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UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS

  • WOJCIECH DZIK (a1) and PIOTR WOJTYLAK (a2)
Abstract

We introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.

Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

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Corresponding author
*INSTITUTE OF MATHEMATICS, UNIVERSITY OF SILESIA BANKOWA 14, KATOWICE 40-132, POLAND E-mail: wojciech.dzik@us.edu.pl
INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF OPOLE OLESKA 48, OPOLE 45-052, POLAND E-mail: pwojtylak@uni.opole.pl
References
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Aczel, P. H. G. (1968). Saturated intuitionistic theories. In Arnold Schmidt, H., Schütte, K., and Thiele, H.-J., editors. Contributions to Mathematical Logic: Proceeding of the Logic Colloquium, Hannover 1966. Amsterdam: North-Holland, pp. 111.
Baader, F. & Ghilardi, S. (2011). Unification in modal and description logics. Logic Journal of the IGPL, 19 (5), 705730.
Baader, F. & Snyder, J. (2001). Unification theory. In Robinson, A. and Voronkov, A., editors. Handbook of Automated Reasoning, Chapter 6. Amsterdam: Elsevier Science B.V., pp. 439526.
Baaz, M., Preining, N., & Zach, R. (2007). First-order Gödel logics. Annals of Pure and Applied Logic, 147(1), 2347.
Beth, E. W. (1970). Aspects of Modern Logic. Dordrecht-Holland: D. Reidel Publishing Company.
Cabrer, L. & Metcalfe, G. (2015). Exact unification and admissibility. Logical Methods in Computer Science, 11(3), 115.
Casari, E. & Minari, P. (1983). Negation-free intermediate predicate logics. Bolletino dell’Unione Matematica Italiana, 6(2-B), 499536.
Church, A. (1956). Introduction to Mathematical Logic I . Princeton, New Jersey: Princeton University Press.
Corsi, G. (1992). Completeness theorem for Dummett’s LC quantified and some of its extensions. Studia Logica, 51, 317335.
Corsi, G. & Ghilardi, S. (1992). Semantical aspects of quantified modal logic. In Bicchieri, C. and Dalla Chiara, M. L., editors. Knowledge, Belief and Strategic Action. Cambridge: Cambridge University Press, pp. 167195.
Dzik, W. (1975). On structural completeness of some nonclassical predicate calculi. Reports on Mathematical Logic, 5, 1926.
Dzik, W. (2004). Chains of structurally complete predicate logics with the application of Prucnal’s substitution. Reports on Mathematical Logic, 38, 3748.
Dzik, W. (2006). Splittings of lattices of theories and unification types. Contributions to General Algebra, 17, 7181.
Dzik, W. (2007). Unification Types in Logic . Katowice: Silesian University Press.
Dzik, W. & Wojtylak, P. (2012). Projective unification in modal logic. Logic Journal of the IGPL, 20(1), 121153.
Dzik, W. & Wojtylak, P. (2015). Infinitary modal logics extending S4.3. Logic Journal of the IGPL, 23(4), 640661.
Dzik, W. & Wojtylak, P. (2016) Modal consequence relations extending S4.3. An application of projective unification. Notre Dame Journal of Formal Logic, 57, 523549.
Gabbay, D., Shehtman, V., & Skvortsov, D. (2009). Quantification in Nonclassical Logic, Vol. 1. Studies in Logic and the Foundations of Mathematics, Vol. 153. Amsterdam: Elsevier Science B.V.
Ghilardi, S. (1997). Unification through projectivity. Journal of Symbolic Computation, 7, 733752.
Ghilardi, S. (1999). Unification in intuitionistic logic. Journal of Symbolic Logic, 64(2), 859880.
Ghilardi, S. (2000). Best solving modal equations. Annals of Pure and Applied Logic, 102, 183198.
Ghilardi, S. & Sacchetti, L. (2004). Filtering unification and most general unifiers in modal logic. The Journal of Symbolic Logic, 69, 879906.
Goldblatt, R. (2011). Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics. Lecture Notes in Logic No. 38. Cambridge: Cambridge University Press.
Goudsmit, J. & Iemhoff, R. (2014). On unification and admissible rules in Gabbay-de Jongh logics. Annals of Pure and Applied Logic, 165(2), 652672.
Harrop, R. (1960). Concerning formulas of the types in intuitionistic formal systems. Journal of Symbolic Logic, 25(1), 2732.
Iemhoff, R. (2006). On the rules of intermediate logics. Archive for Mathematical Logic, 45, 581599.
Iemhoff, R. (2015). On rules. Journal of Philosophical Logic, 80(3), 713729.
Iemhoff, R. & Roziere, P. (2015). Unification in intermediate logics. Journal of Symbolic Logic, 80(3), 713729.
Kleene, S. C. (1962). Disjunction and existence under implication in elementary intuitionistic formalism. Journal of Symbolic Logic, 27(1), 1118.
Kreisel, G. (1958). The non-derivability of , A primitive recursive, in intuitionistic formal systems. Journal of Symbolic Logic, 23, 456457.
Minari, P. (1986). Disjunction and existence properties in intermediate predicate logics. In Abrusci, V. M., editor. Proceedings of the Meeting “Logica e Filosofia della Scienza, oggi”, Vol. I. Bologna: CLUEB, pp. 199203.
Minari, P. & Wroński, A. (1988). The property (HD) in intuitionistic logic. A partial solution of a problem of H. Ono. Reports on Mathematical Logic, 22, 2125.
Ono, H. (1972/73). A study of intermadiate predicate logics. Publications RIMS, Kyoto University, 8, 619643.
Ono, H. (1987). Some problems in intermediate predicate logics. Reports on Mathematical logic, 21, 5567.
Pogorzelski, W. A. (1971). Structural completeness of the propositional calculus. Bulletin de l’Académie Polonaise des Sciences, série des Sciences Mathématiques, Astronomiques et Physiques, 19, 349351.
Pogorzelski, W. A. & Prucnal, T. (1975). Structural completeness of the first-order predicate calculus. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 21, 315320.
Prucnal, T. (1979). On two problems of Harvey Friedman. Studia Logica, 38, 257262.
Rasiowa, H. (1954). Algebraic models for axiomatics theories. Fundamenta Mathematicae, 4, 291310.
Rybakov, V. V., Terziler, M., & Gencer, C. (1999). An essay on unification and inference rules for modal logic. Bulletin of the Section of Logic, 28(3), 145157.
Skvortsov, D. P. (1988). On axiomatizability of some intermediate logics. Reports on Mathematical Logic, 22, 115116.
Skvortsov, D. P. (2006). On non-axiomatizability of superintuitionistic predicate logics of some classes of well-founded and dually well-founded Kripke frames. Journal of Logic and Computation, 16, 685695.
Takano, M. (1987). Ordered sets R and Q as bases of Kripke models. Studia Logica, 46, 137148.
Umezawa, T. (1959). On logics intermediate between intuitionistic and classical predicate logic, Journal of Symbolic Logic, 24, 141153.
van Benthem, J. F. A. K. (2010). Frame correspondences in modal predicate logic. In Feferman, S. and Sieg, W., editors. Proofs, Categories and Computations: Essays in Honor of Grigori Mints. London: College Publications, pp. 287310.
Wroński, A. (1995). Transparent unification problem. Reports on Mathematical Logic, 29, 105107.
Wroński, A. (2008). Transparent verifiers in intermediate logics. Abstracts of the 54th Conference in History of Mathematics. Cracow: The Jagiellonian University Press, p. 16.
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