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

[Rupert McCallum]

In "The Logic of Provability", George Boolos shows how to investigate
issues like those under discussion in this thread via techniques of
modal logic. 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


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