Paraconsistent logic from a modal viewpoint

In this paper we study paraconsistent negation as a modal operator, considering the fact that the classical negation of necessity has a paraconsistent behavior. We examine this operator on the one hand in the modal logic S5 and on the other hand in some new four-valued modal logics.     

Modal logic and model theory

We propose a first order modal logic, theQS4E-logic, obtained by adding to the well-known first order modal logicQS4 arigidity axiom schemas:A → □A, whereA denotes a basic formula. In this logic, thepossibility entails the possibility of extending a given classical first order model. This allows us to express some important concepts of classical model theory, such as existential completeness and the state of being infinitely generic, that are not expressibile in classical first order logic. Since they can be expressed in -logic, we are also induced to compare the expressive powers ofQS4E and . Some questions concerning the power of rigidity axiom are also examined.     

Paraconsistent algebras

The prepositional calculiC _n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra forC _n . C. Mortensen settled the problem, proving that no equivalence relation forC _n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC _n , as well as the concept of paraconsistent closure system, are introduced in this paper.We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.     

弗协调的弗雷格(英文)

本文表明,二阶弗协调概括与弗雷格的第五公理是足道的。也表明,如果等数关系是初始符号,那么通过弗协调推理可以从第五公理可以推出休谟原则。最后表明,弗协调的休谟原则不能用作逻辑主义数学的基础。     

A theorem in 3-valued model theory with connections to number theory,type theory,and relevant logic

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).     

Paraconsistent Logics and Translations

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we present some conservative translations involving classical logic, Lukasiewicz three-valued system L ₃, the intuitionistic system I ¹ and several paraconsistent logics, as for instance Sette's system P ¹, the D'Ottaviano and da Costa system J ₃ and da Costa's systems C _n, 1≤ n≤ω.     

Paraconsistent Logic

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go.     

Ideal Paraconsistent Logics

We define in precise terms the basic properties that an ??ideal propositional paraconsistent logic?? is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.     

Paraconsistent Informational Logic

We introduce a Paraconsistent Informational Logic that formalizes the idea of conjectures which are acceptable as to the quality and the variety of the information that they convey with respect to a given theory T, even if they are classically inconsistent with T. The work constitutes an extension of a previously developed Informational Logic for classical frameworks, where a new notion of logical entropy measure H on formulas and on proofs plays a central role.     

Set theory as modal logic

A logical systemBM ⁺ is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, and These modalities are submitted to a finite number of axioms. is the usual sign of necessity, corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbedding of the usual theory of classesM intoBM ⁺ is constructed, such that a formulaA is provable inM if and only if(A) is provable inBM ⁺. There is also an inverse imbedding with an analogous property.