We analyze the variety of A. Monteiro's tetravalent modal algebras under the perspective of two logic systems naturally associated to it. Taking profit of the contrapositive implication introduced by A. Figallo and P. Landini, sound and complete Hilbert-style calculi for these logics are presented.

Involutive Stone algebras (or S-algebras) were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic (with six truth values) that can be determined by a finite number of matrices (four matrices). Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic.

This paper aims at discussing the importance of Leibniz Law to getting models for Paraconsistent Set Theories.

Abstract. Lukasiewicz residuation algebras with an underlying ordered structure of meet semilattice (or iLR-algebras) are studied. These algebras are the algebraic counterpart of the {->, A}-fragment of Lukasiewicz's many-valued logic. An equational basis for this class of algebras is shown. In addition, the subvariety of (n+ 1)-valued iLR-algebras for 0 < n < u> is considered. In particular, the structure of the free finitely generated (n + 1)-valued iLR-algebra is described. Moreover, a formula to compute its cardinal number in terms of n and the number of free generators is obtained. PreliminaresB. Bosbach ([5, 6]) undertook the investigation of a class of residuated structures that were related to but considerably more general than Brouwerian semilattices and the algebras associated with {-A}-fragment of Lukasiewicz's many valued logic.In a manuscript by J. Büchi and T. Owens ([8]) devoted to a study of Bosbach's algebras, written in the mid-seventies, the commutative members of this equational class were given the name hoops. More precisely, they are algebras (A,-•, 1) of type (2, 2,0) that satisfy:(HI) (A, •, 1) is a commutative monoid, (H2) x^x = 1,An important subclass of the variety of hoops is the variety of Wajsberg hoops, so named and studied by W. Blok and I. Ferreirim in [3]. These Key words and phrases: Free algebras, Lukasiewicz residuation algebras, hoop residuation algebras, BCK-algebras with supremum, Lukasiewicz's many-valued logic.

The tetravalent modal logic ($${\mathcal {TML}}$$ TML ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic$${{\mathcal {TML}}}^N$$ TML N ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.