Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we (...) call Gentzen-regular connectives. For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations. (shrink)

The article explores the following question: which among the most often examined in the literature method of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasi...

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's -conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values . The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic.

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

In this paper the completeness theorems for the finite-valued ukasiewicz logics are proved with the use of the Lindenbaum algebra.