Polish logic

The term ‘Polish logic’ was coined by McCall to signal the important contributions to modern logic by logicians from Poland between the wars. There were several centres of research, of which the Warsaw school, which grew out of the earlier Lwów–Warsaw philosophical movement, was the most significant. Its development was closely connected with the Warsaw school of mathematics, which gave it its characteristic mathematical bent. Polish logic took as its point of departure the main trends in logical research of the time and it has influenced both subsequent logical research and subsequent work in the Western analytic tradition of philosophy. Its chief contributions were: (1) an enrichment of existing logical theory (including work on Boolean algebras, the sentential calculus, set theory, the theory of types); (2) new logical theories (for example, Leśniewski’s systems, Łukasiewicz’s many-valued logics, Tarski’s theory of truth, theory of the consequence operation and the calculus of systems); (3) new methods and tools as well as improvements of existing methods (for example, the matrix method of constructing sentential calculi, axiomatizability of logical matrices, algebraic and topological interpretations of deductive systems, permutation models for set theory, the application of quantifier elimination to decidability and definability problems); and (4) the application of formal methods to the study of the history of logic, resulting in a new understanding of the logics of Aristotle, the Stoics and the medievals.

Formal languages and systems

Formal languages and systems are concerned with symbolic structures considered under the aspect of generation by formal (syntactic) rules, that is, irrespective of their or their components’ meaning(s). In the most general sense, a formal language is a set of expressions. The most important way of describing this set is by means of grammars. Formal systems are formal languages equipped with a consequence operation yielding a deductive system. If one further specifies the means by which expressions are built up (connectives, quantifiers) and the rules from which inferences are generated, one obtains logical calculi of various sorts, especially Frege–Hilbert-style and Gentzen-style systems.

A Syntactic Approach to Closure Operation

In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined.

Many-valued logics—implications and semantic consequences

In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication. In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

Deduction theorems within RM and its extensions

In [13], M. Tokarz specified some infinite family of consequence operations among all ones associated with the relevant logic RM or with the extensions of RM and proved that each of them admits a deduction theorem scheme. In this paper, we show that the family is complete in a sense that if C is a consequence operation with CRM ≤ C and C admits a deduction theorem scheme, then C is equal to a consequence operation specified in [13]. In algebraic terms, this means that the only quasivarieties of Sugihara algebras with the relative congruence extension property are the quasivarieties corresponding, via the algebraization process, to the consequence operations specified in [13].

The deduction theorem for quantum logic—some negative results

We prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).