# Semantic entailment in non classical logics based on proofs found in classical logic

## Abstract

A particular way of relating logics, specially useful in the framework of automated theorem proving is proposed. From the definition of the semantics of a logic (called *source logic* and abbreviated henceforth SL) in another logic (called *target logic* and abbreviated henceforth TL or TLS), we translate formulas of SL into TL using known techniques. Then we show how to partially translate proofs found in TL into SL. More precisely, the main theoretical result of the paper is a theorem establishing that for a class of non-classical logics — taking first-order sorted logic with equality as target logic — given a formula f in SL, it is possible from a proof P of f (obtained in TL) to backward translate into SL some (sometimes all) formulas in P. This set of backward translated formulas are proved to be semantically related each other and to define a partial consequence relation in SL. We get therefore an entailment sequence for f *in SL*. Our approach is applicable to source logics either without “computationally interesting” proof systems or without proof systems at all. One running example is fully treated. We compare the results of our method with the ones of a specialized tableaux-based theorem prover for the logic S4(p). Some hints of future work are given.

## Keywords

Classical Logic Theorem Prove Proof System Predicate Symbol Automate Theorem Prove## Preview

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## References

- [1]A. Avron. Simple Consequence Relations.
*Information and Computation*, 92:105–139, 1991.Google Scholar - [2]T. Boy de la Tour, R. Caferra, and G. Chaminade. Some tools for an Inference Laboratory (ATINF). In
*CADE-9*, pages 744–745. Springer-Verlag, LNCS 310, 1988.Google Scholar - [3]R. Caferra and S. Demri. Semantic entailment in non classical logics based on proofs found in classical logic, 1992. Extended version to appear.Google Scholar
- [4]R. Caferra, S. Demri, and M. Herment. Logic morphisms as a framework for the backward transfer of lemmas and strategies in some modal and epistemic logics. In
*AAAI-9*, pages 421–426. AAAI, MIT Press, July 1991.Google Scholar - [5]R. Caferra, M. Herment, and N. Zabel. User-oriented theorem proving with the ATINF graphic proof editor. In
*FAIR' 91*, pages 2–10. Springer-Verlag, LNAI 535, 1991.Google Scholar - [6]H. D. Ebbinghaus. Extended logics: the general framework. In J. Barwise and Feferman S., editors,
*Model theoretic logics*, pages 25–76. Springer-Verlag, 1985.Google Scholar - [7]M. C. Fitting.
*Proof methods for modal and intuitionistic logics*. D. Reidel Publishing Co., 1983.Google Scholar - [8]A. Herzig.
*Raisonnement automatique en logique modale et algorithmes d'unification*. PhD thesis, Université Paul Sabatier, Toulouse, July 1989.Google Scholar - [9]K. Konolige.
*A deduction model of belief*. Pitman, 1986.Google Scholar - [10]C.R. Mann. Equivalence of deduction in proof theory and free cartesian closed categories.
*Journal of Symbolic Logic*, 39:380–381, 1974.Google Scholar - [11]J. Meseguer. General logic. In H-D Ebbinghaus, editor,
*Logic Colloquium '87*, pages 275–330. North-Holland, 1987.Google Scholar - [12]C. Morgan. Methods for automated theorem proving in non classical logics.
*IEEE Transactions on Computers*, 25(8):852–862, August 1976.Google Scholar - [13]H.J. Ohlbach.
*A resolution calculus for modal logics*. PhD thesis, FB Informatik Univ. of Kaiserslautern, 1988.Google Scholar - [14]H.J. Ohlbach. Context Logic. Technical report, FB Informatik Univ. of Kaiserlautern, 1989.Google Scholar
- [15]E. Orlowska. Resolution systems and their applications I.
*Fundamenta Informaticae*, 3:253–268, 1979.Google Scholar - [16]E. Orlowska. Resolution systems and their applications II.
*Fundamenta Informaticae*, 3:333–362, 1980.Google Scholar - [17]D. Scott. Completeness and axiomatizability in many-valued logic. In L. Henkin et al., editor,
*Tarski Symposium*, pages 411–35, 1974.Google Scholar