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On quasivariety semantics of fragments of intuitionistic propositional logic without exchange and contraction rules. (English) Zbl 0946.03027
Summary: Let \(H\) be the Hilbert-style intuitionistic propositional calculus without exchange and contraction rules (as given by H. Ono and Y. Komori [J. Symb. Log. 50, No. 1, 169-202 (1985; Zbl 0583.03018)]). An axiomatization of \(H\) with the separation property is provided. It is proved that just two of the superimplicational fragments of \(H\) fail to be finitely axiomatized, and that all are algebraizable. The paper is a study of these fragments, their equivalent algebraic (quasivariety) semantics and their axiomatic extensions.

MSC:
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25 Other algebras related to logic
03B20 Subsystems of classical logic (including intuitionistic logic)
06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
08C15 Quasivarieties
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