MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

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42 Robert GoldblattDiodorean interpretation of ✷ to validate the formula ✸✷p → ✷✸p, which is itselfequivalent in the field of S4 to the formula ✷¬✷p ∨ ✷✸✷p that arises by theMcKinsey–Tarski translation of the intuitionistically invalid ¬p ∨ ¬¬p. Adding✸✷p → ✷✸p to S4 gives the logic S4.2. Both S4.2 and S4.3 were introduced in[Dummett and Lemmon, 1959], and shown to have the finite model property in[Bull, 1964].In [Goldblatt, 1980] a completeness proof is given to show that S4.2 is exactlythe Diodorean logic of n-dimensional Minkowski spacetime for all n ≥ 2, as wellas being the logic of the product structure (R, ≤) × (R, ≤). 43 But the problem ofaxiomatising the P F -calculi characterised by these spacetimes remains open.5.3 Segerberg’s EssayKrister Segerberg’s dissertation, An Essay in Classical Modal Logic [1971], provideda comprehensive semantic analysis of whole families of modal logics, as wellas developing important new concepts, some of which had been announced in hispapers of [1968a] and [1970]. These works established some notational and terminologicalconventions that have been lasting. For instance the term frame 44was used in place of model structure, and the Lemmon–Scott satisfaction notation|= M x α was used throughout in place of Kripke’s Φ(α, x) = ⊤, where M = (S, Φ).Later authors have tended to reduce the use of superscripts and write M |= x αinstead of |= M x α. M |= α then means that α is true in M, i.e. true at all pointsof M, and S |= α means that α is valid in the frame S.The weakest system discussed in the Essay is E, the smallest logic that is closedunder the rule from α ↔ β infer ✷α ↔ ✷β. An algebraic semantics for this logicwould employ algebras A = (B, f) having f as a unary function on B satisfying noparticular conditions. The corresponding “relational” models use neighbourhoodsemantics, the idea of which is attributed to Montague [1968] and Scott [1970].Segerberg presents this by the device of a neighbourhood frame S = (K, N), whereN, the neighbourhood system, is a function assigning to each x ∈ K a collectionN x of subsets of K, called neighbourhoods of x. 45 Writing M(α) for the “truthset” {y ∈ K : M |= y α} interpreting α in M, the satisfaction clause for ✷ in amodel M on such a frame S isM |= x ✷α iff M(α) ∈ N x .A topology on K has a naturally associated neighbourhood system in which X ∈N x iff x is interior to X, i.e. x ∈ U ⊆ X for some open set U. In this caseM(✷α) is the topological interior of M(α), and the result is an S4-model. Butdifferent logics can be characterised by validity in frames with weaker conditionsimposed on their neighbourhoods. A relational frame (K, R) is equivalent to theneighbourhood frame (K, N) having U ∈ N x iff {y : xRy} ⊆ U.43 The latter result was obtained independently by V. B. Shehtman [1983].44 This term was suggested to Segerberg by Scott.45 Some authors use a relation R ⊆ K × P(K) in place of N, where xRU iff U ∈ N x.
Mathematical Modal Logic: A View of its Evolution 43Any neighbourhood frame (K, N) has an associated algebra (P(K), f N ), wherethe operation f N , interpreting ✷ on the powerset algebra P(K), is given byf N (X) = {x ∈ K : X ∈ N x }.Inversely, any function f : P(K) → P(K) induces the neighbourhood system N fon K, whereX ∈ N f x iff x ∈ f(X).Thus, whereas Jónsson and Tarski’s analysis shows that relational semantics correspondsto completely additive and normal operators on powerset algebras (seesection 3.3), neighbourhood systems can be used to represent arbitrary operationson such algebras. The relationship between neighbourhood frames and modalalgebras has been systematically investigated by Kosta Došen [1989].Filtration (see section 5.1) was used extensively by Segerberg to prove completenesstheorems. This technique can be effective in dealing with logics whosecanonical model does not satisfy some desired property, and comes into its ownwhen seeking to axiomatise logics defined by some condition on finite frames. Forexample, Segerberg showed [1971, p. 68] that the normal logic K4W, 46 with axioms4 : ✷p → ✷✷pW :✷(✷p → p) → ✷p,is characterised by the class of finite frames (K, R) in which R is transitive andirreflexive, i.e. a strict ordering. (This logic later proved important in studies ofthe provability interpretation of modality. See section 7.5.) The basic method wasto obtain a falsifying model for a given non-theorem by filtration of the canonicalmodel, and then to “deform” this into a model of the desired kind without affectingthe truth value of the formula concerned. This involved an analysis of the waya transitive relations presents itself as an ordered set of connected components,called clusters. The method was applied in the Essay and the [1970] paper toaxiomatise a whole range of logics, including those characterised by the classes offinite partial orderings, finite linear orderings (both irreflexive and reflexive), andthe modal and tense logics of the structures (K, R) where K is any of ω, Z, Q,and R, while R is any of , ≤, and ≥.The logic characterised by the class of all finite partial orderings is particularlysignificant. Segerberg proved [1971, p. 101] that it is S4Grz, the normal logicaxiomatised by adding to S4 the axiomGrz : ✷(✷(p → ✷p) → p) → p.He named this for Andrzej Grzegorczyk whose paper [1967] added a further insightto the relationship between intuitionistic and modal logic. Grzegorczyk showed46 K4W could be called KW, since the axiom 4: ✷p → ✷✷p is deducible from W, as was shownindependently by several people, including de Jongh, Kripke and Sambin.
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Page 24 and 25: 24 Robert Goldblatt“true” or
Page 26 and 27: 26 Robert GoldblattPrior’s articl
Page 28 and 29: 28 Robert GoldblattThis proposal be
Page 30 and 31: 30 Robert Goldblattwhere α ′ is
Page 32 and 33: 32 Robert Goldblattif α is atomic
Page 34 and 35: 34 Robert GoldblattHintikka gives a
Page 36 and 37: 36 Robert Goldblattbetween worlds a
Page 38 and 39: 38 Robert Goldblattnormal (“queer
Page 40 and 41: 40 Robert Goldblattinterpreting for
Page 44 and 45: 44 Robert Goldblattthat the formula
Page 46 and 47: 46 Robert Goldblatt6.1 Incompletene
Page 48 and 49: 48 Robert Goldblatttions: every nor
Page 50 and 51: 50 Robert Goldblatttrue at some poi
Page 52 and 53: 52 Robert Goldblattof the monadic s
Page 54 and 55: 54 Robert Goldblattversion [van Ben
Page 56 and 57: 56 Robert Goldblatt6.5 Duality and
Page 58 and 59: 58 Robert Goldblattfrom a suitably
Page 60 and 61: 60 Robert GoldblattAnother way to d
Page 62 and 63: 62 Robert Goldblattwhether a variet
Page 64 and 65: 64 Robert Goldblattatomic commands
Page 66 and 67: 66 Robert Goldblattmodalities 〈 i
Page 68 and 69: 68 Robert Goldblatt[Hennessy and Li
Page 70 and 71: 70 Robert GoldblattThe logic CTL* w
Page 72 and 73: 72 Robert GoldblattThe meaning of
Page 74 and 75: 74 Robert Goldblattwhich shows that
Page 76 and 77: 76 Robert Goldblattmodal formulas s
Page 78 and 79: 78 Robert GoldblattGrothendieck gen
Page 80 and 81: 80 Robert GoldblattNow if Y and Z a
Page 82 and 83: 82 Robert Goldblatt7.7 Modal Logic
Page 84 and 85: 84 Robert GoldblattThis abstracts t
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Page 90 and 91: 90 Robert Goldblatt[Gerson, 1976] M
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92 Robert Goldblatt[Hoare, 1969] C.
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98 Robert Goldblatt[Tarski, 1956] A
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