Studia Logica

, Volume 66, Issue 2, pp 253-271

First online:

The Classification of Propositional Calculi

  • Alexander S. KarpenkoAffiliated withDepartment of Logic Institute of Philosophy, Russian Academy of Sciences

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We discuss Smirnov’s 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.

implicational logics combinators independence finite Boolean lattice maximal lattice of logics full logic