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UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS

Published online by Cambridge University Press:  03 December 2018

WOJCIECH DZIK  and
PIOTR WOJTYLAK
WOJCIECH DZIK*
Affiliation:
Institute of Mathematics, University of Silesia in Katowice
PIOTR WOJTYLAK*
Affiliation:
Institute of Mathematics and Computer Science, University of Opole
*
*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
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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.

Keywords

03B5503B3503G25superintuitionistic logicpredicate logicunificationadmissible ruleHarrop formulaprojective formulastructural completenessunification type
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 1 , March 2019 , pp. 37 - 61
Copyright
Copyright © Association for Symbolic Logic 2018 

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