References in:
A Gentzen Calculus for Nothing but the Truth
Journal of Philosophical Logic 45 (4):451465 (2016)
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In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of NonContradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the CraigLyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right (...) 

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a natural dual (...) 

Kleene’s strong threevalued logic extends naturally to a fourvalued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it fourvalued analogs of Kleene’s weak threevalued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences. 

Logical pluralism has been in vogue since JC Beall and Greg Restall 2006 articulated and defended a new pluralist thesis. Recent criticisms such as Priest 2006a and Field 2009 have suggested that there is a relationship between their type of logical pluralism and the meaningvariance thesis for logic. This is the claim, often associated with Quine 1970, that a change of logic entails a change of meaning. Here we explore the connection between logical pluralism and meaningvariance, both in general and (...) 

One of the problems we face in manyvalued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as fourvalued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of BelnapDunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, (...) 

A curious feature of Belnap’s “useful fourvalued logic”, also known as firstdegree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic. 



The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. (...) 



ABSTRACT In this paper we study 12 fourvalued logics arisen from Belnap's truth and/or knowledge fourvalued lattices, with or without constants, by adding one or both or none of two new nonregular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice fourvalued logic with bilattice constants are exactly the regular fourvalued operations. Moreover, we prove that its expansion by any nonregular connective is strictly functionally complete. Further, finding axiomatizations of the quasi varieties generated by the (...) 

This book radically simplifies Montague Semantics and generalizes the theory by basing it on a partial higher order logic. The resulting theory is a synthesis of Montague Semantics and Situation Semantics. In the late sixties Richard Montague developed the revolutionary idea that we can understand the concept of meaning in ordinary languages much in the same way as we understand the semantics of logical languages. Unfortunately, however, he formalized his idea in an unnecessarily complex way  two outstanding researchers in (...) 



This paper contains some contributions to the study of Belnap's fourvalued logic from an algebraic point of view. We introduce a finite Hilbertstyle axiomatization of this logic, along with its wellknown semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their “logic of firstdegree entailments”. We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbertstyle presentation allows us to characterize (...) 

This paper contains some contributions to the study of Belnap's fourvalued logic from an algebraic point of view. We introduce a finite Hilbertstyle axiomatization of this logic, along with its wellknown semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their “logic of firstdegree entailments”. We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbertstyle presentation allows us to characterize (...) 