The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in truth conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.

This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e. S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g. 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness (...) theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox. (shrink)

A relativized version of Tarski's T-scheme is introduced as a new principle of the truth predicate. Under the relativized T-scheme, the paradoxical objects, such as the Liar sentence and Jourdain's card sequence, are found to have certain relative contradictoriness. That is, they are contradictory only in some frames in the sense that any valuation admissible for them in these frames will lead to a contradiction. It is proved that for any positive integer n, the n-jump liar sentence is contradictory in (...) and only in those frames containing at least an n-jump odd cycle. In particular, the Liar sentence is contradictory in and only in those frames containing at least an odd cycle. The Liar sentence is also proved to be less contradictory than Jourdain's card sequence: the latter must be contradictory in those frames where the former is so, but not vice versa. Generally, the relative contradictoriness is the common characteristic of the paradoxical objects, but different paradoxical objects may have different relative contradictoriness. (shrink)

There has been much debate recently as to whether the notion of truth, as applied to one's home language, is metaphysically neutral, the interesting metaphysical questions arising elsewhere (in relation to such notions as mind-independence or objectivity or existence). ' On one side, the minimalists, as they have come to be known, favour deflationary accounts of truth such as the redundancy or disquotational theories and conclude that the notion of truth is applicable to declarative sentences in general - at least (...) where there are fairly definite standards of appropriateness for the assertion or denial of such sentences - irrespective of the metaphysical commitments of the propositions the sentences express. Opponents have criticized the minimalist account of truth or denied that it follows from this account that truth is metaphysically neutral. My sympathies in this debate lie with the minimalist but I wish to focus on a serious problem for the minimalist - that posed by the Liar paradox. This problem has featured very little in the recent debate but seems to me to yield the strongest objection to minimalism. I develop the objection in the next section, in §2 sketching what I feel to be the minimalist's best response, namely developing the position not in terms of Tarskian classical biconditionals but rather in terms of the naive rules for truth. It emerges, then, that the minimalist will be forced to repudiate classical logic. The last section, §3, deals with objections to the strategy of §2. (shrink)

Tarski’s Convention T is often taken to claim that it is both sufficient and necessary for adequacy in a definition of truth that it imply instances of the T-schema where the embedded sentence translates the mentioned sentence. However, arguments against the necessity claim have recently appeared, and, furthermore, the necessity claim is actually not required for the indefinability results for which Tarski is justly famous; indeed, Tarski’s own presentation of the results in the later Undecidable Theories makes no mention of (...) an assumption to the effect that the definition of truth implies the biconditionals. This raises a question: was Tarski in fact committed to the necessity claim in the important papers of the 1930s and 40s? I argue that he was not. The discussion of this apparently esoteric interpretive issue in fact gets to the heart of many important questions about truth, and in the final sections of the paper I discuss the importance of the T-biconditionals in the theory of meaning and the relation of deflationary and inflationary theories of truth to the semantic paradoxes. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

In Kripke’s classic paper on truth it is argued that by adding a new semantic category different from truth and falsity it is possible to have a language with its own truth predicate. A substantial problem with this approach is that it lacks the expressive resources to characterize those sentences which fall under the new category. The main goal of this paper is to offer a refinement of Kripke’s approach in which this difficulty does not arise. We tackle this characterization (...) problem by letting certain sentences belong to more than one semantic category. We also consider the prospect of generalizing this framework to deal with languages containing vague predicates. (shrink)

In this paper we shall introduce two types of contextual-hierarchical approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r n and T r n+1 of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r n+1-schema should hold for all sentences of \. In contrast, (...) the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r n and T r n+1 are coherent in the sense that the same sentences of \ should be true according to T r n and T r n+1. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator. (shrink)

El dialeteísmo es la posición que afirma que hay contradicciones verdaderas. Este artículo versará sobre esa posición. En la primera sección, mencionaré los principales aportes que, en mi perspectiva, el dialeteísmo ha hecho a la lógica filosófica. En la segunda sección, analizaré el principal problema del dialeteísmo. En la tercera sección, mostraré que los argumentos a favor del dialeteísmo no llegan a establecer la verdad de esta posición. Finalmente, explicaré cuál es el tipo de paraconsistencia que considero adecuada y la (...) compararé con la posición dialeteísta. Dialetheism is the position which claims that there are true contradictions. This paper deals with this position. In the first section, I will mention the main contributions that, in my perspective, dialetheism has made to philosophical logic. In the second section, I will analyze the main problem of dialetheism. In the third section, I will show that the arguments in favor of dialetheism are not conclusive. Finally, I will explain which kind of paraconsistency I consider adequate, and I will compare it with the dialetheist position. (shrink)

Truth-value gaps have received little attention from a foundational perspective, a fact which has rightfully opened up gap theories to charges of vacuousness. This paper develops an account of the foundations of gap-like behavior which has some hope of avoiding such charges. I begin by reviewing and sharpening a powerful argument of Dummett’s to constrain the options that gap theorists have to make sense of their views. I then show that within these strictures, we can give an account of gaps (...) by drawing on elements of a broadly Stalnakerian framework for assertion and using gaps to track an amalgamation of assertoric effects. The discussion reveals that we may need special resources in our theories of assertion to posit gaps, that gaps may be unusable in characterizing the structure of mental states, and that gaps may have heterogeneous linguistic sources that result in equally heterogeneous projective and inferential behavior. (shrink)

The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P$ satisfying $P\rightarrow \varphi $ and $T\vdash \varphi \,\therefore\,T\vdash P$ is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension $\mathcal {QLP}$ of Artemov’s logic of proofs. $\mathcal {QLP}$ contains both explicit modalities $t:\varphi $ (...) and also proof quantifiers $x:\varphi $. In this system, the basis for the rule NEC is decomposed into a number of distinct principles governing how various modes of reasoning about proofs and provability can be internalized within the system itself. A conceptually motivated resolution to the paradox is proposed in the form of an argument for rejecting the unrestricted rule NEC on the basis of its subsumption of an intuitively invalid principle pertaining to the interaction of proof quantifiers and the proof-theorem relation expressed by explicit modalities. (shrink)

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a further development (...) of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt. (shrink)

Kripke showed how to restrict Tarski’s schema to grounded sentences. I examine the prospects for an analogous approach to the paradoxes of naive class comprehension. I present new methods to obtain theories of grounded classes and test them against antecedently motivated desiderata. My findings cast doubt on whether a theory of grounded classes can accommodate both the extensionality of classes and allow for class definition in terms of identity.

I will discuss Fields Outline of a Theory of Truth. I will point out important properties of Kripkeleast fixed points constructions and theory. I do this not to demean Field’s superb work on truth but rather to suggest that there may be no really satisfactory conditional connective for languages containing their own truth predicates.

I argue for an account of semantic paradox that requires minimal logical revision. I first consider a phenomenon that is common to the paradoxes of definability, Russell’s paradox and the Liar. The phenomenon—which I call Repetition—is this: given a paradoxical expression, we can go on to produce a semantically unproblematic expression composed of the very same words. I argue that Kripke’s and Field’s theories of truth make heavy weather of Repetition, and suggest a simpler contextual account. I go on to (...) outline a ‘singularity’ theory of semantical predicates in the spirit of remarks of Gödel. According to this theory, ‘denotes’, ‘extension’ and ‘true’ are context-sensitive expression that apply almost everywhere on a given occasion of use, except for certain singular points. I then turn to revenge paradoxes, and argue that even the dialetheist is subject to revenge. I then examine how the singularity theory responds to revenge. (shrink)

Suppose our ordinary notion of truth is ‘inconsistent’ in the sense that its meaning is partly given by principles that classically entail a logical contradiction. Should we replace the notion with a consistent surrogate? This paper begins by defusing various arguments in favor of this revisionary proposal, including Kevin Scharp’s contention that we need to replace truth for the purposes of semantic theorizing . Borrowing a certain conservative metasemantic principle from Matti Eklund, the article goes on to build a positive (...) case for the opposite policy: retaining truth as-is. The thought is basically that bivalence for the bulk of what we say in the course of ordinary, scientific, and philosophical inquiry should suffice to justify keeping ‘true’. Two versions of the story are told: one more philosophical, drawing on an analogy to Lewis’ response to Putnam’s paradox; the other more technical, invoking a deviant stran.. (shrink)

This paper is a sequel to Beall (2011), in which I both give and discuss the philosophical import of a result for the propositional (multiple-conclusion) logic LP+. Feedback on such ideas prompted a spelling out of the first-order case. My aim in this paper is to do just that: namely, explicitly record the first-order result(s), including the collapse results for K3+ and FDE+.

This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.

We discuss four distinct semantic consequence relations which are based on Strong Kleene theories of truth and which generalize the notion of classical consequence to 3-valued logics. Then we set up a uniform signed tableau calculus, which we show to be sound and complete with respect to each of the four semantic consequence relations. The signs employed by our calculus are,, and, which indicate a strict assertion, strict denial, tolerant assertion and tolerant denial respectively. Recently, Ripley applied the strict–tolerant account (...) of assertion and denial to develop a new approach to truth and alethic paradox, which we call the Strict–Tolerant Conceptionof Truth. The paper aims to contribute to our understanding of STCT in at least three ways. First, by developing the strict–tolerant calculus.Second, by developing a semantic version of the strict–tolerant calculus, which informs us about the assertoric possibilities relative to a fixed ground model. Third, by showing that the strict–tolerant calculus and assertoric semantics jointly suggest that STCT's claim that the strict and tolerant can be understood in terms of one another has to be reconsidered. (shrink)

In a recent paper, Philip Kremer proposes a formal and theory-relative desideratum for theories of truth that is spelled out in terms of the notion of ‘no vicious reference’. Kremer’s Modified Gupta-Belnap Desideratum (MGBD) reads as follows: if theory of truth T dictates that there is no vicious reference in ground model M, then T should dictate that truth behaves like a classical concept in M. In this paper, we suggest an alternative desideratum (AD): if theory of truth T dictates (...) that there is no vicious reference in ground model M, then T should dictate that all T-biconditionals are (strongly) assertible in M. We illustrate that MGBD and AD are not equivalent by means of a Generalized Strong Kleene theory of truth and we argue that AD is preferable over MGBD as a desideratum for theories of truth. (shrink)

There is one truth, but many truths: i.e., one unambiguous, non-relative truth-concept, but many and various propositions that are true. One truth-concept: to say that a proposition is true is to say (not that anyone, or everyone, believes it, but) that things are as it says; but many truths: particular empirical claims, scientific theories, historical propositions, mathematical theorems, logical principles, textual interpretations, statements about what a person wants or believes or intends, about grammatical and legal rules, etc., etc. But, as (...) Frank Ramsey once said, “There is no platitude so obvious that eminent philosophers have not denied it”; and as soon as you ask why anyone would deny that there is one truth-concept, or that there are many true propositions, it becomes apparent that my initial, simple formula disguises many complexities. (shrink)

Van Heijenoort’s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schröder, and Löwenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then (...) I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2). (shrink)

Is there a key for ‘translating' some set-theoretical paradoxes into counterpart semantical paradoxes and vice-versa? There is, and this encourages the hope of a unified solution. The solution turns not on inventing new axioms that do not entail contradiction, but on imposing a completely intuitive restriction on the comprehension axiom of naive set theory in order to avoid illegitimate (circular) stipulation.

An attractive approach to the semantic paradoxes holds that cases of semantic pathology give rise to indeterminacy. What attitude should a rational agent have toward a proposition that it takes to be indeterminate in this sense? Orthodoxy holds that rationality requires that an agent disbelieve such a proposition. I argue that a rational agent should be such that it is indeterminate whether it believes the proposition in question. For rational agents, indeterminacy in the objects of their attitudes will filter up (...) to the attitudes themselves. (shrink)

Read as a comment on Crispin Wright's \"Vagueness: A Fifth Column Approach\", this paper defends a form of supervaluationism against Wright's criticisms. Along the way, however, it takes up the question what is really wrong with Epistemicism, how the appeal of the Sorities ought properly to be understood, and why Contextualist accounts of vagueness won't do.

John Barker, in two recent essays, raises a variety of intriguing criticisms to challenge my interpretation of the liar paradox and the type of solution I proposein ‘Denying the Liar’ and ‘Denying the Liar Reaffirmed.’ Barker continues to believe that I have misunderstood the logical structure of the liar sentence and itsexpression, and that as a result my solution misfires. I shall try to show that on the contrary my analysis is correct, and that Barker does not properly grasp what (...) mysolution to the liar paradox involves. Additionally, I argue that Barker makes fundamental errors in the explanation of liar sentence formulations in intensional contexts and in the classical metatheory he invokes to support his criticisms. (shrink)

This is a collection of eighteen solicited papers on the topics of the title: circularity, definition, and truth. The papers are loosely connected in subject matter, but present a great variety of issues, theories, and approaches. Amongst the many subjects discussed are: the revision theory of truth and applications of revision rules, partiality and fixed point constructions, substitutional quantification, fuzzy logic, negation, belief revision, context dependence, hierarchies, Tarski on truth, deflationism, correspondence theories of truth, and normative aspects of truth. The (...) Liar paradox figures prominently in this collection, but is not alone. Other familiar paradoxes are also discussed, including paradoxes of definability, the Sorites paradox, and the Surprise Exam paradox. So-called paradoxes of rationality, such as the Prisoner’s Dilemma, receive attention as well. (shrink)

The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overview of recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

We consider a language containing partial predicates for subjective knowability and truth. For this language, inductive hierarchy rules are proposed which build up the extension and anti-extension of these partial predicates in stages. The logical interaction between the extension of the truth predicate and the anti-extension of the knowability predicate is investigated.

This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions (...) in logic, by means of ‘translations’ of the medieval logical theories into the modern framework of symbolic logic, i.e. formalizations. One of its conclusions is that, when properly understood within their own framework, the interest of medieval logical theories for modern investigations go beyond mere historical interest, but that a thorough conceptual analysis of such theories must be undertaken in order to avoid conceptual misprojections. While such translations of medieval into modern logic have been attempted before, the approach presented here is innovative in that attention is paid to the similarities as well as to the dissimilarities between the two traditions, and to what can be learned from the medieval masters for modern investigations in logic and semantics. (shrink)

The purpose of this paper is to defend a conception of language that does not rely on linguistic meanings, and use it to address the Sorites and Liar paradoxes.

Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.

This paper spells out the positive theory sketched at the end of "Against Stepping Back".): According to deflationists, [p] is true is in some sense equivalent to p. The problem that the semantic paradoxes pose for the deflationist is to explicate this equivalence without relying on a semantics grounded in the sort of real reference relations that a deflationist thinks do not exist. More generally, the deflationist is challenged to give an account of logical validity that does not force us (...) to countenance such relations. (The usual model-theoretic definition seems to presuppose that there is some special interpretation, the intended interpretation, such that truth simpliciter is truth on that intended interpretation. So if the deflationist adopts this sort of definition, the deflationist will be challenged to identify the intended interpretation without positing real reference relations.) Fortunately, a precise semantics compatible with the deflationist philosophy can be had as follows: First, we define a context as a certain sort of set constructed from a basis of literals (atomic sentences and negations of atomic sentences). This formal account of contexts has to be supplemented with an account of the conditions under which a structure satisfying the formal definition is the structure of that kind pertinent ot a given conversation. For each syntactic type of sentence, we define the conditions under which a sentence of that type is assertible relative to a context. In particular, we define the conditions under which sentences of the form " [p] is true" are assertible in a context, and we define the conditions under which sentences of the form "[p] is assertible in context G" are assertible in a context. Finally, logical validity is defined as preservation of assertibility in a context. It is demonstrated that this approach to semantics resists the semantic paradoxes. (shrink)

Consideration of a paradox originally discovered by John Buridan provides a springboard for a general solution to paradoxes within the Liar family. The solution rests on a philosophical defence of truth-value-gaps and is consistent (non-dialetheist), avoids ‘revenge’ problems, imports no ad hoc assumptions, is not applicable to only a proper subset of the semantic paradoxes and implies no restriction of the expressive capacities of language.

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...) (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a operator can be introduced in other settings. (shrink)

We introduce an epistemic theory of truth according to which the same rational degree of belief is assigned to Tr(. It is shown that if epistemic probability measures are only demanded to be finitely additive (but not necessarily σ-additive), then such a theory is consistent even for object languages that contain their own truth predicate. As the proof of this result indicates, the theory can also be interpreted as deriving from a quantitative version of the Revision Theory of Truth.

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic -- Deflationism (...) -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)