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A syntactic machinery is developed for π-institutions based on the notion of a category of natural transformations on their sentence functors. Rules of inference, similar to the ones traditionally used in the sentential logic framework to define the best known sentential logics, are, then, introduced for π-institutions. A π-institution is said to be rule-based if its closure system is induced by a collection of rules of inference. A logical matrix-like semantics is introduced for rule-based π-institutions and a version of Bloom's (...) |
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The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) |
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An important part of the theory of algebraizable sentential logics consists of studying the algebraic semantics of these logics. As developed by Czelakowski, Blok, and Pigozzi and Font and Jansana, among others, it includes studying the properties of logical matrices serving as models of deductive systems and the properties of abstract logics serving as models of sentential logics. The present paper contributes to the development of the categorical theory by abstracting some of these model theoretic aspects and results from the (...) |
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The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) |
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The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-Institutions. More precisely, the notion of an N-equivalence system for a given π-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-Institutions I having an N-equivalence system, the maximum such system is singled out (...) |
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