It is argued that the ‘inner’ negation ∼ familiar from 3-valued logic can be interpreted as a form of ‘conditional’ negation: ∼A is read “A is false if it has a truth value”. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language ...

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional log...

While the purpose of conventional proof calculi is to axiomatise the set of valid sentences of a logic, refutation systems axiomatise the invalid sentences. Such systems are relevant not only for proof-theoretic reasons but also for realising deductive systems for nonmonotonic logics. We introduce Gentzen-type refutation systems for two basic three-valued logics and we discuss an application of...

Introduction. Classical logics are based on the bivalence principle, that is, the set of truth-values V has cardinality |V | = 2, usually with V = {T,F} where T and F stand for truthhood and falsity, respectively. Many-valued logics generalize this requirement to more or less arbitrary sets of truth-values, rather referred to as truth-degrees in that context. Popular examples of many-valued log...

A drawback of many logics for reasoning on Semantic Web ontologies is the Principle of Explosion in case of inconsistent data. Considering a four-valued semantics, inconsistencies can be tolerated such that paraconsistency is obtained. We extend Belnap’s four-valued logic to a four-valued first-order logic with implication and equality, and we develop a method for translating formulas to a repr...

Abstract This paper shows how to transform explosive many-valued systems into paraconsistent logics. We investigate mainly the case of three-valued exhibiting non-explosive logics can be obtained from them.

The paper investigates the characterisation of vague concepts within the framework of modal logic. This work builds on the su-pervaluation approach of Fine and exploits the idea of a precisiication space. A simple language is presented with two modalities: a necessity operator and an operatorìt is unequivocal that' which is used to articulate the logic of vagueness. Both these operators obey th...

In this paper we study the logic Lωω, which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of Lωω with respect to models in Grothendieck toposes, which can be sharpened t...