In this paper we define and study a generalized notion of a logical system that covers on an equal formal basis sentential, equational and sequential systems. We develop a general theory of equivalence between generalized logics that provides, first, a conception of algebraizable logic , second, a formal concept of equivalence between sequential systems and, third, a notion of equivalence between sentential and sequential systems. We also use our theory of equivalence for developing a general algebraic approach to conjunctive non-pseudo-axiomatic (...) self-extensional sentential logics. Finally, we consider within the framework of the mentioned approach various sequential formulations for some well-known sentential logics. (shrink)

Two characterizations are given of those structural consequence operations on a propositional language which can be defined via proofs from a finite number of polynomial rules.

The main result of the present paper is equivalence of the following conditions, for any k-dimensional logic L : L has a full-replacement implication system, i.e., a finite set of k-dimensional formulas with 2k variables that in a natural way adopts the Identity axiom and the Modus Ponens rule for the ordinary implication connective; L has an unary-replacement implication system, i.e., a finite set of k-dimensional formulas with k+1 variables that in a different way adopts the Identity axiom and the (...) Modus Ponens rule for the ordinary implication connective; L has a parameterized local deduction theorem ; L has the syntactic correspondence property that is essentially the restriction of the filter correspondence property to deductive L-filters over the formula algebra alone; L is protoalgebraic in the sense that the Leibniz operator is monotonic on the set of deductive L -filters over every algebra; L has a system of equivalence formulas with parameters that defines the Leibniz operator on deductive L -filters over every algebra.We also present a family of specific examples which collectively show that the above metaequivalence doesn't remain true when in ``2k'' ``k+1'') is replaced by ``2k-1''. This, in particular, disproves the statement of [4], Theorem 13.2. (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...) of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)