Beyond Rasiowan Systems: Unital Deductive Systems

Consequence Relations An Introduction to the Tarski-Lindenbaum Method

“… [ ] is a congruence class with respect to the congruence on the algebra of  [ ], generated by the set Cn  ( )×Cn  ( ). As shown in (Muravitsky, 2014a), the class of unital logics properly contains the class of implicative logics introduced by H. Rasiowa (Rasiowa, 1974), among which we find classical proposi-tional logic, intuitionistic propositional logic, modal logic S4 and many others, and the class of Fregean logics introduced in (Czelakowski and Pigozzi, 2004).…”

“…(our translation; comp. (Muravitsky, 2014a)) Although in case of structural abstract logic , a Lindenbaum matrix  [ ] is a separating tool to decide whether ⊢  , however, the algebra of this matrix can be very complex; therefore, the direct application of…”

“…This perspective prompted A. Muravitsky to introduce in (Muravitsky, 2014a) a class of unital abstract logics, that is, the structural abstract logics  with the property: for every set of formulas, the logical filter Cn  ( ) of the Lindenbaum matrix…”

“…The class of unital logics was defined in (Muravitsky, 2014a), although the name for this class of logics was suggested by an anonymous referee of this paper. In fact, this notion is rooted in Definition 4 of (Muravitsky, 2014b), where "a calculus [that] admits the Lindenbaum-Tarski algebra" was defined, which is somewhat weaker than the notion of a unital logic.…”

“…It was established in (Muravitsky, 2014a) that the class of implicative logics (studied in (Rasiowa, 1974)) and that of Freagean logics (studied in (Czelakowski and Pigozzi, 2004)) are properly included in the class of unital logics.…”