Equivalential Logics (After 25 Years of Investigations)

The notion of equivalential logic is due to Tadeusz Prucnal and Andrzej Wroński. The short note “An algebraic characterization of the notion of structural completeness” appeared in 1974 in Bulletin of the Section of Logic, Vol. 3, 30-33. The work dealt mainly with structural completeness, the property of deductive systems which was systematically investigated in Poland in the beginning of the 70ties. One may say that singling out the class of equivalential logics was a side-effect of the Prucnal and Wroński’s investigations of structural completeness. The term “equivalential logic” is usually associated with the purely equivalential fragment of the classical or intuitionistic propositional logics. In the Prucnal Wroński’s paper the notion of an equivalential logic is understood much broader. The fragments mentioned above fall under this general notion of an equivalential logic. Received September 15, 1999 1991 AMS Subject Classification 03B22 24 JANUSZ CZELAKOWSKI Logic is defined here in accordance with the Tarskian, purely consequential paradigm of logic which is dominant in the Polish logical tradition (see Tarski [1956]). Thus a logic (when restricted to sentential languages) is viewed as a pair