Non-deterministic matrices are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above (...) types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics. (shrink)

Non-deterministic matrices, a natural generalization of many-valued matrices, are semantic structures in which the value assigned to a complex formula may be chosen non-deterministically from a given set of options. We show that by combining non-deterministic matrices and distance-based considerations, one obtains a family of logics that are useful for reasoning with uncertainty. These logics are a conservative extension of those that are obtained by standard distance-based semantics, and so usual distance-based methods are easily simulated within our framework. We investigate (...) the basic properties of the distance-preferential non-deterministic logics, consider their application for reasoning with incomplete and inconsistent information, and show the correspondence between some particular entailments in our framework and well-known problems like max-SAT. (shrink)

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) (...) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)

A growing interest in many-valued logic has developed which to a large extent is based on applications, intended as well as already realised ones. These applications range from the field of computer science, e.g. in the areas of automated theorem proving, approximate reasoning, multi-agent systems, switching theory, and program verification, through the field of pure mathematics, e.g. in independence of consistency proofs, in generalized set theories, or in the theory of particular algebraic structures, into the fields of humanities, linguistics and (...) philosophy. (shrink)

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.. (shrink)

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. (shrink)

We consider the very weak paracomplete and paraconsistent logics that are obtained by a straightforward weakening of Classical Logic, as well as some of their maximal extensions that are a fragment of Classical Logic. We prove that these logics may be faithfully embedded in Classical Logic, and that the interpolation theorem obtains for them.