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Axiomatizing non-deterministic many-valued generalized consequence relations

  • S.I.: Varieties of Entailment
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Abstract

We discuss the axiomatization of generalized consequence relations determined by non-deterministic matrices. We show that, under reasonable expressiveness requirements, simple axiomatizations can always be obtained, using inference rules which can have more than one conclusion. Further, when the non-deterministic matrices are finite we obtain finite axiomatizations with a suitable generalized subformula property.

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Notes

  1. Note that, in general, \(\vartriangleright ^\Upsilon _R\) may not be a generalized consequence relation.

References

  • Areces, C., & ten Cate, B. (2006). Hybrid logics. In P. Blackburn, F. Wolter, & J. van Benthem (Eds.), Handbook of modal logics. Amsterdam: Elsevier.

    Google Scholar 

  • Avron, A. (1991). Simple consequence relations. Information and Computation, 92(1), 105–139.

    Article  Google Scholar 

  • Avron, A. (1994). What is a logical system? In D. Gabbay (Ed.), What is a logical system? (pp. 217–259). Oxford: Oxford Science Publications.

    Google Scholar 

  • Avron, A., Ben-Naim, J., & Konikowska, B. (2007). Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics. Logica Universalis, 1(1), 41–70.

    Article  Google Scholar 

  • Avron, A., & Konikowska, B. (2005). Multi-valued calculi for logics based on non-determinism. Logic Journal of the IGPL, 13(4), 365–387.

    Article  Google Scholar 

  • Avron, A., & Lev, I. (2005). Non-deterministic multiple-valued structures. Journal of Logic and Computation, 15(3), 241–261.

    Article  Google Scholar 

  • Avron, A., & Zamansky, A. (2011). Non-deterministic semantics for logical systems. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 16, pp. 227–304). Berlin: Springer.

    Chapter  Google Scholar 

  • Baaz, M., Fermüller, C., & Salzer, G. (2001). Automated deduction for many-valued logics. In A. Robinson & A. Voronkov (Eds.), Handbook of automated reasoning (pp. 1355–1402). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Baaz, M., Lahav, O., & Zamansky, A. (2013). Finite-valued semantics for canonical labelled calculi. Journal of Automated Reasoning, 51(4), 401–430.

    Article  Google Scholar 

  • Belnap, N. (1977). A useful four-valued logic. In J. M. Dunn & G. Epstein (Eds.), Modern uses of multiple-valued logic. Dordrecht: D. Reidel.

    Google Scholar 

  • Blok, W., & Pigozzi, D. (1989). Algebraizable logics. Number 396 in Memoirs of the AMS. Providence: American Mathematical Society.

    Google Scholar 

  • Caleiro, C., Marcelino, S., & Marcos, J. (2017). Merging fragments of classical logic. In C. Dixon & M. Finger (Eds.), 11th international symposium on FroCoS 2017: Frontiers of combining systems, Volume 10483 of LNCS (pp. 298–315). Berlin: Springer.

  • Caleiro, C., Marcelino, S., & Rivieccio, U. (2018). Characterizing finite-valuedness. Fuzzy Sets and Systems, 345, 113–125.

    Article  Google Scholar 

  • Caleiro, C., Marcos, J., & Volpe, M. (2015). Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics. Theoretical Computer Science, 603, 84–110.

    Article  Google Scholar 

  • Czelakowski, J. (1983). Some theorems on structural entailment relations. Studia Logica, 42(4), 417–429.

    Article  Google Scholar 

  • Font, J. (2016). Abstract algebraic logic: An introductory textbook. London: College Publications.

    Google Scholar 

  • Hähnle, R. (1999). Tableaux for many-valued logics. In M. D’Agostino, D. Gabbay, R. Hähnle, & J. Posegga (Eds.), Handbook of tableau methods (pp. 529–580). Berlin: Springer.

    Chapter  Google Scholar 

  • Lahav, O., & Y. Zohar (2014). On the construction of analytic sequent calculi for sub-classical logics. In U. Kohlenbach, Pablo Barceló, R. de Queiroz (Eds.), Logic, language, information, and computation, WoLLIC 2014, volume 8652 of Lecture Notes in Computer Science (pp. 206–220). Berlin: Springer.

  • Pałasińska, K. (1994). Three-element nonfinitely axiomatizable matrices. Studia Logica, 53, 361–372.

    Article  Google Scholar 

  • Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics. Warszawa: Polish Scientific Publishers (PWN).

    Google Scholar 

  • Rautenberg, W. (1981). 2-Element matrices. Studia Logica, 40(4), 315–353.

    Article  Google Scholar 

  • Restall, G. (2005). Multiple conclusions. In P. Hájek, L. Valdés-Villanueva, & D. Westerståhl (Eds.), Logic, methodology and philosophy of science. London: College Publications.

    Google Scholar 

  • Rosser, J., & Turquette, A. (1952). Many-valued logics. Amsterdam: North Holland.

    Google Scholar 

  • Scott, D. (1974). Completeness and axiomatizability in many-valued logic. In L. Henkin, J. Addison, C. Chang, W. Craig, D. Scott, & R. Vaught (Eds.), Proceedings of the Tarski symposium, volume XXV of proceedings of symposia in pure mathematics (pp. 411–435). Providence: American Mathematical Society.

  • Shoesmith, D., & Smiley, T. (1978). Multiple-conclusion logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Steinberger, F. (2011). Why conclusions should remain single. Journal of Philosophical Logic, 40(3), 333–355.

    Article  Google Scholar 

  • Wójcicki, R. (1998). Theory of logical calculi, volume 199 of synthese library. South Holland: Kluwer.

  • Wroński, A. (1979). A three element matrix whose consequence operation is not finitely based. Bulletin of the Section of Logic, 2(8), 68–70.

    Google Scholar 

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Correspondence to Sérgio Marcelino.

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This research was done under the scope of R&D Unit 50008, funded by the applicable financial framework (FCT/MEC, UID/EEA/50008/2013, through national funds and when applicable co-funded by FEDER/PT2020), and is part of the MoLC project of SQIG at Instituto de Telecomunicacoes. The authors are grateful to the anonymous referees for their valuable comments that helped improve an earlier version of the paper.

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Marcelino, S., Caleiro, C. Axiomatizing non-deterministic many-valued generalized consequence relations. Synthese 198 (Suppl 22), 5373–5390 (2021). https://doi.org/10.1007/s11229-019-02142-8

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