[FOM] The liar and the semantics of set theory (expansion)

Richard Heck heck at fas.harvard.edu
Mon Sep 30 23:16:35 EDT 2002

Roger Bishop Jones wrote:

>On Monday 23 September 2002  1:47 am, Rupert McCallum wrote:
>>We can re-interpret the diamond of modal propositional logic to mean "true in some V_kappa with kappa inaccessible" and the box to mean "true in all V_kappa with kappa inaccessible". As Boolos discusses in Chap. 13 of the cited work, Solovay proved in 1975, assuming infinitely many inaccessibles, that the following is a complete axiomatization of the modal logic of these notions:
>>(1) all tautologies
>>(2) box(a implies b) implies (box a implies box b)
>>(3) box(box a implies a) implies box a
>>(4) box(box a implies b) or box((box b and b) implies a)
>>Rules of inference: modus ponens and necessitation
>This is certainly of great interest and I will spend some time on it. On the briefest perusal however, its not obvious how this helps to settle the question at hand.
Even if "true in all V_kappa for kappa inaccessible" is definable in ZFC 
(I expect Solovay did resolve the question of definability, but I don't 
have Boolos's book with me), Solovay's result shows that there is 
something strange about treating this notion as a truth-predicate. For 
the conditions mentioned already imply that "~[]A" can never be a 
theorem. If it were, then of course "[]A --> A" would be a theorem; by 
necessitation, "[]([]A --> A)" would be a theorem; and so, by (3), "[]A" 
would be a theorem, and so the theory would be inconsistent. So nothing 
can ever be proven not to be true.

Obviously, the argument here is similar to that by which we infer the 
unprovability of consistency from Loeb's theorem, which is roughly the 
import of (3). That suggests that "true in all V_kappa for kappa 
inaccessible" is a notion more closely related to notions like 
provability than it is to semantic notions like truth: Indeed, the 
axioms governing the box, so interpreted, simply include those that 
govern the box, interpreted as just meaning provability: The modal logic 
GL, which characterizes provability, just omits (4). (Note that the 
condition "[][]A --> []A", familiar from Loeb's version of the 
derivability conditions, is itself derivable from the others.)

Richard Heck

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