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On the degrees of unsolvability of modal predicate logics of provability

Published online by Cambridge University Press:  12 March 2014

Vann McGee
Vann McGee*
Affiliation:
Department of Philosophy, Rutgers University, New Brunswick, New Jersey 08903, E-mail: VMCGEE@ZODIAC.RUTGERS.EDU
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Extract

The modal predicate logic of provability identifies the “□” of modal logic with the “Bew” of proof theory, so that, where “Bew” is a formula representing, in the usual way, provability in a consistent, recursively axiomatized theory Γ extending Peano arithmetic (PA), an interpretation of a language for the modal predicate calculus is a map * which associates with each modal formula an arithmetical formula with the same free variables which commutes with the Boolean connectives and the quantifiers and which sets (□ϕ)* equal to Bew(⌈ϕ*⌉). Where Δ is an extension of PA (all the theories we discuss will be extensions of PA), MPL(Δ) will be the set of modal formulas ϕ such that, for every interpretation *, ϕ* is a theorem of Δ. Most of what is currently known about the modal predicate logic of provability consists in demonstrations that MPL(Δ) must be computationally highly complex. Thus Vardanyan [11] shows that, provided that Δ is 1-consistent and recursively axiomatizable, MPL(Δ) will be complete , and Boolos and McGee [5] show that MPL({true arithmetical sentences}) is complete in {true arithmetical sentences}. All of these results take as their starting point Artemov's demonstration in [1] that {true arithmetical sentences} is 1-reducible to MPL({true arithmetical sentences}).

The aim here is to consolidate these results by providing a general theorem which yields all the other results as special cases. These results provide a striking contrast with the situation in modal sentential logic (MSL); according to fundamental results of Solovay [8], provided Γ does not entail any falsehoods, MSL({true arithmetical sentences}) and MSL(PA) (which is the same as MSL(Γ)) are both decidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 253 - 261
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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