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Prooftheoretic semantics is an alternative to modeltheoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation (...) 

If a certain semantic relation (which we call 'local consequence') is allowed to guide expectations about which rules are derivable from other rules, these expectations will not always be fulfilled, as we illustrate. An alternative semantic criterion (based on a relation we call 'global consequence'), suggested by work of J.W. Garson, turns out to provide a much better  indeed a perfectly accurate  guide to derivability. 

This paper presents and defends a way to add a transparent truth predicate to classical logic, such that and A are everywhere intersubstitutable, where all Tbiconditionals hold, and where truth can be made compositional. A key feature of our framework, called STTT (for StrictTolerant Transparent Truth), is that it supports a nontransitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STTT allows for arise in paradoxical cases. 

I investigate syntactic notions of theoretical equivalence between logical theories and a recent objection thereto. I show that this recent criticism of syntactic accounts, as extensionally inadequate, is unwarranted by developing an account which is plausibly extensionally adequate and more philosophically motivated. This is important for recent antiexceptionalist treatments of logic since syntactic accounts require less theoretical baggage than semantic accounts. 

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a metainferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is (...) 

ABSTRACTWe will present 12 different mixed metainferential consequence relations. Each one of them is specified using two different inferential Tarskian or nonTarskian consequence relations: or. We will show that it is possible to obtain a Tarskian logic with nonTarskian inferential logics, but also a nonTarskian logic with Tarskian inferential logics. Moreover, we will show how some of these metainferential logics work better than the corresponding inferential rivals. Finally, we will show how these logics prove that it is not enough to (...) 

In a previous paper (see ‘Tolerant, Classical, Strict’, henceforth TCS) we investigated a semantic framework to deal with the idea that vague predicates are tolerant, namely that small changes do not affect the applicability of a vague predicate even if large changes do. Our approach there rests on two main ideas. First, given a classical extension of a predicate, we can define a strict and a tolerant extension depending on an indifference relation associated to that predicate. Second, we can use (...) 

The paper is concerned with a logical difficulty which Lionel Shapiro’s deflationist theory of logical consequence (as well as the author’s favoured, nondeflationist theory) gives rise to. It is argued that Shapiro’s noncontractive approach to solving the difficulty, although correct in its broad outlines, is nevertheless extremely problematic in some of its specifics, in particular in its failure to validate certain intuitive rules and laws associated with the principle of modus ponens. An alternative noncontractive theory is offered which does not (...) 



One of the most fruitful applications of substructural logics stems from their capacity to deal with selfreferential paradoxes, especially truththeoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot (...) 

Substructural theories of truth are theories based on logics that do not include the full complement of usual structural rules. Existing substructural approaches fall into two main families: noncontractive approaches and nontransitive approaches. This paper provides a sketch of these families, and argues for two claims: first, that substructural theories are betterpositioned than other theories to grapple with the truththeoretic paradoxes, and second—more tentatively—that nontransitive approaches are in turn betterpositioned than noncontractive approaches. 



Logical pluralism is the claim that diﬀerent accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truthpreservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic speciﬁes a class of cases, but diﬀers over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truthpreservation, provided this is properly understood. Once understood, (...) 

We will present a threevalued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid metametainference. We will afterwards develop a hierarchy of consequence relations (...) 

This paper shows how to conservatively extend classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth—involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system allows for Cut—elimination, but the (...) 



This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaningone which takes meaning to be constituted by assertibility and deniability conditionsand shows how the usual multipleconclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical systemSTthat conservatively extends classical (...) 

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the StrictTolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) 

This paper is based on Lectures 1, 2 and 4 in the series of ten lectures titled “Algebraic Structures for Logic” that Professor Blok and I presented at the Twenty Third Holiday Mathematics Symposium held at New Mexico State University in Las Cruces, New Mexico, January 812, 1999. These three lectures presented a new approach to the algebraization of deductive systems, and after the symposium we made plans to publish a joint paper, to be written by Blok, further developing these (...) 

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is wellknown, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a nontransitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this (...) 

By using the notions of exact truth and exact falsity, one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the Strong Kleene schema. Besides familiar logics such as Strong Kleene logic, the Logic of Paradox and First Degree Entailment, the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar (...) 