Reduction and Elimination in Philosophy and the Sciences

62
The Knower Paradox and the Quantified Logic of Proofs — Walter Dean / Hidenori Kurokawa
arithmetic, 10) is already suggestive of quantification over
a domain of informal proofs — cf. (Tait 2001).
In order to reconstruct the Knower in a system of
explicit modal logic, we need a version of LP which
contains quantifiers ranging over proofs. Such a system is
presented in (Fitting 2004) under the name QLP. The
language of QLP is given by first specifying a class of
proof terms
TermQLP = c | x | !t | t1 ⋅ t2 | t1 + t2 |
The class of formulas of LP is then specified as follows:
FormLP = P | t: φ | ¬φ | φ → ψ | (∀x)φ | (∃x)φ
The axioms of QLP are as follows:
LP1 all tautologies of classical propositional logic
LP2 t:( φ → ψ) → (s: φ → t·s:ψ)
LP3 t: φ → φ
LP4 t: φ →!t:t: φ
LP5 t: φ → t+s: φ and s: φ → t+s: φ
QLP1 (∀x) φ (x) → φ (t)
QLP2 (∀x)(ψ → φ(x)) → (ψ → (∀x) φ (x))
QLP3 φ(t) → (∃x)φ(x)
QLP4 (∀x)(φ(x) → ψ) → ((∃x)φ(x) → ψ)
UBF (∀x)t:φ(x) → :(∀x)φ(x), x ∉ FV(t)
Axioms LP1-LP5 correspond to versions of the S4 axioms
wherein instances of � have been “realized” by proof
terms. Axioms QLP1-QLP4 correspond to a set of axioms
adequate for classical predicate calculus and to which the
usual free variable restrictions apply. UBF is an explicit
form of the Barcan formula and is justified on the basis of
the observation that if we possess a proof term t which
serves to uniformly verify φ(x) for all x, then there should
be a proof (denoted by the complex proof term )
which serves to justify (∀x) φ(x). The rules of QLP consist
of modus ponens and universal generalization together
with a rule known as axiom necessitation. This rule says
that if φ is an axiom of QLP, then we may introduce c:φ
where c is a so-called proof constant — i.e. an unstructured
proof term introduced as an atomic justification for φ.
Before reconstructing the derivation of the Knower
in QLP, it will be useful to record the following technical
result:
Theorem (Lifting) [Artemov/Fitting]
If QLP ⊢ φ, then for some proof term t, QLP ⊢ t:φ.
The Lifting Theorem reports that if a statement φ is derivable
in QLP, its derivation may be internalized within the system
so as to yield a proof term t which exhibits its structure. As
such, the Lifting Theorem serves as a sort of explicit counterpart
to the S4 necessitation rule (i.e. ⊢ F / ⊢ �F) which
itself is an implicit form of the rule Nec used to justify the
step 5)-6).
The final step which we must undertaken before
reconstructing the Knower is to introduce some means of
introducing an explicit analog of a self referential statement
which mirrors (*). The most straightforward way to proceed
is to simply consider the result of adjoining a statement of
the form
12) d:(¬(∃x)x:D ↔ D)
which formalizes “d is a proof of ‘there does not exist a
proof of D iff D’.” Reasoning in QLP from 12) as a premise,
we may now derive a contradiction as follows:
13) ¬(∃x)x:D → D left to right direction of 11)
14) (∃x)x:D → ¬D right to left direction of 11)
15) (∃x)x:D → D derivable in QLP
16) ¬(∃x)x:D propositional logic
17) D
18) t : D for some term t obtainable via Lifting
19) t : D → (∃x)x:D QLP3
20) (∃x)x:D
21) ⊥
The step 17)-18) is analogous to the step 5)-6) in the original
derivation. In the case of QLP, however, this step is
elliptical in the sense that although we know a term t exists
via the Lifting Theorem, such a term must be explicitly
constructed by internalizing steps 13)-18). Constructing t
requires not only constants d1 and d2 such that
21) d1:(¬(∃x)x:D → D)
22) d2:((∃x)x:D → ¬D)
(which may be constructed from d in 12)) but also a proof
term which serves as a verification of 16). Note that while
this statement is a explicit analog of an instance of the
reflection axiom T, it is not an axiom of QLP. Not only must
this statement be derived in QLP, but to construct t, its
proof must also be lifted. This may be done as follows:
24) x:D →D LP3
25) r:(x:D → D) axiom necessitation
26) (∀x)r:(x:D → D) universal generalization
27) (∀x)r:(x:D → D) → :∀x:(x:D → D) UBF
28) :(∀x):(x:D → D)
29) (∀x)r:(x:D → D) → ((∃x):D → D) QLP4
30) q:[(∀x)r : (x:D → D) → ((∃x):D → D)] axiom necessitation
31) q ·:((∃x):D → D) LP2
With this derivation in hand, it is then easy to see that we
may take
31) t ≡ d1 · ((a · (q · )) ·d2)
where a is a proof constant for the tautology (ϕ → ψ) →
((ϕ → ¬ ψ) → ¬ ϕ).
The insight which we think QLP provides into the
Knower can now be framed by considering the role which
UBF plays in the foregoing derivation. For note that QLP
includes neither a general necessitation rule analogous to
Nec, nor even a local instance of this principle akin to U.
As we have just seen, however, the explicit forms of both
principles — i.e.
32) (∃x):D → D and
33) q · :((∃x):D → D)
— are derivable in QLP. Both of these principles are required
in order for the derivation 13)-21) to go through.
However UBF turns out to essential to the derivation of 33)
as it may be shown , without UBF, no statement of the
form t:((∃x):D → D) is derivable in QLP. 2
This is significant for diagnosing how the principles
involved in the original derivation of the Knower conflict.
Several recent commentators have proposed that the
paradox should be resolved by rejecting U. 3 However, the
2 This follows from the fact that UBF is not conservative over the QLP-UBF for
statements not containing terms of the of . In particular, it may be
shown that for no φ, do we have QLP−UBF ⊢ (∃y)y:((∃x): φ → φ).
3 More specifically, among the three principles employed in the original M&K
derivation (i.e. T, U and I), the consensus among recent commentators has