Many-Valued Non-deterministic Semantics for First-Order Logics of Formal (In)consistency

Abstract

Abstract.A paraconsistent logic is a logic which allows non-trivial inconsistent theories. 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 non-deterministic semantics for a very large family of first-order LFIs (which includes da Costa's original system C * 1 , as well as thousands of other logics). We show that our semantics is effective and modular, and we use this effectiveness to derive some important properties of logics in this family.

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

“…For instance, in [9] Nmatrices are utilized for knowledge-base integration, and in [3] they are used in the context of distance-based reasoning. In [8,11] Nmatrices have been used to provide a simple and modular non-deterministic semantics for LFIs [15]. Although the syntactic formulations of the propositional LFIs are relatively simple, the previously known semantic interpretations were more complicated: the vast majority of LFIs cannot be characterized by means of finite deterministic matrices.…”

Cut-free sequent calculi for C-systems with generalized finite-valued semantics

“…To the best of our knowledge, currently there are no known analytic systems available on the first-order level. However, [12] provided finite non-deterministic semantics for first-order C-systems, which could be exploited along the lines of the approach presented here.…”

Non-deterministic Multi-valued Matrices for First-Order Logics of Formal Inconsistency

“…We obtain a large family of first-order LFIs by adding to the basic system QB different combinations of the following schemata: 4 In [4] the name QB is used for a slightly different first-order system. 5 The schemata (c), (e), (i 1 ) and (i 2 ) were used in [1,10] (with the dual operator •).…”

“…29 in [4] and is omitted here. The modular approach of Nmatrices provides some important insights into the semantic role of each of the above schemata.…”

“…The first steps in this direction were taken in [3] for a very restricted family of paraconsistent logics (no consistency operators), and in [10,4], where finite non-deterministic semantics was provided for a family of LFIs with the consistency operator •. In this paper we study the semantic effects of 18 new axioms, capturing de Morgan principles and inconsistency propagation both for the propositional connectives and quantifiers.…”