The classification of propositional calculi

Studia Logica 66 (2):253-271 (2000)

Abstract
We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.
Keywords Philosophy   Logic   Mathematical Logic and Foundations   Computational Linguistics
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Reprint years 2004
DOI 10.1023/A:1005292130553
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An Algebraic Approach to Non-Classical Logics.Helena Rasiowa - 1974 - Warszawa, Pwn - Polish Scientific Publishers.
Relevant Entailment--Semantics and Formal Systems.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (2):334-342.
The Weak Theory of Implication.Alonzo Church - 1951 - In A. Menne, A. Wilhelmy & H. Angsil (eds.), KontrolIiertes Denken: Untersuchungen zum Logikkalk\"{u}l und der Logik der Einzelwissenschaften. Munich: Kommissions-verlag Karl Alber. pp. 22-37.

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