In this paper we show that the Gupta-Belnap systems S# and S* are П12. Since Kremer has independently established that they are П12-hard, this completely settles the problem of their complexity. The above-mentioned upper bound is established through a reduction to countable revision sequences that is inspired by, and makes use of a construction of McGee.

This paper extends Kripke’s theory of truth to a language with a variably strict conditional operator, of the kind that Stalnaker and others have used to represent ordinary indicative conditionals of English. It then shows how to combine this with a different and independently motivated conditional operator, to get a substantial logic of restricted quantification within naive truth theory.

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A (...) within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language. (shrink)

The Revision Theory of Truth has been challenged in A. M. Yaqūb's recent book The Liar Speaks the Truth. Yaqūb suggests some non-trivial changes in the original theory - changing the limit rule - to avoid certain artifacts. In this paper it is shown that the proposed changes are not sufficient, i.e., Yaqūb's system also produces artifacts. An alternative solution is proposed and the relation between it and Yaqūb's solution is explored.

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

Gupta has proposed a definition of strategic rationality cast in the framework of his revision theory of truth. His analysis, relative to a class of normal form games in which all players have a strict best reply to all other players’ strategy profiles, shows that game-theoretic concepts have revision-theoretic counterparts. We extend Gupta’s approach to deal with normal form games in which players’ may have weak best replies. We do so by adapting intuitions relative to Nash equilibrium refinements to the (...) revision-theoretic framework. We prove that there is a precise equivalence between trembling-hand perfect equilibria in two-player normal games and a revision-theoretic property. We then introduce lexicographic choice of action as a way to represent players’ expectations, which allows our analysis to reach full generality. Finally, we provide an example of the versatility of revision theory as applied to strategic interaction by formalizing a risk-and-compensation procedure of strategic choice in the revision-theoretic framework. (shrink)

The paper introduces Hilbert– and Gentzen-style calculi which correspond to systems ${\mathsf{C}_{n}}$ from Gupta and Belnap [3]. Systems ${\mathsf{C}_{n}}$ were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems ${\mathsf{GC}_{n}}$ admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.

Gupta’s Rule of Revision theory of truth builds on insights to be found in Martin and Woodruff and Kripke in order to permanently deepen our understanding of truth, of paradox, and of how we work our language while our language is working us. His concept of a predicate deriving its meaning by way of a Rule of Revision ought to impact significantly on the philosophy of language. Still, fortunately, he has left me something to.

Theories of truth and vagueness are closely connected; in this article, I draw another connection between these areas of research. Gupta and Belnap’s Revision Theory of Truth is converted into an approach to vagueness. I show how revision sequences from a general theory of definitions can be used to understand the nature of vague predicates. The revision sequences show how the meaning of vague predicates are interconnected with each other. The approach is contrasted with the similar supervaluationist approach.

The purpose of this note is to acknowledge a gap in a previous paper — “The Complexity of Revision”, see [1] — and provide a corrected version of argument. The gap was originally pointed out by Francesco Orilia (personal communication and [4]), and the fix was developed in correspondence with Vann McGee.

The purpose of this note is to acknowledge a gap in a previous paper, "The complexity of revision," and to provide a corrected version of the argument.

In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of definition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a different perspective on mathematically more “respectable” entities. Revision Rules were first introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they (...) have been further developed to provide the foundations for a general theory of (possibly circular) definitions. Revision Rules are non-monotonic inductive operators that are iterated into the transfinite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then define a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta. (shrink)

In this book, Yaqub describes a simple conception of truth and shows that it yields a semantical theory that accommodates the whole range of our seemingly conflicting intuitions about truth. This conception takes the Tarskian biconditionals as correctly and completely defining the notion of truth. The semantical theory, which is called the revision theory, that emerges from this conception paints a metaphysical picture of truth as a property whose applicability is given by a revision process rather than by a fixed (...) extension. The main advantage of this revision process is its ability to explain why truth seems in many cases almost redundant, in others substantial, and yet in others paradoxical. Yaub offers a comprehensive defense of the revision theory of truth by developing consistent and adequate formal semantics for languages in which all sorts of problematic sentences can be constructed. Yaqub concludes by introducing a logic of truth that further demonstrates the adequacy of the revision theory. (shrink)

This volume reprints eight of Anil Gupta's essays, some with additional material. The essays bring a refreshing new perspective to central issues in philosophical logic, philosophy of language, and epistemology.

This book offers a novel account of the relationship of experience to knowledge. The account builds on the intuitive idea that our ordinary perceptual judgments are not autonomous, that an interdependence obtains between our view of the world and our perceptual judgments. Anil Gupta shows in this important study that this interdependence is the key to a satisfactory account of experience. He uses tools from logic and the philosophy of language to argue that his account of experience makes available an (...) attractive and feasible empiricism. (shrink)

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

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. Understanding this paradox is very important because belief is a central notion of psychologically motivated semantic theories in philosophy, linguistics, and cognitive science, and this paradox poses serious problems for these theories. In this dissertation I criticize previous proposals and offer a new proposal, which I call a 'revision theory of belief'. -/- My revision theory of belief is in many respects an application of Gupta's and (...) Belnap's revision theory of truth, which is proposed to deal with the Liar paradox. They argue that truth is a circular notion because Tarski biconditionals provide circular, partial definitions of truth. And this circularity of truth is the source of the Liar paradox. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion of belief involved in the paradox of belief instability is circular, and that is why this paradox arises. -/- If the notion of belief involved in the paradox is circular, we have to employ a hypothetical evaluation of the belief predicate, and if we do so through a revision process, we can avoid the contradiction which seems to be generated by some paradoxical belief sentences. But it is not easy to show that our notion of belief is circular. Especially, there is nothing like Tarski biconditionals in the case of belief. Nonetheless, I show in this dissertation that there is a certain schema, which I call the B-schema, whose instances can be taken as circular, partial definitions of a notion of belief, and it is that notion of belief which is involved in the paradox of belief instability. In this dissertation I also show that the revision approach can be extended for handling the Knower paradox. I argue that the Knower paradox arises because our notion of knowledge involved in the paradox is circular. (shrink)

Many different modes of definition have been proposed over time, but none of them allows for circular definitions, since, according to the prevalent view, the term defined would then be lacking a precise signification. I argue that although circular definitions may at times fail uniquely to pick out a concept or an object, sense still can be made of them by using a rule of revision in the style adopted by Anil Gupta and Nuel Belnap in the theory of truth.

It is “the received wisdom” that any intuitively natural and consistent resolution of a class of semantic paradoxes immediately leads to other paradoxes just as bad as the first. This is often called the “revenge problem”. Some proponents of the received wisdom draw the conclusion that there is no hope of any natural treatment that puts all the paradoxes to rest: we must either live with the existence of paradoxes that we are unable to treat, or adopt artificial and ad (...) hoc means to avoid them. Others (“dialetheists”) argue that we can put the paradoxes to rest, but only by licensing the acceptance of some contradictions (presumably in a paraconsistent logic that prevents the contradictions from spreading everywhere). (shrink)