According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non- existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity (...) of the consequence relation. A lattice-theoretical framework for a whole family of “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic. (shrink)

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth predicate is merely a device that allows us to express a certain kind of generality. I argue that a parallel approach is available in the case of logical consequence. Just as deflationism about truth offers an alternative to accounts of truth's nature in terms of correspondence or justification, deflationism about consequence promises an alternative to model-theoretic or proof-theoretic accounts of consequence's nature. I then (...) argue, against considerations put forward by Field and Beall, that Curry's paradox no more rules out deflationism about consequence than the liar paradox rules out deflationism about truth. (shrink)

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 (...) other does not.). (shrink)

It might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Gödel's second incompleteness theorem any such argument must break down, but just how it breaks down depends on the kind of theory of truth that is built into T. The paper surveys the possibilities, and suggests that some theories of truth give far more intuitive diagnoses of (...) the breakdown than do others. The paper concludes with some morals about the nature of validity and about a possible alternative to the idea that mathematical theories are indefinitely extensible. (shrink)

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, (...) which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, the substructure—of the validity relation itself.

In Spandrels of Truth, Beall concisely presents and defends a modest, so-called dialetheic theory of transparent truth.

A selective background -- Broadly classical approaches -- Paracompleteness -- More on paracomplete solutions -- Paraconsistent dialetheism.

According to the dominant approach in the theory of vagueness, the nature of the vagueness of an expression ‘F’ consists in its presenting borderline cases in an appropriately ordered series: objects which are neither definitely F nor definitely not F (where the notion of definiteness can be semantic, ontic, epistemic, psychological or primitive). In view of the various problems faced by theories of vagueness adopting the dominant approach, the thesis proposes to reconsider the naive theory of vagueness, according to which (...) the nature of the vagueness of an expression consists in its not drawing boundaries between any neighbouring objects in an appropriately ordered series. It is argued that expressions and concepts which do present this feature play an essential role in our cognitive and practical life, allowing us to conceptualize---in a way which would otherwise be impossible---the typically coarse-grained distinctions we encounter in reality. Despite its strong initial plausibility and ability to explain many phenomena of vagueness, the naive theory is widely rejected because thought to be shown inconsistent by the sorites paradox. In reply, it is first argued that accounts of vagueness based on the dominant approach are themselves subject to higher-order sorites paradoxes. The paradox is then solved on behalf of the naive theory by rejecting the unrestricted transitivity of the consequence relation on a vague language; a family of logics apt for reasoning with vague expressions is proposed and studied (using models with partially ordered values). The characteristic philosophical and logical consequences of this novel solution are developed and defended in detail. In particular, it is shown how the analysis of what happens in the attempt of surveying a sorites series and deciding each case allows the naive theory to recover a "thin" notion of a borderline case. (shrink)