreticdogmaaccordingtowhichthecategoricalnotionhasprimacyoverthehypotheticalone. 

Bydoingthiswecouldreallyembody 

Tennant’sintuitionaccordingtowhichclosedproofsarejustlimitcasesof 

openones.Intuitively,thismeansthatthegroundconceptofproof-theoretic 

semanticsistherecognitionofdeductivelinksamongsentences,thatonly 

inveryspecialoccasionscanbetakentobeorientedbytheconclusionor 

bytheassumption. Thisideacanbeseenatworkinreadingtherulefor 

makingassumptionsinnaturaldeduction: 

forany A 

isadeductionhaving Aasconclusionand Aasassumption 

A 

Bothverificationismandfalsificationismareforcedtoreadtheruleasproducing“incomplete”derivations,inthesenseofeitheranincompleteproof 

of Aoranincompleterefutationof A.Fromtheunifiedperspective,therule 

forassumptionisinterpretedsimplyas‘Consider A’:inconsidering Awe 

areneithercommittedtotheexpectationofaproofnortotheexpectation 

ofarefutationofit,weareopentoseewhatwillhappenatlaterstagesof 

thedevelopmentofthedeductiveprocess. 

Formalmodelswhichexplicitlyendorsethisintuitionaresequentcalculi. 

Insuchsystemsthefullsymmetrybetweenassumptionsandconclusion, 

i.e.assertionanddenial,isembodiedinthesymmetrybetweenleftand 

rightsideoftheturnstile.Comingbacktovalidity,itisinterestingtonote 

thatnoquestionofvalidityhaseverbeenaddressedforsequentcalculiand 

itisnotcompletelyclearhowtoformulateit. Itwouldnotbesurprising


thatratherthanaglobaldefinitionofvaliditywhatisneededaresimply 

localcriteriatobeimposedonrules. 6 Butwedonotpushtheissuefurther. 

AbsurdityandConsistency 

Aswesaw,inordertofixthemeaningof ⊥viadeductiverules,theverificationisthastorequirethatallatomicsentencesoflanguagecan’tbe 

simultaneouslyasserted. Otherwisenothingbansthepossibilityofasserting 

⊥,violatingtheBHKclausethatstatesthat ⊥can’tbeassertedin 

anysituation. 

Theinterpretationweareproposingclearlymakestheproblemofthe 

assertionof ⊥disappear,as ⊥isnolongertobeconsideredasentential 

contentcapableofbeingasserted(ordenied). Nonethelesstheintuition 

that ⊥can’tbeassertedcanbereformulatedasfollows. Wenotedthata 

sentencecanbeassertedwhenweareinpossessionofaderivation,having 

thesentenceasconclusion,withnoopenassumptions. So,theexpression 

“⊥canbeasserted”appearsasaroughwaytorefertoasituationthatwe 

canschematicallyrepresentinthisway: 

⊤. 

⊥ 

Howisthispatterntoberead? Itlookslikeaderivationinwhichboth 

assumptionsandconclusionhavebeenclosed. Tobetterunderstandit, 

considerasentence Afiguringinthederivation: 

⊤. A. 

⊥ 

Ifwesplitupthisdeductivepatternwefindourselveswiththefollowing: 

⊤. A 

Accordingtothereadingof ⊥and ⊤,thesederivationsamounttoaproof 

of Aandtoarefutationof A.Thatis,thepossessionofboththetop-and 

bottom-closedderivationsallowsboththeassertionandthedenialof A. 

Ifwetakethesentence Atobeanatomicsentence,e.g.,‘Thecupon 

thetableisred’,thetop-andbottom-closeddeductivepatternisavailable 

6 ThisdirectionisstronglycalledforbySchroeder-Heister(seeforinstance(Schröder- 

Heister,2009)). 

A. 

⊥


onlyifweareinpossessionbothofsupportingandrefutingevidenceforit. 

Obviously,thefactthatthesentence‘Thecuponthetableisred’canbe 

neitherprovednorrefuteddoesnotdependondeduction,butratheronthe 

factthatitisnotpossibletoseearedcupandagreencuponthetableat 

thesametime. 

Usuallyconsistencyistakentobetheimpossibilityofassertingtheabsurdity.Intheframeworkwearedevelopinganalternativenotionofconsistencycanbeputforward:namely,theimpossibilityofbeinginthepositionofassertinganddenyingasentenceatthesametime.Thisnotionofconsistencyamountstotheimpossibilityofobtainingdeductivepatternshaving 

bothassumptionsandconclusionsthatareclosed. 

JustlikeDummett,topreserveconsistencywehavetoimposearestrictiononatomicsentences: 

namely,wehavetorequirethateveryatomic 

sentencecan’tbebothassertedanddenied(wewillrefertothisasatomic 

consistency). Ourrestrictioncan’tbejustifiedonlogicalbasis,justas 

Dummett’sone.Nonetheless,itismuchmoreplausibletorequirethateach 

atomicsentencecan’tbeassertedanddeniedatthesametimeratherthan 

torequirethattheremustbemutuallyincompatibleatomicsentences. In 

particular,sucharestrictioncouldbefullyarguedfor,onthebackground 

ofconsiderationsonhumancognition. 

5 Conclusions 

Traditionally,thepossibilitiesofdevelopinganaccountofassertionandan 

accountofdenialhavebeenconsideredtwodifferententerprises.Togivean 

accountofthemeaningofnegationwesuggestedtodevelopauniqueframeworkinwhichthecentralroleisplayedbythenotionofopendeduction. 

Bymeansofthe ⊤and ⊥signswecangiveanaccountinwhichdeductive 

patternscountasproofsandrefutationsofsentences,i.e.,allowtheirassertionanddenial.Aswesaw,introductionrulesareatthecoreoftheprocess 

ofproof,whileeliminationsareatthecoreoftheprocessofrefutation. 

AtthispointwecanreconsiderthealternativetotheBHKclausefor 

negationTennantproposed: 

Aprooffor ¬Aisarefutationof A 

Inthelightoftheconventionsintroducedwecanschematizetheequivalence 

as: 

A. 

⊤. 

⊥ = ¬A 

Itseemsthatwearefacedwithasortofgeometricaloperationonderivation:byrotating180 

◦ arefutationof Acanbeturnedintoaproofof ¬A.


Hence,negationappearsasalinguisticdevicethatstatesinanexplicitway 

theimplicitharmony,embodiedintheinversionprinciple,holdingbetween 

groundsforasentenceandconsequencesofasentence. Indeed,inversion 

governstherelationshipbetweenintroductionandeliminationrulesand 

negationtheonebetweenproofsandrefutationswhicharedirectlyconnectedtothetwosetsofrules. 

Afurtherquestionnaturallyarises,namelywhetherthestandardrules 

fornegationdoproperlyseizethiscrucialfeatureoftheconnective. If 

theanswertobegivenwerenegative,thanmovingapartfromstandard 

intuitionisticlogicwouldbenecessary. 

Inconclusion,webelievetohaveisolatedtheroleofnegationinthe 

architectureofdeductiveactivityasbeingradicallydifferentfromthatof 

otherconnectives.Eventhoughthedevelopmentofaunifiedframework,in 

whichtoaccountforboththeactivitiesofproofandrefutation,seemsto 

requirefurtherinvestigation,webelieveittobeanimportantsteptofully 

developaproof-theoreticaccountofthemeaningoflogicalconstants,asthe 

analysisofnegationemergingfromitshows. 

LucaTranchini 

Wilhelm-SchickardInstitutfürInformatik,TübingenUniversity 

Sand13,72076Tübingen,Germany 

luca.tranchini@gmail.com 

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Brouwer,L.E.J.(1908).Theunreliabilityofthelogicalprinciples.InA.Heyting 

(Ed.),Collectedworks(Vol.I,pp.443–446). 

Dummett,M.(1991).Thelogicalbasisofmetaphysics.London:Duckworth. 

Prawitz,D.(1987).Dummettonatheoryofmeaninganditsimpactonlogic.In 

B.M.Taylor(Ed.),MichealDummett. 

Schröder-Heister,P. (2009). Hypotheticalreasoning: AcritiqueofDummett- 

Prawitz-styleproof-theoreticsemantics.InTheLogicaYearbook2008.(Sequent 

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Tennant,N. (1999). Negation,absurdityandcontrariety. InD.M.Gabbay& 

H.Wansing(Eds.),Whatisnegation? KluwerAcademicPublishers.

Oiva Ketonen’s Logical Discovery 

Michael von Boguslawski 

1 ShortbiographyofOivaKetonen 

OivaToivoKetonenwasbornJanuary21,1913,inthemunicipalityofTeuva 

intheSouthernOstrobothniaregionofFinland. 1 Hewaschildnumbereight 

inafamilythatraisedaltogether13children.Alreadyatayoungagethe 

law-governednessofnaturemadeadeepimpressiononhimandapparently 

plantedtheseedforaninterestinthenaturalsciences. Ketonenwasthe 

onlyoneofthefamily’schildrentogetanyformofhighereducation. 

KetonengraduatedfromKristiinakaupunginLukio(roughlyequivalent 

tohighschool)in1932,andenrolledintothedepartmentofhistoryand 

linguistics(wherephilosophyinHelsinkiwastaughtatthattime)atthe 

universityofHelsinki. TheprofessorofphilosophyatthattimewasEino 

Kaila,whohadcloseconnectionswiththeWienerKreisanditwasdue 

tohispersonaleffortsthatlogicarrivedinFinland. Ketonenswitchedto 

thedepartmentofmathematicsayearlater,despitehavingdoubtsthat 

mathematicsalonewouldsatisfyhisacademicinterests.Ketonen’steacher 

inmathematicsbecameRolfNevanlinna,thefamouscomplexfunctiontheoretician,andwecantellfrompreservedcorrespondencethatNevanlinna 

wasextremelyimpressedbyKetonen’smathematicalabilities. 

Therewasonlyonetext-bookonlogicavailableinFinnishatthattime 

—ThiodolfRein’sMuodollinenlogiikka—Formallogic(freetranslation 

fromFinnish)whichtreatedonlyAristotelianlogic.Therewasachangein 

thecurriculum,however,andBertrandRussell’sTheproblemsofphilosophyandKaila’sNykyinenmaailmankäsitys—Thepresentworld-view(free 

translationfromFinnish),amongothers,wereintroduced.Theteachingof 

logicwas,accordingtoKetonen,confinedtothebasicsandcouldnotas 

1 AnextendedversionofthisarticlewillappearintheYearbookoftheViennaCircle 

institute.IwouldalsoliketothankOiva’ssonTimoforgenerouslyprovidingmewitha 

copyofOiva’sunfinishedautobiography.

290 MichaelvonBoguslawski 

such,Ketonenspeculates,causeanyinterest.WereadinKetonen’sstudy 

bookthathedidnottakeasinglecourseinlogic. 

AccordingtoTimoKetonen,Oiva’searlyinterestswerealgebraandnumbertheory,whichpavedthewayforthehugeinterestinGödel’sfirstincompletenesstheorem,ofwhichhewasmadeawarebyhisfellowstudent,Max 

Söderman.Nevanlinnaalsolatermentionedthetheorem. 2 Gödel’sfantastic 

resultwasprobablywhatignitedKetonen’sinterestinformallogic.Ketonenwritesintheautobiographythathefrequentlywenttoeveningmeetings 

ofwhathecalled“Thephilosophicalclub.” Thesemeetingsseemtohave 

beenquiteunofficial,usuallythegroupgatheredatthehomeofoneofthe 

professors,e.g.,KailaorYrjöReenpääandlogicwasamongthetopicsdiscussed.TheyalsogatheredatleastonceatSöderman’shome.Inthestudy 

diarywecanreadthathelateralsospentsomeeveningsattendingwhat 

hecalls“mathematical-logicalconferences”. Itisunclearatthismoment 

whethertheseconferencesandthemeetingsofthe“philosophicalclub”were 

thesame. 

NevanlinnatriedtoconvinceKetonentotakeupfunctiontheory— 

anotherwitnessofNevanlinna’sfaithinKetonen’sabilities—butKetonen, 

aftersomecontemplation,decidedtoworkonlogic.Hewrotehismaster’s 

thesisonaxiomaticlogic,arithmetic,andGödel’stheorem. Thefirstpart 

waspublished(Ketonen,1938)andusedbyKailaasatextbookforlogic 

courses. KetonenhadreceivedtheimpressionfromNevanlinnathatsome 

mathematicianssuspectedthattherewassomefaultinGödel’sproof,and 

thatthisfaultmightbeworthuncovering. Ketonenbelievedthatasa 

resultofhisworkwiththethesis,hesucceededinstreamliningGödel’s 

proofsomewhat. 3 

KetonenkeptworkingonGödel’sresultsandmadeasmallimprovementtoGödel’scompletenesstheoremforthepredicatecalculusin1941 

(Ketonen,1941). Gödelshowedthatthateitheraproposition Aisprovable,oritisimpossiblethattheredoesnotexistacounterexample.Ketonenimprovedthisresultsothatthiscounterexamplecanbefounddirectly. 

Söderman,whoresidedinViennaatthetime,reportedKetonen’sresultto 

Gödel,whoadmittedthatitwasindeedanimprovement(vonPlato,2004). 

2 TheDissertation—UntersuchungenzumPrädikatenkalkül 

Accordingtohisautobiography,Ketonenhaddecidedalreadyinthespring 

of1938togoforadissertationimmediately. Hewenttotheuniversityin 

Göttingen,mostprobablywiththeaidofNevanlinna’scontacts,whohad 

2 

HowwellNevanlinnawasacquaintedwithlogic,andwhathethoughtoftheatthetime 

completelynewdiscipline,remainsdebated. 

3 

Wehopetoinvestigatethisstreamlininginalaterwork.

Ketonen’sDiscovery 291 

workedattheuniversityasavisitingprofessorin1936–1937.Kailahadmet 

GentzeninMünsterin1936aswell.KetonenalsowenttoMünsterwhere 

hemet—amongothers—HeinrichScholzwithwhomtherewassome 

correspondence. Shockingly,theverysamenightthatKetonenarrivedin 

Göttingen,9–10November1938,laterbecameinfamousasthe“Kristallnacht”—“crystalnight”. 

InGöttingen,intheautumnof1938,Ketonen 

becameGerhardGentzen’spresumablyfirst—andalsolast—student, 

althoughKetonenhadtowaituntilChristmastoreceiveaproblemfrom 

Gentzentoworkon.HerecallsGentzenasasympatheticyoungmanwho 

“didnottalkmuch”butmentionedthathischiefassignmentasHilbert’s 

assistantwasthereading(apparentlyaloud)of“popular”scientificpublicationstohisprofessor.Thedissertation(Ketonen,1944),UntersuchungenzumPrädikatenkalkül,isdividedintothreeparts.ThefirstpartpresentsandimprovesGerhard 

Gentzenssequentcalculusbyintroducinginvertiblerulesforthecalculus’ 

propositionalparts, 4 parttwodiscussesacertainSkolemnormalizationof 

derivations,andthethirdpartappliestheresultsfrompartsoneandtwoto 

produceaproofoftheunderivabilityofEuclid’sparallellpostulatefromthe 

restoftheSkolem-axiomsforEuclideangeometry.Ketonenwasthefirstto 

continueSkolem’sworkongeometry(vonPlato,2007b). Theinvertibility 

resultwillnowbepresentedindetail. 

InvertibilityofRulesinGentzen’sLK 

Asequentisoftheform A1,A2,... ,Am → B1,B2,...,Bn. Capitallatin 

letters A,B,C,...willbeusedtodenoteformulas,capitalgreekletters 

Γ,∆,Θ,...willbeusedtodenotethe(possible)contextofaderivation. 

Contextsaretreatedaslistsofformulas. Theformulastotheleftofthe 

sequentarrow →makeuptheantecedent,theformulastotherightthe 

succedent. Thesequentarrowcanconvenientlybereadas“gives”. Thus 

thesequent A&B → Cmeansthatfromtheassumptions Aand Btogether, 

theconclusion Cfollows. Thesequentisreadas“Aand Bgives C”. A 

sequentshouldbeviewedasageneralizationoftheconceptofderivability, 

withoneormoreassumptionsintheantecedentgivingoneormorepossible 

casesinthesuccedent. Weusetheparenthesesintheusualway,andall 

theconnectives ¬, ∨,&,and ⊃. Forthefalsesentence(andtodenotea 

contradiction),Gentzenusesaspecialsymbolbutwewillnotneedithere. 

Theonlyaxiomistheinitialsequent A → A. Tobeabletocarryout 

derivationsandproofswithinthesystem,weneedlogicalandstructural 

rules.Thelogicalrulesmanipulateconnectiveswhereasthestructuralrules 

manipulateformulas.Derivationsarein“tree-form”andbeginfrominitial 

4 Obviously,invertibilityinKetonen’ssensecannotholdforthepredicatepart.


sequents(andpossiblycontexts)attheendofbranches,andendwiththe 

provensequentatthebottomofthetree,the“root”. 5 Below 6 aregiventhe 

structuralandlogicalrulesofGentzen’sfirstsystemofsequentcalculus, 

whichwetodaycallGentzenLK: 

StructuralrulesforGentzenLK 

Γ → Θ 

LW 

A,Γ → Θ 

Γ → Θ 

RW 

Γ → Θ,A 

Leftweakening Rightweakening 

A,A,Γ → Θ 

LC 

A,Γ → Θ 

Γ → Θ,A,A 

RC 

Γ → Θ,A 

Leftcontraction Rightcontraction 

∆,B,A,Γ → Θ 

LE 

∆,A,B,Γ → Θ 

Γ → Θ,B,A,Λ 

RE 

Γ → Θ,A,B,Λ 

Leftexchange Rightexchange 

Γ → Θ,B B,∆ → Λ Cut 

Γ,∆ → Θ,Λ 

Cut 

LogicalrulesforGentzenLK 

Γ → Θ,A Γ → Θ,B 

R& 

Γ → Θ,A&B 

A,Γ → Θ B,Γ → Θ 

L∨ 

A ∨ B,Γ → Θ 

Rightconjunction Leftdisjunction 

A,Γ → Θ 

L&1 

A&B,Γ → Θ 

B,Γ → Θ 

L&2 

A&B,Γ → Θ 

Leftconjunction1 Leftconjunction2 

Γ → Θ,A 

R∨1 

Γ → Θ,A ∨ B 

Γ → Θ,B 

R∨2 

Γ → Θ,A ∨ B 

Rightdisjunction1 Rightdisjunction2 

A,Γ → Θ 

R¬ 

Γ → Θ, ¬A 

Γ → Θ,A 

L¬ 

¬A,Γ → Θ 

Rightnegation Leftnegation 

A,Γ → Θ,B R ⊃ 

Γ → Θ,A B,∆ → Λ L ⊃ 

A ⊃ B,Γ,∆ → Θ,Λ 

Γ → Θ,A ⊃ B 

Rightimplication Leftimplication 

5 Aderivationmayofcoursehaveonlyonebranch,i.e.,haveonlyoneinitialsequentfrom 

whichsomeothersequentisproven. 

6 See(Gentzen,n.d.)fordetails.


Withinvertibilityismeantthatifasequentmatchestheconclusionofa 

rule,andifitisderivable,thenthecorrespondingpremissesarederivable. 

Gentzen’sLKisnotinvertible.ConsiderruleR∨2,forexample.Ifitwere 

invertible,thenthesequent A → Bwouldbederivablebecause A → A ∨ B 

isderivablefromtheinitialsequent A → A. A → Bisnotatallatautology 

soitclearlyshouldnotbederivablewithoutassumptionsinacompleteand 

consistentsystem. Thus,thelogicalrulesforleftconjunctionandright 

disjunctionneedtobereplacedwithinvertibleones,andKetonennotes 

thattheruleforleftimplicationwillhavetobereplacedwitharulewhich 

hasthesamecontextsinitstwopremisses: 

A,B,Γ → ∆ L& 

A&B,Γ → ∆ 

Ketonen’sinvertiblerulesforGentzenLK 

Γ → ∆,A,B R∨ 

Γ → ∆,A ∨ B 

Γ → ∆,A B,Γ → ∆ L ⊃ 

A ⊃ B,Γ → ∆ 

Theproofsoftheinvertibilityoftherulesareeasyandshort.Theones 

givenherediffersomewhatfromthosegivenbyKetonen,specificallysothat 

whenKetonenintroducestheconclusionofaruletheinvertibilityofwhich 

istobeprovedthroughaninstanceofitsnon-invertiblecounterpart,we 

simplyintroducetheconclusionaftertheverticaldotsthatindicatesome 

possiblederivation. 

A → A LW 

B,A → A LE 

ProofofinvertibilityofruleL& 

B → B LW 

A,B → B R& 

A,B → A . 

A,B → A&B A&B,Γ → Ω 

Cut 

A,B,Γ → Ω 

TheproofoftheinvertibilityofruleR∨issimplyahorizontal“mirror 

image”oftheproofabove. InordertoprovetheinvertibilityofruleL⊃, 

weshowthatbothpremissesarederivablefromtheconclusionbycut: 

A → A RW 

A → A,B R ⊃ 

. 

→ A,A ⊃ B A ⊃ B,Γ → Ω 

Cut,RE 

Γ → Ω,A 

B → B LW 

A → A,B R ⊃ 

. 

B → A ⊃ B A ⊃ B,Γ → Ω 

Cut 

B,Γ → Ω 

Withtheinvertiblerules,wecancarryouta“root-first”proofsearch 

inanalgorithmicfashion,beginningwiththesequentwewanttoprove,


andthenapplyingtherulesinreverseuntilwereachasituationwithonly 

initialsequents(andpossiblycontexts).Thisproofsearchwillterminate,so 

itcanintheorybedonebyacomputer.IndeeditispossiblethatKetonen’s 

sequentcalculusisthefirstsystemthatwouldpermitacomputertoproduce 

proofs.Itdoesnotmatterinwhichordertherulesareappliedinreverse, 

asthetwoproofsof → (A ⊃ B) ⊃ (¬B ⊃ ¬A)belowillustrate: 

A → A LW 

¬B,A → A R¬ 

¬B → ¬A,A R ⊃ 

B → B RW 

B → B, ¬A L¬ 

B, ¬B → ¬A R ⊃ 

B → ¬B ⊃ ¬A L ⊃ 

→ ¬B ⊃ ¬A,A 

A ⊃ B → ¬B ⊃ ¬A 

→ (A ⊃ B) ⊃ (¬B ⊃ ¬A) 

A → A LW 

¬B,A → A R¬ 

¬B → ¬A,A 

R ⊃ 

B → B RW 

B → B, ¬A L¬ 

B, ¬B → ¬A L ⊃ 

A ⊃ B, ¬B ⊃ → A R ⊃ 

A ⊃ B → ¬B ⊃ ¬A 

→ (A ⊃ B) ⊃ (¬B ⊃ ¬A) 

Themodificationoftherulesdoesnothamperthepropertiesofthe 

system,theHauptsatz,forexample,stillholds. KurtSchütteandHaskell 

Currygavecut-freeproofsofinvertibilityin1950and1963respectively, 

Currywiththeaddedresultthatinversionsareheightpreserving. 7 

Reactionstothethesisandfollow-up 

PaulBernays(Bernays,1945)wroteafavorablereviewofKetonen’sthesisinTheJournalofSymbolicLogicin1945,andKleenenotes(Kleene, 

1952)thatheknowsofKetonen’scalculusonlythroughthisreview. We 

knowthroughseveralsources,forexample(vonWright,1951),thatseveralresearchersincludingRichardFeys,andthealreadymentionedCurry, 

Kleene,andBernaysheldKetonen’sworkinhighregard.Curryreportedly 

(vonPlato,2004)heldKetonen’sworktobethebestthinginprooftheory 

sinceGentzen,andthepresentwriterhasseenaletterfromCurrytoKetonenwheretheformerasksforeverythingKetonenhaswrittenonlogic, 

eveninFinnish. ArendHeytingwroteareviewofthethesisin1947,but 

apparentlyfailedtoseeitsmainpointandappearsinsteadtoviewitas 

aworkongeometryratherthanonprooftheory. Thefirstinternational 

referencetoKetonen’sworkseemstobebyKarlPopperin1947(Popper, 

1947),andBethusespartsofKetonen’scalculusinhistableaumethod 8 

butcitesKleeneandGentzen,butnotKetonen. 

7 See(vonPlato,2007a). 

8 Seeforexample(Beth,1962). 

R ⊃


NomoreoriginalworkonlogicbyKetonenappearedafterthethesis, 

andexactlywhythisissoisnotcompletelyclear. Heisknowntohave 

beenworkingonforcinginsettheoryandevenrelativitytheory(inspired 

bypreviousworkonthesubjectbyKaila)butdidnotpublishanyown 

resultsevenifsurvivedcorrespondencesuggeststhatheindeedhadworked 

outsomeresultsofhisown. Hehasalsoworkedontheinterpretationof 

consistencyproofs,many-valuedlogics,andtheapplicationofsomeofthe 

resultsofthethesisonepistemology(vonWright,1951).Apossiblereason 

astowhyhedidnotcontinuewithlogiccouldbetheseveredisappointment 

heexperiencedwithphilosophyofscienceingeneralduringhisvisittothe 

UnitedStatesinthe1950’s(Ketonen&vonWright,1950)and,asishinted 

atintheautobiography,theeffectsthatthesecondWorldWarbrought 

withitwhichpossiblysteeredalsohisphilosophicalinterestsawayfrom 

theworldofmathematicstowardsbroaderphilosophicalenquiries. Only 

oneworkonlogicafterthedissertationhasbeenfoundasaveryrough 

manuscriptofabouttenpages,writtenonatypewriterbutwithseveral 

hand-writtencorrections,andcontainingsomenotesonepistemologyand 

geometry,butisnothinglikesuchapolishedversionmentionedbyvon 

Wright(Ketonen,1944–1950).Weknowforcertainhowever,fromsurvived 

correspondence,thatatleaststillinthelate1960’sKetonentriedtostay 

up-to-datewithrecentlogicalresearch.Healsogavelecturesinbasiclogic 

forstudentsattheuniversity. Whenhewasaskedinhislateryearswhy 

hehadabandonedlogic,Ketonenalwaysremarkedabruptly“logicgivesme 

suchheadache”.Onecouldperhapsspeculatethatlogicbecamesomething 

ofaspare-timeactivity,whilethemainattentionwasonuniversitypolitics, 

hisprofessorshipthatheheldforover25years,between1951–1977,andon 

aphilosophyincorporatingelementswhichfalloutsidethoseofthenatural 

sciences. 

MichaelvonBoguslawski 

Departmentofphilosophy,UniversityofHelsinki 

Siltavuorenpenger20A,P.O.Box9,00014Helsinki,Finland 

michael.vonboguslawski@helsinki.fi 

References 

Bernays, P. (1945). Review: Oiva Ketonen, Untersuchungen zum 

Prädikatenkalkül.TheJournalofSymbolicLogic,10(4),127–130. 

Beth,E.(1962).Formalmethods.Dordrecht:D.Reidel. 

Gentzen,G.(n.d.).UntersuchungenüberdaslogischeSchliessen.InM.E.Szabo 

(Ed.),ThecollectedpapersofGerhardGentzen. 

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vonPlato,J.(2004).Einleben,einWerk.Gedankenüberdaswissenschaftliche 

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