Reduction and Elimination in Philosophy and the Sciences

The Knower Paradox and the Quantified Logic of Proofs — Walter Dean / Hidenori Kurokawa
original motivation for adopting this principle over the Nec
rule seems to be mainly to reduce the strength of the
assumptions required to develop the paradox. (Note in
particular that the epistemic rational which is commonly
given for adopting U seems to be a special case of that
which is given for Nec.) If we develop the Knower in the
context of QLP, not only is neither principle accorded
elementary status, but the foregoing observations
demonstrate that if we think of knowledge in terms of proof
existence, that there is an implicit interaction between the
knowledge modality and proof quantification implicit in the
original derivation. It is precisely this interaction which is
exposed by the role of UBF in QLP derivation.
This observation prompts a reconsideration of UBF
itself. The original motivation for its inclusion in QLP was to
preserve the Lifting Theorem (a version of which also
holds for LP). However, in light of the original setting of the
Knower, one might also inquire into its arithmetic
significance. In this regard, a parallel may be drawn
between UBF and implicit form of the Barcan formula —
i.e.
34) (∀x)�φ(x) → �(∀x)φ(x)
— in the context of Quantified Provability Logic. As (Boolos
1993) observes, if we take Φ(x) ≡ ¬ProofPA(x,⊥), then it
may readily be seen that 34) is not arithmetically valid. For
note that on this interpretation, the antecedent expresses
the fact that no natural number is provably a proof of ⊥,
while the consequent expresses the fact that it is provable
that there is no proof of ⊥. But of course the former statement
is true (in the standard model), whereas, per the
second incompleteness theorem, the latter is false (assuming
that PA is consistent).
Now define an arithmetic interpretation of QLP to be
a mapping (⋅) * which i) replaces every propositional letter
P with an arithmetic sentence (P) * and every proof term t
with a natural number or variable according to its type, ii) is
such that (x:φ)* = ProofPA(x,φ*), iii) commutes with
connectives, and iv) is such that ((∀x)φ)* = (∀x)[Pf(x) → φ*]
(where Pf(x) expresses that x is a code of a proof). On the
basis of such an interpretation, it may similarly be shown
that UBF is not arithmetically valid. In particular, the
interpretation of UBF for Φ(x) = ¬x:⊥ corresponds to the
claim that if for all natural numbers x, (b) * is a proof that if x
codes a proof, then ¬ProofPA(x,⊥), then ()* is a
proof that there is no proof of ⊥. On the assumption that
()* denotes a standard natural number (and that PA
is consistent), this conclusion also violates the second
incompleteness theorem. And from this it follows that there
can be no uniform means of arithmetically interpreting
proof terms of the form .
been to blame the paradox on either U or I. (Maitzen 1998) argues that the
paradox may be resolved by rejecting the assumption that knowledge is closed
under deductive consequence as embodied by I. However, (Cross, 2001)
shows that a version of the Knower may be developed by using a modified
knowledge predicate which is not assumed to be deductively closed. This
observation appears to lay the blame squarely on the principle U -- a point of
view which is adopted by both Cross and Érgé. We take our explicit reconstruction
of the Knower to deepen the motivation for adopting this position.
We take the foregoing observations to highlight the
applicability of explicit modal logic to the Knower, but also
point to a more precise diagnosis of the root of the
paradox. For not only does the use of constructive
necessitation in the derivation allow us to see logical
structure which is hidden by the use of principles like U or
Nec in the original derivation, but it also appears that there
are good reasons to be suspicious of at least one of the
principles which is suppressed in the original derivation —
i.e. UBF — at least if we wish to assign it an arithmetic
interpretation.
This desire may be reasonable if we look to
arithmetic for the source of self reference required to
develop the Knower. However, if our aim is merely to
reason about justified knowledge more generally, there
may also be good reasons to retain UBF. For not only
does it arises naturally out of reflection on the notion of
informal proof and provability, it also allows us to prove the
provable consistency of our reasoning about these
concepts. Both facts appear to have been foreseen by
(Gödel 1938, p. 101-103). Much more can be said about
these issues, but doing so is outside the scope of the
current paper.
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