Skip to main content
Log in

Proof-Theoretic Modal PA-Completeness I: A System-Sequent Metric

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we calculate a canonical characteristic fomula H of S (char(S)) so that ⊢G ∼ H → (∼S) and ⊢GL-LIN ∼ H, and the complexity σ of ∼ H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of ∼ H with H = char(S) and σ(∼ H) = d(S, G).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernardi, C., F. Montagna, ‘Equivalence relations induced by extensional formulae: classification by means of a new fixed point property’, Fundamenta Mathematicae 124 (1984), 221–233.

    Google Scholar 

  2. Boolos, G., The Unprovability of Consistency, Cambridge University Press, Cambridge, 1979.

    Google Scholar 

  3. Borga, M., ‘On some proof theoretical properties of the modal logic GL’, Studia Logica 42 (1983), 453–459.

    Google Scholar 

  4. De Jongh, D. H. J., F. Montagna, ‘Generic Generalized Rosser Fixed Points’, Studia Logica 46 (1987), 193–203.

    Google Scholar 

  5. Gentilini, P., ‘Provability Logic in the Gentzen formulation of arithmetic’, Zeitsch. F. Math. Logik und Grundlagen der Math. 38 (1992), 535–550.

    Google Scholar 

  6. Gentilini, P., ‘Syntactical results on the arithmetical completeness of modal logic’, Studia Logica 52,4 (1993), 549–564.

    Google Scholar 

  7. Gentilini, P., PA-Completeness of Modal Logic: Constructive Reduction of PA-Proofs to GL-LIN-Proofs, pubblicazioni dell' Istituto per la Matematica Applicata del CNR, n. 2, January 1997.

  8. Gentilini, P., Theoremi di completezza aritmetica della logica modale: una trattazione sintattica, Genova 1990: Doctoral Thesis for Doctorate in Mathematical Research, discussed and approved by the national board of italian Ministry of Research.

  9. Gentilini, P., ‘Proof-theoretic modal PA-completeness II: the syntactic countermodel’, in the next issue of this journal.

  10. Gentilini, P.: ‘Proof-theoretic modal PA-completeness III: the syntactic proof’, forthcoming in Studia Logica.

  11. Gentilini, P., ‘Informational proof-theory’, contribution to International Conference on Logic Methodology and Philosophy of Science (LMPS 1995), Florence 1995.

  12. Gentzen, G., Collected Papers, Szabo (ed.), North Holland, 1969.

  13. Girard, J. Y., Proof Theory and Logical Complexity, Bibliopolis, Napoli, 1987.

    Google Scholar 

  14. Sambin, G., S. Valentini, ‘A modal sequent calculus for a fragment of arithmetic’, Studia Logica 39 (1980), 245–256.

    Google Scholar 

  15. Sambin, G., S. Valentini, ‘The modal logic of the provability. The sequential approach’, Journal of Philosophical Logic 11 (1982), 311–342.

    Google Scholar 

  16. SmoryŃski, C., Selfreference and Modal Logic, Springer Verlag, New York, 1985.

    Google Scholar 

  17. Solovay, R., ‘Provability interpretations of modal logic’, Israel Jour. of Math. 25 (1976), 287–304.

    Google Scholar 

  18. Takeuti, G., Proof Theory, North-Holland, 1987 (second editio).

  19. Valentini, S., U Solitro, ‘The modal logic of consistency assertions of Peano Arithmetic’. Zeitsch. F. Math. Logik und Grundlagen der Math. 29 (1983), 25–32.

    Google Scholar 

  20. Valentini, S., ‘A syntactic proof of cut-elimination for GL-LIN’, Zeitsch. F. Math. Logik und Grundlagen der Math. 32 (1986), 137–144.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gentilini, P. Proof-Theoretic Modal PA-Completeness I: A System-Sequent Metric. Studia Logica 63, 27–48 (1999). https://doi.org/10.1023/A:1005203020301

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005203020301

Navigation