Notes on the semantics for the logic with semi-negation

Studia logica, 1977

2009

In this chapter we determine the classes of S-algebras and of full models for several logics, especially for some which do not fit into the classical approaches to the algebraization of logic. We classify them according to several of the criteria we have been considering, i.e., the properties of the Leibniz, Tarski and Frege operators, which determine the classes of selfextensional logics, Fregean logics, strongly selfextensional logics, protoalgebraic logics, etc. We also study the counterexamples promised in the preceding chapters of this monograph. It goes without saying that the number of cases we have examined is limited, and that many more are waiting to be studied 32. In our view this is an interesting program, especially for non-algebraizable logics. Among those already proven in Blok and Pigozzi [1989a] not to be algebraizable we find many quasi-normal and other modal logics like Lewis' S1, S2 and S3, entailment system E, several purely implicational logics like BCI, the system R → of relevant implication, the "pure entailment" system E → , the implicative fragment S5 → of the Wajsbergstyle version of S5, etc. Other non-algebraizable logics not treated in the present monograph are Da Costa's paraconsistent logics C n (see Lewin, Mikenberg, and Schwarze [1991]), and the "logic of paradox" of Priest [1979] (see Pynko [1995]). This program is also interesting for some algebraizable logics whose class of Salgebras is already known, but whose full models have not yet been investigated; this includes Łukasiewicz many-valued logics (see Rodríguez, Torrens, and Verdú [1990]), BCK logic and some of its neighbours (see Blok and Pigozzi [1989a] Theorem 5.10), the equivalential fragments of classical and intuitionistic logics 32 The full models of several subintuitionistic logics have been determined in Bou [2001]; those of

2009

In this chapter we include the main definitions, notations, and general properties concerning logical matrices, abstract logics and sentential logics. Most of the results reproduced here are not new; however, those concerning abstract logics are not well-known, so it seems useful to recall them in some detail, and to prove some of the ones that are new. Useful references on these topics are Brown and Suszko [1973], Burris and Sankappanavar [1981] and Wójcicki [1988]. Algebras In this monograph (except in Chapter 5, where we deal with examples) we will always work with algebras A = A,. .. of the same, arbitrary, similarity type; thus, when we say "every/any/some algebra" we mean "of the same type". By Hom(A, B) we denote the set of all homomorphisms from the algebra A into the algebra B. The set of congruences of the algebra A will be denoted by ConA. Many of the sets we will consider have the structure of a (often complete, or even algebraic) lattice, but we will not use a different symbol for the lattice and for the underlying set, since no confusion is likely to arise. Given any class K of algebras, the set Con K A = {θ ∈ ConA : A/θ ∈ K} is called the set of K-congruences of A; while this set is ordered under ⊆, in general it is not a lattice. This set will play an important role in this monograph.

Notre Dame Journal of Formal Logic, 1990

The main purposes of this paper are to provide an algebraic analysis of a certain class of constructive logics with strong negation, and to investigate algebraically the relations between strong and nonconstructible negations definable in these logics. The tools used in this analysis are varieties of algebraic structures of ordered pairs called special NΊattices which were first introduced as algebraic models for Nelson's constructive logic with strong negation (CLSN). Via suitable restrictions of the domains, algebras of this type are shown to be algebraic models for the propositional fragments of E°, CLSN, Intuitionistic Logic, and E%. The differences between E° and CLSN are then studied, via the interrelations that the related algebras exhibit among strong and nonconstructible negations, properties of filters, and their behavior with respect to classically valid formulas.

arXiv: Logic, 2020

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations closed under substitutions. This allows to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C1 ) but the whole hierarchy of da Costa's calculi Cn. This produces a novel and simple decision procedure for these logics. Finally, we generalize these finite RNmatrices to others based on arbitrary Boolean algebras, through the notion of restricted swap structures.

Reports on Mathematical Logic, 2002

Archive for Mathematical Logic, 2006

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [Höh95], intuitionistic logic without contraction [AV00], H BCK [OK85] (nowadays called FL ew by Ono), etc. In this paper 1 we study the ∨, * , ¬, 0, 1-fragment and the ∨, ∧, * , ¬, 0, 1-fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It results that this notion of fragment is axiomatized by the rules of the sequent calculus FL ew for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable.

Zeitschrift fuer Math. Logik, Vol. 33, 433-439, 1988

Fundamenta Informaticae

Our aim in this article is to present a method for classifying and characterizing the various different semantics of logic programs with negation that have been considered in the last years. Instead of appealing to more or less questionable intuitions, we take a more structural view: our starting point is the observation that all semantics induce in a natural way non-monotonic entailment relations " j ". The novel idea of our approach is to ask for the properties of these j-relations and to use them for describing all possible semantics. The main properties discussed in this paper are adaptations of rules that play a fundamental rôle in general non-monotonic reasoning: Cumulativity and Rationality. They were introduced and investigated by Gabbay, Kraus, Lehmann, Magidor and Makinson. We show that the 3-valued version COMP 3 of Clark's completion, the stratified semantics M supp P as well as the well-founded semantics WFS and two extensions of it behave very regular: they are cumulative, rational and one of them is even supraclassical. While Pereira's recently proposed semantics O-SEM is not rational it is still cumulative. Cumulativity fails for the regular semantics REG-SEM of You/Yuan (recently shown to be equivalent to three other proposals). In a second article we will supplement these strong rules with a set of weak rules and consider the problem of uniquely describing a given semantics by its strong and weak properties together.

1980