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The degree of the set of sentences of predicate provability logic that are true under every interpretation1

Published online by Cambridge University Press:  12 March 2014

George Boolos  and
Vann McGee
George Boolos
Affiliation:
Department of Linguistics and Philosophy, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Vann McGee
Affiliation:
Department of Philosophy, University of Arizona, Tucson, Arizona 85721
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Extract

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows.

Pr(x), alias Bew(x), is the standard provability predicate of PA. For any formula F of PA, Pr[F] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F, i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[F], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[F] = Pr(‘F’), the sentence that expresses the PA-provability of F.

Let υ 1, υ 2,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P* of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 165 - 171
Copyright
Copyright © Association for Symbolic Logic 1987

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Footnotes

1

Similar proofs of the main result of this paper, Theorem 1, were independently discovered by the authors (McGee first). It appears that V. A. Vardanyan has also found a (different) proof of Theorem 1. We are grateful to S. N. Artemov, V. A. Vardanyan, and Scott Weinstein for comments and suggestions.

References

REFERENCES

[A] Artemov, S. N., Nonarithmeticity of truth predicate logics of provability, Doklady Akademii Nauk SSSR, vol. 284 (1985), pp. 270–271 (in Russian); English translation, Soviet Mathematics—Doklady , vol. 32 (1986), pp. 403–405.Google Scholar
[B1] Boolos, George, On deciding the truth of certain statements involving the notion of consistency, this Journal, vol. 41 (1976), pp. 779–781.Google Scholar
[B2] Boolos, George, The unprovability of consistency, Cambridge University Press, Cambridge, 1979.Google Scholar
[D] Davis, Martin, Computability and unsolvability, McGraw-Hill, New York, 1958.Google Scholar
[F] Friedman, Harvey, One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113–129.Google Scholar
[S] Solovay, Robert M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25(1976), pp. 287–304.CrossRefGoogle Scholar
[V] Vardanyan, V. A., O predikatnoĭ logike dokazuemosti, preprint, Nauchnyĭ Sovet po Kompleksnoĭ Probleme “Kibernetika”, Akademiya Nauk SSSR, Moscow, 1985. (in Russian)Google Scholar