In this paper, I will describe a technique for generating a novel kind of semantics for a logic, and explore some of its consequences. It would be natural to call the semantics produced by the technique in question ‘many-valued'; but that name is, of course, already taken. I call them, instead, ‘plurivalent'. In standard logical semantics, formulas take exactly one of a bunch of semantic values. I call such semantics ‘univalent'. In a plurivalent semantics, by contrast, formulas may take one (...) or more such values. The construction I shall describe can be applied to any univalent semantics to produce a corresponding plurivalent one. In the paper I will be concerned with the application of the technique to propositional many-valued logics. Sometimes going plurivalent does not change the consequence relation; sometimes it does. I investigate the possibilities in detail with respect to small family of many-valued logics. (shrink)

In this paper, I will describe a technique for generating a novel kind of semantics for a logic, and explore some of its consequences. It would be natural to call the semantics produced by the technique in question ‘many-valued'; but that name is, of course, already taken. I call them, instead, ‘plurivalent'. In standard logical semantics, formulas take exactly one of a bunch of semantic values. I call such semantics ‘univalent'. In a plurivalent semantics, by contrast, formulas may take one (...) or more such values. The construction I shall describe can be applied to any univalent semantics to produce a corresponding plurivalent one. In the paper I will be concerned with the application of the technique to propositional many-valued logics. Sometimes going plurivalent does not change the consequence relation; sometimes it does. I investigate the possibilities in detail with respect to small family of many-valued logics. (shrink)

Kevin Scharp proposes an original theory of the nature and logic of truth on which truth is an inconsistent concept that should be replaced for certain theoretical purposes. He argues that truth is best understood as an inconsistent concept, and proposes a detailed theory of inconsistent concepts that can be applied to the case of truth. Truth also happens to be a useful concept, but its inconsistency inhibits its utility; as such, it should be replaced with consistent concepts that can (...) do truth's job without giving rise to paradoxes. To this end, Scharp offers a pair of replacements, which he dubs ascending truth and descending truth, along with an axiomatic theory of them and a new kind of possible-worlds semantics for this theory. He goes to develop Davidson's idea that truth is best understood as the core of a measurement system for rational phenomena, and offers a semantic theory that treats truth predicates as assessment-sensitive and solves the problems posed by the liar and other paradoxes. (shrink)

In this book, Scott Soames illuminates the notion of truth and the role it plays in our ordinary thought as well as in our logical, philosophical, and scientific theories. Soames aims to integrate and deepen the most significant insights on truth from a variety of sources. He powerfully brings together the best technical work and the most important philosophical reflection on truth and shows how each can illuminate the other. Investigating such questions as whether we need a truth predicate at (...) all, what theoretical tasks it allows us to accomplish, and how we are to understand the content of any predicate capable of accomplishing these tasks, Soames organizes his discussion into three parts. Part I addresses crucial foundational issues as it identifies the bearers of truth, provides a basis for distinguishing truth from other notions, and formulates positive responses to well-known forms of truth-skepticism. Part II explicates the formal theories of Alfred Tarski and Saul Kripke and evaluates the philosophical significance of their work. It discusses their treatments of the Liar paradox, the relationship between truth and proof, the notion of a partially defined predicate, the concepts of logical truth and logical consequence, and the connection between truth and meaning. Part III extends important lessons drawn from Tarski and Kripke into new domains: vague predicates, the Sorites paradox, and the development of a larger, deflationary perspective on truth. Throughout the book, Soames examines a wide range of deflationary theories of truth, and attempts to separate what is correct and worth preserving in them from what is not. In doing so, he seeks to clear up many of the most significant philosophical doubts about truth. Written for a general audience while offering engaging material to the specialist, this rich study will be profitably read by both. (shrink)

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 meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion 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 system---ST---that conservatively extends classical (...) logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered. (shrink)

I distinguish paradoxes and hypodoxes among the conundrums of time travel. I introduce ‘hypodoxes’ as a term for seemingly consistent conundrums that seem to be related to various paradoxes, as the Truth-teller is related to the Liar. In this article, I briefly compare paradoxes and hypodoxes of time travel with Liar paradoxes and Truth-teller hypodoxes. I also discuss Lewis’ treatment of time travel paradoxes, which I characterise as a Laissez Faire theory of time travel. Time travel paradoxes are impossible according (...) to Laissez Faire theories, while it seems hypodoxes are possible. (shrink)

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

Roy Sorenson offers a unique exploration of an ancient problem: vagueness. Did Buddha become a fat man in one second? Is there a tallest short giraffe? According to Sorenson's epistemicist approach, the answers are yes! Although vagueness abounds in the way the world is divided, Sorenson argues that the divisions are sharp; yet we often do not know where they are. Written in Sorenson'e usual inventive and amusing style, this book offers original insight on language and logic, the way world (...) is, and our understanding of it. (shrink)

Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.

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)

What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...) so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. (shrink)

In Vagueness and Contradiction (2001), Roy Sorensen defends and extends his epistemic account of vagueness. In the process, he appeals to connections between vagueness and semantic paradox. These appeals come mainly in Chapter 11, where Sorensen offers a solution to what he calls the no-no paradox—a “neglected cousin” of the more famous liar—and attempts to use this solution as a precedent for an epistemic account of the sorites paradox. This strategy is problematic for Sorensen’s project, however, since, as we establish, (...) he fails to resolve the semantic pathology of the no-no paradox. (shrink)

Hilary Putnam's famous arguments criticizing Tarski's theory of truth are evaluated. It is argued that they do not succeed to undermine Tarski's approach. One of the arguments is based on the problematic idea of a false instance of T-schema. The other ignores various issues essential for Tarski's setting such as language-relativity of truth definition.

Of the dozens of purported solutions to the liar paradox published in the past fifty years, the vast majority are "traditional" in the sense that they reject one of the premises or inference rules that are used to derive the paradoxical conclusion. Over the years, however, several philosophers have developed an alternative to the traditional approaches; according to them, our very competence with the concept of truth leads us to accept that the reasoning used to derive the paradox is sound. (...) That is, our conceptual competence leads us into inconsistency. I call this alternative the inconsistency approach to the liar. Although this approach has many positive features, I argue that several of the well-developed versions of it that have appeared recently are unacceptable. In particular, they do not recognize that if truth is an inconsistent concept, then we should replace it with new concepts that do the work of truth without giving rise to paradoxes. I outline an inconsistency approach to the liar paradox that satisfies this condition. (shrink)

A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered here (...) revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes. (shrink)

Classic work by one of the most brilliant figures in post-war analytic philosophy.

The Russell class does not exist because the conditions purporting to specify that class are contradictory, and hence fail to specify any class. Equally, the conditions purporting to specify the Liar statement are contradictory and hence, although the Liar sentence is grammatically in order, it fails to yield a statement. Thus the common source of these and related paradoxes is contradictory (or tautologous) specifying conditions-for such conditions fail to specify. This is the diagnosis. The cure consists of seeking and destroying (...) the deep-seated preconceptions that make almost irresistible our belief in the existence of items which provably do not exist. (shrink)

This book is about one of the most baffling of all paradoxes – the famous Liar paradox. Suppose we say: 'We are lying now'. Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers (...) his own solutions and in the process assesses other attempts to solve the paradox. Unlike such attempts, Simmons' 'singularity' solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality – the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (shrink)

Ttler=‘Ttler is true’ says of itself that it is true. It is a truth-teller. I argue that we have equally telling arguments (i) to the effect that all truth-tellers must have the same truth-value (ii) and the effect that truth-tellers differ in truth-value. This is what I call the Truth-Tellers paradox. This paradox stems from the fact that the truth-value of a truth-teller like Ttler should be determined by the fact that it says of itself that it is true (which (...) entails (i)) but that it cannot be determined by that fact (as witnessed by (ii)). The Truth-Tellers paradox resem- bles the classical semantic paradoxes like the Liar. In both cases, a form of self-reference allows us to derive a contradiction from oth- erwise plausible semantic and logical principles. Furthermore the Truth-Tellers paradox can be formulated without using sentences which are in an intuitive sense ungrounded, it thus severs the link, almost universally taken for granted, between the semantic para- doxes and ungroundedness. Finally, some classical solutions to the Liar do not generalize to the Truth-Tellers paradox. (shrink)

The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.

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 T-biconditionals hold, and where truth can be made compositional. A key feature of our framework, called STTT (for Strict-Tolerant Transparent Truth), is that it supports a non-transitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STTT allows for arise in paradoxical cases.

We trace self-reference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ,{ }), where D is a non-empty set; for every a∈ D, which is a name of a k-ary function, {a}: Dk → D is the function named by a, and type is the type of a, which tells us if a is a name and, if it is, (...) the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel's proof, Kleene's recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including “undefined” values; we investigate different systems arising out of different ways of dealing with them. Many-sorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene's recursion theorem is obtained along the same lines. (shrink)

This chapter is a detailed study of predicativity in mathematics. It presents a number of historical versions predicativity requirements, looking for unifying ideas. The further development of the notions and requirements up to the present is traced, articulating connections among the different ideas. One underlying theme of the chapter is the motivations for the various requirements for rejecting impredicativity and the various ways of stating the requirement.

There are two paradoxes of satisfaction, and they are of different kinds. The classic satisfaction paradox is a version of Grelling’s: does ‘does not satisfy itself’ satisfy itself? The Unsatisfied paradox finds a predicate, P, such that Px if and only if x does not satisfy that predicate: paradox results for any x. The two are intuitively different as their predicates have different paradoxical extensions. Analysis reduces each paradoxical argument to differing rule sets, wherein their respective pathologies lie. Having different (...) pathologies, they are paradoxes of different kinds. Furthermore, each of these satisfaction paradoxes has an analogue with the same pathology in set theory. Therefore, these analogues are respectively of the same two kinds. This level of abstraction is significant in that it tracks two related but different pathologies. Thus, not all paradoxes of semantics and set theory share the same pathology: there are at least two kinds of paradox cutting across the semantic and set-theoretic distinction. (shrink)

Semantic pathology is most widely recognized in the liar paradox, where an apparent inconsistency arises in ‘‘liar sentences’’ and their ilk. But the phenomenon of semantic pathology also manifests a sibling symptom—an apparent indeterminacy—which, while not largely discussed (save for the occasional nod to ‘‘truthteller sentences’’), is just as pervasive as, and exactly parallels, the symptom of inconsistency. Moreover, certain ‘‘dual symptom’’ cases, which we call naysayers, exhibit both inconsistency and indeterminacy and also manifest a higher-order indeterminacy between them. In (...) this paper, we explore and lay out the full, broader extent of semantic pathology. We then turn to the best-known attempts to diagnose and treat the phenomenon: consistentist approaches, proposed by Alfred Tarski and Saul Kripke, and an inconsistentist approach, originally presented by Graham Priest and dubbed ‘‘dialetheism’’. We explain the basic elements of each view and argue that each fails to provide an adequate way of dealing with the full extent of semantic pathology. We conclude that a new kind of approach is required and briefly sketch the pretense-based view that we favor. (shrink)

This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders the proof of the so–called fixed point theorem transparent. We also point out various historical details and make some observations on circularity and some comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motive for arithmetization and what it should accomplish are made obvious. We suggest this as a way to (...) teach the incompleteness results to students that have had a basic course in logic, which is more efficient than the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. Motivated partly by didactic considerations, the present paper presents things somewhat differently. It also includes various points concerning natural language and circularity that appear only here. (shrink)