In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics of formal (in)consistency (LFIs). In this paper we provide in a modular way simple non-deterministic semantics for 64 of the most important logics from this family. Our semantics is 3-valued for some of the systems, and infinite-valued for the others. We prove that these results cannot be improved: neither of the systems with a three-valued non-deterministic semantics has either a finite characteristic ordinary matrix or a two-valued characteristic non-deterministic matrix, and neither of the other systems we investigate has a finite characteristic non-deterministic matrix. Still, our semantics provides decision procedures for all the systems investigated, as well as easy proofs of important proof-theoretical properties of them.

It is well known that every propositional logic which satisfies certain very

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One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and effectiveness of the use of non-deterministic many-valued semantics. 1

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ABSTRACT. It is well-known that da Costa’s C-systems of paraconsistent logic do not admit a Blok-Pigozzi algebraization. Still, an algebraic flavored semantics for them has been proposed in the literature, namely using the class of so-called da Costa algebras. However, the precise connection between these semantic structures and the C-systems was never established at the light of the theory of algebraizable logics. In this paper we propose to study the C-systems from an algebraic point of view, and to fill in this gap by using the tools and techniques of the newly developed behavioral approach to abstract algebraic logic. As a by-product of the approach, we also rediscover the bivaluation semantics of the logics. KEYWORDS: C-systems, algebraization of logics, behavioral reasoning, da Costa algebra. 1.

This paper solves an old problem: to devise decent inconsistencyadaptive logics that have the Cn logics as their lower limit. Two kinds of logics are presented. Those of the first kind offer a maximally consistent interpretation of the premise set in as far as this is possible in view of logical considerations. At the same time, they indicate at which points further choices may be made on extra-logical grounds. The logics of the second kind allow one to introduce those choices in a defeasible way and handle them. 1 Aim of This Paper Both the structure of the Cn logics and certain statements of da Costa’s seem to suggest a specific application for those logics, viz. to apply a certain stratagem— see Section 3—to theories that turned out inconsistent. Even if da Costa did not have this application in mind, the stratagem is clearly interesting and suggested by the Cn logics. This makes it worthwhile to develop inconsistency-adaptive logics that have the Cn systems as their lower limit. Indeed, the adaptive logics by themselves accomplish most of the task that is served by the stratagem. To be more precise, they accomplish that part of the task which can be accomplished in view of logical considerations. There is a further reason to devise adaptive logics that have Cn logics as their lower limit—this term is explained in Section 4. It is in principle possible to do so for any paraconsistent logic. The Cn logics are the oldest paraconsistent logics that were presented in a direct form, that is by an axiomatic system and not by a translation. So, as one may expect, to use them as lower limit logics has been on the agenda of adaptive logicians for a long time now. The delay is caused by a technical complication. Cn logics introduce dependencies between inconsistencies. Where this is the case, the flip-flop danger lurks. As we shall see in Section 5, flip-flop logics are rather uninteresting adaptive logics. Until recently, no general method was ∗ Research for this paper was supported by subventions from Ghent University and from the Fund for Scientific Research – Flanders. I am indebted to Christian Straßer and to Peter Verdée for comments on a former draft of this paper.

A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the best-known approaches to designing useful paraconsistent logics is da Costa’s approach, which has led to the family of Logics of Formal Inconsistency (LFIs), where the notion of inconsistency is expressed at the object level. In this paper we use non-deterministic matrices, a generalization of standard multi-valued matrices, to provide simple and modular finite-valued semantics for a large family of first-order LFIs. We demonstrate that the modular approach of Nmatrices provides new insights into the semantic role of the studied axioms and the dependencies between them. Furthermore, we study the issue of effectiveness in Nmatrices, a property which is crucial for the usefulness of semantics. We show that all of the nondeterministic semantics provided in this paper are effective. Key words: Many-valued logic, Paraconsistent logic, Non-deterministic matrices

abstract. We construct a modular semantic frameworks for LFIs (logics of formal (in)consistency) which extends the framework developed in [1; 3], but includes Marco’s schema too (and so practically all the axioms considered in [11] plus a few more). In addition, the paper provides another demonstration of the power of the idea of nondeterministic semantics, especially when it is combined with the idea of using truth-values to encode relevant data concerning propositions. 1