Curry’s Paradox and ω -Inconsistency

Abstract

In recent years there has been a revitalised interest in nonclassical solutions to the semantic paradoxes.1 In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but not jointly, lack the problematic feature.

Relevant First-Order Logic LP# and Curry’s Paradox Resolution

“…Although logic Ƚ does not contain K, it does contain . In general the n+1-valued version of Ƚukasiewicz logic, Ƚ , validates and is thus unsuitable for the same reason [22], [23].…”

“…We have dubbed any Logic LP # with the -rule (1.11) by LP # . Note that without the rule of reduction one cannot derive strong -inconsistency from weak -inconsistency [23]. Definition 6.3.…”

“…Definition 6.3. [23]. By a classical "naive truth theory" (CNTT) we shall mean any set of first order sentences in the language of arithmetic with a truth predicate which, in addition to being closed under modus ponens, has the following properties:…”

Paths to Triviality

“…9 T formulated in ∀ L ℵ is, although a non-trivial theory, riddled with ω-troubles. See [1], [27], [29] and [38].…”

“…1 John Slaney showed around the same time that there is a definite limit on how strong such a paraconsistent logic can be ( [51]); although the contraction axiom (A → (A → B)) → (A → B) is not derivable in 1 There were earlier attempts at showing that the naïve theories can be non-trivial, notably [8], [24], [31], [47], [48], [49] and [50]. However, these results either restrict abstraction or lack a decent conditional, one satisfying at least identity and modus ponens-A → A and A, A → B B-and so at best show that A & T A and A(a) & a ∈ {x|A} are intersubstitutable without delivering the biconditionals A ↔ T A and a ∈ {x|A} ↔ A( x /a).…”