Meet-Combination of Consequence Systems
DOI:
https://doi.org/10.12775/LLP.2024.017Keywords
combination of logics, meet-combination, consequence systems, product of matrix semanticsAbstract
We extend meet-combination of logics for capturing the consequences that are common to both logics. With this purpose in mind we define meet-combination of consequence systems. This notion has the advantage of accommodating different ways of presenting the semantics and the deductive calculi. We consider consequence systems generated by a matrix semantics and consequence systems generated by Hilbert calculi. The meet-combination of consequence systems generated by matrix semantics is the consequence system generated by their product. On the other hand, the meet-combination of consequence systems generated by Hilbert calculi is the consequence system generated by their interconnection. We investigate preservation of several properties. Capitalizing on these results we show that interconnection provides an axiomatization for the product. Illustrations are given for intuitionistic and modal logics, Łukasiewicz logic and some paraconsistent logics.
References
Avron, A., and I. Lev, 2005, “Non-deterministic multiple-valued structures”, Journal of Logic and Computation, 15(3): 241–261.
Avron, A., and Y. Zohar, 2019, “Rexpansions of nondeterministic matrices and their applications in nonclassical logics”, The Review of Symbolic Logic, 12(1): 173–200.
Blok, W. J., and D. Pigozzi, 1989, Algebraizable Logics, Memoirs of the American Mathematical Society. AMS.
Bolc, L., and P. Borowik, 1992, Many-Valued Logics, Springer.
Carnielli, W. A., and M. E. Coniglio, 2016, Paraconsistent Logic: Consistency, Contradiction and Negation, Springer.
Carnielli, W. A., J. Rasga, and C. Sernadas, 2002, “Modulated fibring and the collapsing problem” The Journal of Symbolic Logic, 67(4): 1541–1569.
Carnielli, W. A., M. E. Coniglio, and J. Marcos, 2007, “Logics of formal inconsistency”, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, volume 14, 2nd edition, Kluwer.
Carnielli, W. A., M. E. Coniglio, D. Gabbay, P. Gouveia, and C. Sernadas, 2008a, Analysis and Synthesis of Logics, Springer.
Carnielli, W., J. Rasga, and C. Sernadas, 2008b, “Preservation of interpolation features by fibring”, Journal of Logic and Computation, 18(1): 123–151.
Cocchiarella, N. B., and M. A. Freund, 2008, Modal Logic, Oxford University Press.
Czelakowski, J, 1982, “Logical matrices and the amalgamation property”, Studia Logica, 41(4): 329–341.
Czelakowski, J., 2001, Protoalgebraic Logics, Kluwer.
Fariñas del Cerro, L., and A. Herzig, 1996, “Combining classical and intuitionistic logic” , pages 93–102 in F. Baader and K. U. Schulz (eds.), Frontiers of Combining Systems, Springer.
Feitosa, H. A., and I. M. L. D’Ottaviano, 2001, “Conservative translations”, Annals of Pure and Applied Logic, 108(1-3): 205–227.
Feitosa, H., G. Cruz, and A. Golzio, 2015, “Um novo sistema de axiomas para a lógica paraconsistente J3”, CQD: Revista Eletrônica Paulista de Matemática, 4: 16–29.
Filipe, P., S. Marcelino, and C. Caleiro, 2022, “Computational properties of finite PNmatrices”, Journal of Logic and Computation, 32(8): 1694–1719.
Font, J. M., 2016, Abstract Algebraic Logic. An Introductory Textbook, College Publications, London.
Gabbay, D., 1996, “Fibred semantics and the weaving of logics: Modal and intuitionistic logics”, The Journal of Symbolic Logic, 61(4): 1057–1120.
Gabbay, D., 1999, Fibring Logics, Oxford University Press.
Gabbay, D., A. Kurucz, F. Wolter, and M. Zakharyaschev, 2003, Many-Dimensional Modal Logics: Theory and Applications, North Holland.
Gabbay, D., and V. Shehtman, 1998, “Products of modal logics. Part I”, Logic Journal of the IGPL, 6(1): 73–146.
Gottwald, S., 2001, A Treatise on Many-Valued Logics, volume 9 of Studies in Logic and Computation, Research Studies.
Kracht, M., 1999, Tools and Techniques in Modal Logic, North-Holland.
Kracht, M., and F. Wolter, 1991, “Properties of independently axiomatizable bimodal logics”, The Journal of Symbolic Logic, 56(4): 1469–1485.
Łukasiewicz, J., 1970, Selected Works, North-Holland.
Marcelino, S., 2022, “An unexpected Boolean connective”, Logica Universalis, 16(1-2): 85–103.
Marcelino, S., and C. Caleiro, 2017, “On the characterization of fibred logics, with applications to conservativity and finite-valuedness”, Journal of Logic and Computation, 27(7): 2063–2088.
Rescher, N., 1962, “Quasi-truth-functional systems of propositional logic”, The Journal of Symbolic Logic, 27: 1–10.
Rybakov, V., 1997, Admissibility of Logical Inference Rules, North-Holland.
Sernadas, A., C. Sernadas, and J. Rasga, 2012, “On meet-combination of logics”, Journal of Logic and Computation, 22(6): 1453–1470.
Tarski, A., 1956, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford University Press.
Thomason, S. K., 1980, “Independent propositional modal logics”, Studia Logica, 39(2–3): 143–144.
Voutsadakis, G., 2013, “Categorical abstract algebraic logic: meet-combination of logical systems”, Journal of Mathematics, pages Art. ID 126347, 8.
Wójcicki, R., 1973, “Matrix approach in methodology of sentential calculi”, Studia Logica, 32: 7–35.
Wójcicki, R., 1984, Lectures on Propositional Calculi, Ossolineum Publishing Co.
Wójcicki, R., 1988, Theory of Logical Calculi, Kluwer.
Zanardo, A., A. Sernadas, and C. Sernadas, 2001, “Fibring: Completeness preservation”, Journal of Symbolic Logic, 66(1): 414–439.
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Copyright (c) 2024 Paula Gouveia, João Rasga, Cristina Sernadas
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