Reduction and Elimination in Philosophy and the Sciences

The Knower Paradox and the Quantified Logic of Proofs
Walter Dean / Hidenori Kurokawa, New York, USA
The Knower paradox was originally introduced by
(Montague and Kaplan 1960) [M&K]. We will begin by
recording a simple version of the paradox adapted from
(Egré 2005). Suppose that T extends Q and let K(x) be a
(possibly complex) predicate in L T. It follows that T proves
a fixed point theorem of the following form:
(FP) For every open formula φ(x) in LT, there exists a sentence
δ such that
(*) T ⊢ φ(δ) ↔ δ.
Now suppose K(x) additionally satisfies
(T) T ⊢ K(φ) → φ
(Nec) if T ⊢ φ, then T ⊢ K(φ)
Then it may be shown that T is inconsistent by letting δ be
such that
1) T ⊢ ¬K(δ) → δ
2) T ⊢ K(δ) → ¬δ
via (FP) and then arguing as follows
3) K(δ) → δ T
4) ¬K(δ) 2), 3)
5) δ 1), 4)
6) K(δ) 5), Nec
7) ⊥
The foregoing presentation of the Knower departs from
that of M&K in two respects. The first of these is that rather
than using a sentence δ satisfying 1), 2), they use one
satisfying K(¬δ) ↔ δ. The second is that we have employed
the rule Nec, as opposed to assuming that K(x)
satisfies the axioms
(U) K(K φ → φ)
(I) K(φ) & I(φ,ψ) → K(ψ)
wherein I(φ,ψ) expresses that ψ is derivable from φ. It may
reasonably be claimed that the original derivation of M&K
rests on a set of principles which more precisely isolates
the source of the paradox than those we have employed.
We have elected to base our treatment on 1)-7) because
the resolution we suggest below will also be applicable to
the choice of fixed point and weaker principles employed
by M&K.
It is also notable that the Knower was originally
formulated in an arithmetic language as opposed to one
with a propositional operator. This reflects the fact that
M&K assume that such a setting is required in order to
ensure the existence of self-referential statements and
argues that they took the paradox to weaken Quine’s
argument that modal operators must be conceived as
predicates of sentences. As Érgé convincingly argues,
however, the availability of self-reference in a language
with modal operators is essentially independent of whether
we think of these operators as taking sentences or
propositions as arguments.
This observation suggests that by viewing the
foregoing derivation in a modal setting, it may be possible
to isolate the principles which lead to paradox in a manner
that does not depend on the mechanism by which selfreference
is achieved. It is an easy observation that this
derivation remains valid when we reinterpret K(x) as a
propositional operator and treat the arithmetic sentence δ
as a denoting a fixed proposition D of which
8) ¬�D ↔ D
is provable. When recast in this light, the derivation can be
taken to show that there is a general conflict between the
modal reflection axiom T (which is the analogue of T) and
any modal principles which would imply 8).
One means by which this can be demonstrated is to
note that the logic S4 (which includes T) is incompatible
with self-reference in the sense that not only is it incapable
of proving any instance of 8) but also
9) S4 + � (¬�D ↔ D) is inconsistent 1
This result might be taken to bear on the Knower not only
because its proof essentially recapitulates 1)-7), but also
because there is a well-known interpretation of S4
whereby � is assigned the reading
10) �F iff F is informally provable
Such an interpretation was first proposed by (Gödel 1933)
in an attempt to provide a provability semantics for intuitionistic
logic. The details of what follows do not, however,
rely specifically on the relationship between S4 and
intuitionism. Rather, they depend on the availability of socalled
explicit refinements of S4 which can be employed to
reason about knowledge qua provability.
For present purposes, an explicit modal logic can be
taken to be one that possesses an infinite family of
modalities of the form t:F. As opposed to 10), wherein �
expresses a notion of provability in which proofs are kept
implicit, this notation is conventionally assigned the
interpretation
11) t:φ iff t verifies φ
Here t may be a structured term which, in the paradigmatic
case, is taken to denote an explicit mathematical proof. A
system employing this notation was envisioned by (Gödel
1938). However, a complete formalization of a logic of
explicit proof was first provided by (Artemov 2001) under
the name LP (the Logic of Proofs).
LP itself is not sufficient to express the versions of
FP and T which are required to formulate the Knower. For
note that on their intended interpretations, both the
arithmetic knowledge predicate K(x) and the informal
provability predicate � contain implicit quantifiers over
proofs or other evidentiary entities. This is clearest in the
case of K(x), which is standardly taken to extend an
arithmetic provability predicate Bew(y) which itself
abbreviates a statement of the form ∃xProof(x,y).
Although as Gödel already observed, the � of S4 cannot
be interpreted as expressing provability within formal
1 This tension also surfaces with respect to the provability logic GL in which
statements like 8) are provable. However, it may easily be shown that GL + T
is inconsistent.
61