On the way to a wider model theory: completeness theorems for first-order logics of formal inconsistency

sem informaçãoThis paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain nonobvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic.This paper investigates the question of characterizing first-orderLFIs (logics of formal inconsistency) by means of two-valued semantics.LFIs are powerfulparaconsistent logics that encode classical logic and permit a finer distinction be-tween contradictions and inconsistencies, with a deep involvement in philosophicaland foundational questions. Although focused on just one particular case, namely,the quantified logicQmbC, the method proposed here is completely general forthis kind of logics, and can be easily extended to a large family of quantifiedparaconsistent logics, supplying a sound and complete semantical interpretationfor such logics. However, certain subtleties involving term substitution and re-placement, that are hidden in classical structures, have to be taken into accountwhen one ventures into the realm of non-classical reasoning. This paper showshow such difficulties can be overcome, and offers detailed proofs showing that asmooth treatement of semantical characterization can be given to all such logics.Although the paper is well-endowed in technical details and results, it has a sig-nificant philosophical aside: it shows how slight extensions of classical methodscan be used to construct the basic model theory of logics that are weaker than tra-ditional logic due to the absence of certain rules present in classical logic. Severalsuch logics, however, as in the case of theLFIs treated here, are notorious fortheir wealth of models precisely because they do not make indiscriminate use ofcertain rules; these models thus require new methods. In the case of this paper, byjust appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus,some new constructions and crafty solutions to certain non-obvious subtleties areproposed. The result is that a richer extension of model theory can be inaugurated,with interest not only for paraconsistency, but hopefully to other enlargements oftraditional logic.73548578sem informaçãosem informaçãosem informaçãoAlves, E., Paraconsistent logic and model theory (1984) Studia Logica, 43 (1-2), pp. 17-32Avron, A., Lev, I., Canonical propositional Gentzen-type syst ems (2001) Proceedings of the 1st International Joint Conference on Automated Reasoning (IJCAR 2001). Lecture Notes in Artificial Intelligence (LNAI, 2083, pp. 529-544. , In Gore, R., Leitsch, A., & Nipkow, T., editors. Berlin: Springer VerlagAvron, A., Lev, I., Non-deterministic multi-valued structures (2005) Journal of Logic and Computation, 15, pp. 241-261Avron, A., Zamansky, A., Man y-valued non-deterministic semantics for firstorder logics of formal (in)consistency (2007) Algebraic and Proof-theoretic Aspects of Non-classical Logics. Papers in Honor of Daniele Mundici on the Occasion of His 60th Birthday. Lecture Notes in Computer Science, 4460, pp. 1-24. , In Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., & Marra, V., editors. Berlin: Springer VerlagCarnielli, W., Coniglio, M.E., Paraconsistent set theory by predic ating on consistency (2013) Journal of Logic and Computation, , First published online: July 9 2013Carnielli, W., Coniglio, M.E., Swap structures for LFI's (2014) CLE E-Prints, 14 (1). , http://www.cle.unicamp.br/e-prints/vol-14,n-1,2014.html, Available fromCarnielli, W., Coniglio, M.E., Marcos, J., Logics of formal inconsistency (2007) Handbook of Philosophical Logic, 14, pp. 1-93. , In Gabbay, D. & Guenthner, F., editors. 2nd. edition Dordrecht SpringerCarnielli, W.A., Marcos, J., A taxonomy of C-systems (2002) Paraconsistency: The Logical Way to the Inconsistent. Proceedings of the 2nd World Congress on Paraconsistency (WCP 2000). Lecture Notes in Pure and Applied Mathematics, 228, pp. 1-94. , In Carnielli, W. A., Coniglio, M. E., & Ottaviano, I. M. L. D., editors. New York, Marcel DekkerCarnielli, W.A., Marcos, J., De Amo, S., Formal inconsistency and evolutionary databases (2000) Logic and Logical Philosophy, 8, pp. 115-152Chang, C.C., Keisler, H.J., (1990) Model Theory (Third Ed.). Studies in Logic and the Foundations of Mathematics, 73. , Amsterdam ElsevierConiglio, M.E., Silvestrini, L.H., An alternative approach for quasi-truth (2014) Logic Journal of the IGPL, 22 (2), pp. 387-410Da Costa, N.C.A., (1963) Sistemas Formais Inconsistentes (Inconsistent Formal Systems, in Portuguese), , Habilitation thesis Universidade Federal do Paraná , Curitiba, Brazil. Republished by Editora UFPR, Curitiba, Brazil, 1993Da Costa, N.C.A., On the theory of inconsistent formal systems (Lecture delivered at the First Latin-American Colloquium on Mathematical Logic held at Santiago, Chile July 1970) (1974) Notre Dame Journal of Formal Logic, 15 (4), pp. 497-510Da Costa, N.C.A., Alves, E., A semantical analysis of the calculi Cn (1977) Notre Dame Journal of Formal Logic, 18 (4), pp. 621-630D'Ottaviano, I.M.L., (1982) Sobre Uma Teoria de Modelos Trivalente (On A Three-valued Model Theory, in Portuguese), , Ph. D. thesis, IMECC-State University of Campinas, BrazilD'Ottaviano, I.M.L., The completeness and compactness of a three-valued first o rder logic (1985) Revista Colombiana de Matemáticas, 19 (1-2), pp. 31-42D'Ottaviano, I.M.L., The model extension theorems for J3-theories (1985) Methods in Mathematical Logic. Proceedings of the 6th Latin American Symposium on Mathematical Logic Held in Caracas, p. 1983. , In Di Prisco C. A., editor. Venezuela, Aug. 1-6Lecture Notes in Mathematics, 1130, pp. 157-173. , Berlin: Springer-VerlagD'Ottaviano, I.M.L., Definability and quantifier elimination for J3-theories (1987) Studia Logica, 46 (1), pp. 37-54D'Ottaviano, I.M.L., Da Costa, N.C.A., Sur un problème de Jáskowski (1970) Comptes Rendus de l'Académie de Sciences de Paris (A-B, 270, pp. 1349-1353Hodges, W., Scanlon, T., First-order model theory (2013) The Stanford Encyclopedia of Philosophy, , http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=modeltheory-fo, In Zalta, E. N., editor. Fall 2013 ed. Av ailable fromMendelson, E., (1987) Introduction to Mathematical Logic, pp. 41-44. , Third ed New York, Chapman & HallMikenberg, I., Da Costa, N.C.A., Chuaqui, R., Pragmatic truth and approximation to truth (1986) The Journal of Symbolic Logic, 51 (1), pp. 201-221Podiacki, R., (2008) Lógicas da Inconsistência Formal Quantificadas (Quantified Logics of Formal Inconsistency, in Portuguese), , Masters thesis IFCH-State University of Campinas, BrazilPriest, G., (2006) Contradiction: A Study of the Transconsistent, pp. 76-78. , (2nd ed.). Oxford: Oxford University PressRodrigues, T.G., (2010) Sobre Os Fundamentos da Programação Lógica Paraconsistente (On the Foundations of Paraconsistent Logic Programming, in Portuguese)., , Mas ters thesis, IFCH-State University of Campinas, BrazilShoenfield, J.R., (1967) Mathematical Logic., pp. 9-17. , Reading, MA Addison-WesleyWójcicki, R., Lectures on propositional calculi (1984) Theorem 22.2, pp. 54-55. , http://www.ifispan.waw.pl/studialogica/wojcicki/papers.html, Wroclaw Poland: Ossolineum Available fro