Piotr Wojtylak

Projective unification in modal logic

A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A (i.e. a substitution making A a theorem of L) such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all normal modal logics L containing S4.3 are almost structurally complete, i.e. all (structural) admissible rules having unifiable premises are derivable in L. Moreover, L is (hereditarily) structurally complete iff L contains McKinsey axiom M .

Matrix representations for structural strengthenings of a propositional logic

The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.

On structural completeness of many-valued logics

In the paper some consequence operations generated by Łukasiewiczs matrices are examined.

On structural completeness of implicational logics

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.

Generalizing proofs in monadic languages

Abstract This paper develops a proof theory for logical forms of proofs in the case of monadic languages. Among the consequences are different kinds of generalization of proofs in various schematic proof systems. The results use suitable relations between logical properties of partial proof data and algebraic properties of corresponding sets of linear diophantine equations.

Cn-Definitions of Propositional Connectives

We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.

Almost structurally complete infinitary consequence operations extending S4.3

It is shown that almost structurally complete consequence operations coincide with the finitely approximable ones, among all consequence operations, including infinitary, which extend the modal logic S4.3; this gives the semantic characterization of a syntactic notion. An infinite basis for infinitary admissible rules is also provided and it is shown that any formula derivable in any structurally complete consequence operation over S4.3 has a proof (which takes the form of a sequence) of the type of ω+1. Partial descriptions of the lattice EXT(S4.3) (and its distributive sublattice EXTfin(S4.3)) of all (finitary) consequence operations extending S4.3 as well as some other lattices EXT(L), for L extending S4.3, are given. The results strongly rely on projective unification of logics extending S4.3. In the language of algebra the result states that, in the class of S4.3-algebras, a prevariety is almost structurally complete iff it is generated by its finite members.

A proof system for classical logic

An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.