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Unification in intermediate logics. (English) Zbl 1357.03059
Unification is concerned with the problem whether two given terms can be identified via a substitution in a theory of equality. The paper presents a proof-theoretic treatment of unification in intermediate logics.
In this context, unification is the study of substitutions under which a formula becomes provable in logic. In classical propositional logic, every formula has a most general unifier while this is not true in intermediate logics: for example, in intuitionistic logic the formula \(p \vee \neg p\) has not a most general unifier. The purpose of the article is to study when an intermediate logic has finitary of unitary unification.
Although many results are already known, the method to deduce the presented results is new and relies on syntax only, not involving semantics, like, e.g., in S. Ghilardi’s work [J. Symb. Log. 64, No. 2, 859–880 (1999; Zbl 0930.03009)]. In particular, the paper shows how a proof of a formula under a projective unifier can be obtained from the closure under the Visser rules, clarifying the connection between valuations and most general unifiers. So, many existing and well-known results can be extended to logical frameworks containing at least conjunction and implication.
The article is of interest to everyone working in the proof theory of propositional intermediate logics and, although it requires some technical knowledge, it contains a number of useful results, obtained through a uniform method which is flexible enough to allow generalisations.

MSC:
03B55 Intermediate logics
03B20 Subsystems of classical logic (including intuitionistic logic)
03B70 Logic in computer science
03F03 Proof theory, general (including proof-theoretic semantics)
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