Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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62<br />
The Knower Paradox <strong>and</strong> <strong>the</strong> Quantified Logic of Proofs — Walter Dean / Hidenori Kurokawa<br />
arithmetic, 10) is already suggestive of quantification over<br />
a doma<strong>in</strong> of <strong>in</strong>formal proofs — cf. (Tait 2001).<br />
In order to reconstruct <strong>the</strong> Knower <strong>in</strong> a system of<br />
explicit modal logic, we need a version of LP which<br />
conta<strong>in</strong>s quantifiers rang<strong>in</strong>g over proofs. Such a system is<br />
presented <strong>in</strong> (Fitt<strong>in</strong>g 2004) under <strong>the</strong> name QLP. The<br />
language of QLP is given by first specify<strong>in</strong>g a class of<br />
proof terms<br />
TermQLP = c | x | !t | t1 ⋅ t2 | t1 + t2 | <br />
The class of formulas of LP is <strong>the</strong>n specified as follows:<br />
FormLP = P | t: φ | ¬φ | φ → ψ | (∀x)φ | (∃x)φ<br />
The axioms of QLP are as follows:<br />
LP1 all tautologies of classical propositional logic<br />
LP2 t:( φ → ψ) → (s: φ → t·s:ψ)<br />
LP3 t: φ → φ<br />
LP4 t: φ →!t:t: φ<br />
LP5 t: φ → t+s: φ <strong>and</strong> s: φ → t+s: φ<br />
QLP1 (∀x) φ (x) → φ (t)<br />
QLP2 (∀x)(ψ → φ(x)) → (ψ → (∀x) φ (x))<br />
QLP3 φ(t) → (∃x)φ(x)<br />
QLP4 (∀x)(φ(x) → ψ) → ((∃x)φ(x) → ψ)<br />
UBF (∀x)t:φ(x) → :(∀x)φ(x), x ∉ FV(t)<br />
Axioms LP1-LP5 correspond to versions of <strong>the</strong> S4 axioms<br />
where<strong>in</strong> <strong>in</strong>stances of � have been “realized” by proof<br />
terms. Axioms QLP1-QLP4 correspond to a set of axioms<br />
adequate for classical predicate calculus <strong>and</strong> to which <strong>the</strong><br />
usual free variable restrictions apply. UBF is an explicit<br />
form of <strong>the</strong> Barcan formula <strong>and</strong> is justified on <strong>the</strong> basis of<br />
<strong>the</strong> observation that if we possess a proof term t which<br />
serves to uniformly verify φ(x) for all x, <strong>the</strong>n <strong>the</strong>re should<br />
be a proof (denoted by <strong>the</strong> complex proof term )<br />
which serves to justify (∀x) φ(x). The rules of QLP consist<br />
of modus ponens <strong>and</strong> universal generalization toge<strong>the</strong>r<br />
with a rule known as axiom necessitation. This rule says<br />
that if φ is an axiom of QLP, <strong>the</strong>n we may <strong>in</strong>troduce c:φ<br />
where c is a so-called proof constant — i.e. an unstructured<br />
proof term <strong>in</strong>troduced as an atomic justification for φ.<br />
Before reconstruct<strong>in</strong>g <strong>the</strong> derivation of <strong>the</strong> Knower<br />
<strong>in</strong> QLP, it will be useful to record <strong>the</strong> follow<strong>in</strong>g technical<br />
result:<br />
Theorem (Lift<strong>in</strong>g) [Artemov/Fitt<strong>in</strong>g]<br />
If QLP ⊢ φ, <strong>the</strong>n for some proof term t, QLP ⊢ t:φ.<br />
The Lift<strong>in</strong>g Theorem reports that if a statement φ is derivable<br />
<strong>in</strong> QLP, its derivation may be <strong>in</strong>ternalized with<strong>in</strong> <strong>the</strong> system<br />
so as to yield a proof term t which exhibits its structure. As<br />
such, <strong>the</strong> Lift<strong>in</strong>g Theorem serves as a sort of explicit counterpart<br />
to <strong>the</strong> S4 necessitation rule (i.e. ⊢ F / ⊢ �F) which<br />
itself is an implicit form of <strong>the</strong> rule Nec used to justify <strong>the</strong><br />
step 5)-6).<br />
The f<strong>in</strong>al step which we must undertaken before<br />
reconstruct<strong>in</strong>g <strong>the</strong> Knower is to <strong>in</strong>troduce some means of<br />
<strong>in</strong>troduc<strong>in</strong>g an explicit analog of a self referential statement<br />
which mirrors (*). The most straightforward way to proceed<br />
is to simply consider <strong>the</strong> result of adjo<strong>in</strong><strong>in</strong>g a statement of<br />
<strong>the</strong> form<br />
12) d:(¬(∃x)x:D ↔ D)<br />
which formalizes “d is a proof of ‘<strong>the</strong>re does not exist a<br />
proof of D iff D’.” Reason<strong>in</strong>g <strong>in</strong> QLP from 12) as a premise,<br />
we may now derive a contradiction as follows:<br />
13) ¬(∃x)x:D → D left to right direction of 11)<br />
14) (∃x)x:D → ¬D right to left direction of 11)<br />
15) (∃x)x:D → D derivable <strong>in</strong> QLP<br />
16) ¬(∃x)x:D propositional logic<br />
17) D<br />
18) t : D for some term t obta<strong>in</strong>able via Lift<strong>in</strong>g<br />
19) t : D → (∃x)x:D QLP3<br />
20) (∃x)x:D<br />
21) ⊥<br />
The step 17)-18) is analogous to <strong>the</strong> step 5)-6) <strong>in</strong> <strong>the</strong> orig<strong>in</strong>al<br />
derivation. In <strong>the</strong> case of QLP, however, this step is<br />
elliptical <strong>in</strong> <strong>the</strong> sense that although we know a term t exists<br />
via <strong>the</strong> Lift<strong>in</strong>g Theorem, such a term must be explicitly<br />
constructed by <strong>in</strong>ternaliz<strong>in</strong>g steps 13)-18). Construct<strong>in</strong>g t<br />
requires not only constants d1 <strong>and</strong> d2 such that<br />
21) d1:(¬(∃x)x:D → D)<br />
22) d2:((∃x)x:D → ¬D)<br />
(which may be constructed from d <strong>in</strong> 12)) but also a proof<br />
term which serves as a verification of 16). Note that while<br />
this statement is a explicit analog of an <strong>in</strong>stance of <strong>the</strong><br />
reflection axiom T, it is not an axiom of QLP. Not only must<br />
this statement be derived <strong>in</strong> QLP, but to construct t, its<br />
proof must also be lifted. This may be done as follows:<br />
24) x:D →D LP3<br />
25) r:(x:D → D) axiom necessitation<br />
26) (∀x)r:(x:D → D) universal generalization<br />
27) (∀x)r:(x:D → D) → :∀x:(x:D → D) UBF<br />
28) :(∀x):(x:D → D)<br />
29) (∀x)r:(x:D → D) → ((∃x):D → D) QLP4<br />
30) q:[(∀x)r : (x:D → D) → ((∃x):D → D)] axiom necessitation<br />
31) q ·:((∃x):D → D) LP2<br />
With this derivation <strong>in</strong> h<strong>and</strong>, it is <strong>the</strong>n easy to see that we<br />
may take<br />
31) t ≡ d1 · ((a · (q · )) ·d2)<br />
where a is a proof constant for <strong>the</strong> tautology (ϕ → ψ) →<br />
((ϕ → ¬ ψ) → ¬ ϕ).<br />
The <strong>in</strong>sight which we th<strong>in</strong>k QLP provides <strong>in</strong>to <strong>the</strong><br />
Knower can now be framed by consider<strong>in</strong>g <strong>the</strong> role which<br />
UBF plays <strong>in</strong> <strong>the</strong> forego<strong>in</strong>g derivation. For note that QLP<br />
<strong>in</strong>cludes nei<strong>the</strong>r a general necessitation rule analogous to<br />
Nec, nor even a local <strong>in</strong>stance of this pr<strong>in</strong>ciple ak<strong>in</strong> to U.<br />
As we have just seen, however, <strong>the</strong> explicit forms of both<br />
pr<strong>in</strong>ciples — i.e.<br />
32) (∃x):D → D <strong>and</strong><br />
33) q · :((∃x):D → D)<br />
— are derivable <strong>in</strong> QLP. Both of <strong>the</strong>se pr<strong>in</strong>ciples are required<br />
<strong>in</strong> order for <strong>the</strong> derivation 13)-21) to go through.<br />
However UBF turns out to essential to <strong>the</strong> derivation of 33)<br />
as it may be shown , without UBF, no statement of <strong>the</strong><br />
form t:((∃x):D → D) is derivable <strong>in</strong> QLP. 2<br />
This is significant for diagnos<strong>in</strong>g how <strong>the</strong> pr<strong>in</strong>ciples<br />
<strong>in</strong>volved <strong>in</strong> <strong>the</strong> orig<strong>in</strong>al derivation of <strong>the</strong> Knower conflict.<br />
Several recent commentators have proposed that <strong>the</strong><br />
paradox should be resolved by reject<strong>in</strong>g U. 3 However, <strong>the</strong><br />
2 This follows from <strong>the</strong> fact that UBF is not conservative over <strong>the</strong> QLP-UBF for<br />
statements not conta<strong>in</strong><strong>in</strong>g terms of <strong>the</strong> of . In particular, it may be<br />
shown that for no φ, do we have QLP−UBF ⊢ (∃y)y:((∃x): φ → φ).<br />
3 More specifically, among <strong>the</strong> three pr<strong>in</strong>ciples employed <strong>in</strong> <strong>the</strong> orig<strong>in</strong>al M&K<br />
derivation (i.e. T, U <strong>and</strong> I), <strong>the</strong> consensus among recent commentators has