In an interesting experimental study, Bonini et al. (1999) present partial support for truth-gap theories of vagueness. We say this despite their claim to find theoretical and empirical reasons to dismiss gap theories and despite the fact that they favor an alternative, epistemic account, which they call ‘vagueness as ignorance’. We present yet more experimental evidence that supports gap theories, and argue for a semantic/pragmatic alternative that unifies the gappy supervaluationary approach together with its glutty relative, the subvaluationary approach.

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n -valued logics, each one of which is not (...) equivalent to any k -valued logic with k < n. (shrink)

In ‘A Consistent Way with Paradox’, Laurence Goldstein (2009) clarifies his solution to the liar, which he touts as revenge immune . In addition, he (Ibid.) responds to one of the objections that Armour-Garb and Woodbridge (2006) raise against certain solutions to the open pair and argues that his proffered solution to the liar family of paradoxes undermines what they (Ibid.) call the dialetheic conjecture . In this paper, after critically evaluating Goldstein’s response to A-G&W, I turn to his proposed (...) solution to the liar paradox, where I show that it is difficult to see how it manages to avoid that conjecture. (shrink)

Philosophical work on truth covers two streams of inquiry, one concerning the nature (if any) of truth, the other concerning truth-related paradox, especially the Liar. For the most part these streams have proceeded fairly independently of each other. In his Deflationary Truth and the Liar (JPL 28:455–488, 1999) Keith Simmons argues that the two streams bear on one another in an important way; specifically, the Liar poses a greater problem for deflationary conceptions of truth than it does for inflationist conceptions. (...) We agree with Simmons on this point; however, we disagree with his main conclusion. In a nutshell, Simmons' main conclusion is that deflationists can solve the Liar only by compromising deflationism. If Simmons is right, then deflationists cannot solve the Liar paradox. In this paper we argue that, pace Simmons, there is an approach to the Liar that is available to deflationists, namely dialetheism. (shrink)

Over the past 25 years, Graham Priest has ably presented and defended dialetheism, the view that certain sentences are properly characterized as true with true negations. Our goal here is neither to quibble with the tenability of true, assertable contradictions nor, really, with the arguments for dialetheism. Rather, we wish to address the dialetheist's treatment of cases of semantic pathology and to pose a worry for dialetheism that has not been adequately considered. The problem that we present seems to have (...) broader bite, afflicting both consistent and inconsistent proposals for resolving semantic pathology. Thus, while our primary goal is to uncover some important connections between dialetheism, semantic pathology, and other, more general issues, the problem that we pose might be a worry for anyone who aims to resolve semantic pathology - consistently or not. (shrink)

It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so immune from revision. The other, radical (...) reading of Quine does not exempt logic from the attack on analyticity and a priority. Logical truths and inferences are themselves part of the web of belief, and the same global methodology applies to logic as to any other part of the web, such as theoretical chemistry or ordinary beliefs about ordinary objects. Everything, including logic, is up for grabs in our struggle for holistic confirmation. The purpose of this paper is to examine the law of non-contradiction, and the concomitant principle of ex falso quodlibet, from the perspective of the principles advocated by the radical Quine. I show that he has no compelling reason to accept either of these. To put it bluntly, neither the law of non-contradiction nor the rule of ex falso quodlibet is empirically confirmed, and these principles fare poorly on the various criteria for theory acceptance on the methodology of the radical Quine. So the radical Quine is led rather quickly and rather directly into something in the neighborhood of Graham Priest’s dialetheism. (shrink)

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

A common and much-explored thought is ?ukasiewicz's idea that the future is ?indeterminate??i.e., ?gappy? with respect to some claims?and that such indeterminacy bleeds back into the present in the form of gappy ?future contingent? claims. What is uncommon, and to my knowledge unexplored, is the dual idea of an overdeterminate future?one which is ?glutty? with respect to some claims. While the direct dual, with future gluts bleeding back into the present, is worth noting, my central aim is simply to sketch (...) and briefly explore an alternative glutty-future view, one that is conservative?indeed, entirely classical?with respect to the present. The structure of the paper runs as follows. ?1 briefly sketches the target gap picture of an indeterminate future yielding gappy claims at the present. ?2 presents the direct dual idea?a glut picture of an overdeterminate future yielding glutty claims at present. ?3 sketches the central idea, a more interesting glut picture in which the future contains contradictory states but the present remains entirely classical. ?4 contains a general defence of the idea, leaving it open as to whether the gappy-future view enjoys substantive virtues over the proposed glutty-future view of ?3. (shrink)

There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does not (...) hold unrestrictedly – that in peculiar circumstances the same thing may at the same time be and not be, and contradictions may obtain in the world. This book opens with an examination of the famous logical paradoxes that appear to speak on behalf of contradictions (e.g., the Liar paradox, the set-theoretic paradoxes such as Cantor’s and Russell’s), and of the reasons for the failure of the standard attempts to solve them. It provides, then, an introduction to paraconsistent logics – non-classical logics in which the admission of contradictions does not lead to logical chaos –, and their astonishing applications, going from inconsistent data base management to contradictory arithmetics capable of circumventing Gödel’s celebrated Incompleteness Theorem. The final part of the book discusses the philosophical motivations and difficulties of dialetheism, and shows how to extract from Aristotle’s ancient words a possible reply to the dialetheic challenge. How to Sell a Contradiction will appeal to anyone interested in non-classical logics, analytic metaphysics, and philosophy of mathematics, and especially to those who consider challenging our most entrenched beliefs the main duty of philosophical inquiry. (shrink)

In his famous work on vagueness, Russell named “fallacy of verbalism” the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories – the non-adjunctive and relevant approaches –, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of (...) their underlying semantics. The meaning of “intuitive reading” is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense – therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency – and this could be a welcome result for a dialetheist. (shrink)

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.

The paper discusses which modal principles should hold for a truth operator answering to the truth theory of internal realism. It turns out that the logic of truth in internal realism is isomorphic to the modal system S4.

In the first part the paper rehearses the main arguments why to be a dialetheist (i.e. why to assume that some contradictions are true). Dialetheism, however, has been criticised as irrational or self-refutating. Therefore the second part of the paper outlines one way to make dialetheism rational assertable. True contradictions turn out to be both believable and assertable. The argument proceeds by setting out basic principles of assertion and denial, and employing bivalent truth value operators.

Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (shrink)

Doris Olin's Paradox is a very helpful book for those who want to be introduced to the philosophical treatment of paradoxes, or for those who already have knowledge of the general area and would like to have a helpful resource book.

Beginning with the paradoxes of zombie twins, we present an argument that dualism is both true and false. We show that avoiding this contradiction is impossible. Our diagnosis is that consciousness itself engenders this contradiction by producing contradictory points of view. This result has a large effect on the realism/anti-realism debate, namely, it suggests that this debate is intractable, and furthermore, it explains why this debate is intractable. We close with some comments on what our results mean for metaphysics and (...) philosophy, in general. (shrink)

It has been proposed that the law of non-contradiction be revised to permit the simultaneous truth and falsity of the key sentences of the logical paradoxes, e.g., This sentence is false. In an attempt to show to what extent this bizarre suggestion of inconsistent models or truth-value gluts is a coherent suggestion it is proved that a first-order language for number theory can be semantically closed by having its own global truth predicate under some non-standard interpretation and thus that it (...) actually can contain the Liar sentence. It is proved that in this interpretation the Liar sentence is both true and false, although not every sentence is. (shrink)

This article argues that Nietzsche's transvaluation project refers not to a mere inversion or negation of a set of values but, instead, to a different conception of what a value is and how it functions. Traditional values function within a standard logical framework and claim legitimacy and bindingness based on exogenous authority with absolute extension. Nietzsche regards this framework as unnecessarily reductive in its attempted exclusion of contradiction and real opposition among competing values and proposes a nonstandard, dialetheic model of (...) valuation. (shrink)

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these (...) systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity". (shrink)

The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox is a version of the Liar, (...) unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias. (shrink)

In this article, I consider the possibility of interpreting Hegel's dialectic as dialetheism. After a first basic recapitulation about the meaning of the words ?dialetheism? and ?dialectic? and a consideration of Priest's own account of the relation between dialectical and dialetheic logic in 1989, I discuss some controversial issues, not directly considered by Priest. As a matter of fact, the reflection on paraconsistent logics and dialetheism has enormously grown in recent years. In addition, the reception of Hegel's logic and metaphysics (...) has also impressively improved. So I suggest that the discussion about the binomial dialectic/dialetheism should be reopened, on these new bases. (shrink)

I defend the consistency of affirming the right to life while rejecting universal healthcare and liveable income programmes. I also defend the rationality of accepting inconsistency.

This paper compares two proposed solutions to the liar paradox, both of which involve revisions to classical semantics. The first, that of truth value gaps, denies that all sentences are true or false. The second, that of truth value gluts, asserts that some sentences are true and false. A natural question about these proposals is, ?Do they offer equally good (or bad) solutions, or is one better than the other?? Parsons 1990 suggested an answer to this question, arguing that for (...) every problem for one solution, there is a parallel problem for the other. Since then, several people have given arguments for and against the idea that the two face parallel problems. This paper advances the debate in two ways. First, it answers four attempts by Graham Priest to reject the parallel. In particular, it discusses alleged advantages for the gap solution with regard to the teleological account of truth, an extended liar paradox, Berry's paradox, and inexpressibility. Second, it discusses where, if anywhere, the parallel may be broken. (shrink)

Spinoza’s letter of June 2, 1674 to his friend Jarig Jelles addresses several distinct and important issues in Spinoza’s philosophy. It explains briefly the core of Spinoza’s disagreement with Hobbes’ political theory, develops his innovative understanding of numbers, and elaborates on Spinoza’s refusal to describe God as one or single. Then, toward the end of the letter, Spinoza writes: With regard to the statement that figure is a negation and not anything positive, it is obvious that matter in its totality, (...) considered without limitation [indefinitè consideratam], can have no figure, and that figure applies only to finite and determinate bodies. For he who says that he apprehends a figure, thereby means to indicate simply this, that he apprehends a determinate thing and the manner of its determination. This determination therefore does not pertain to the thing in regard to its being [esse]; on the contrary, it is its non-being [non-esse]. So since figure is nothing but determination, and determination is negation [Quia ergo figura non aliud, quam determinatio, et determinatio negatio est], figure can be nothing other than negation, as has been said. Arguably, what is most notable about this letter is the fate of a single subordinate clause which appears in the last sentence of this passage: et determinatio negatio est. That clause was to be adopted by Hegel and transformed into the slogan of his own dialectical method: Omnis determinatio est negatio (Every determination is negation). Of further significance is the fact that, while Hegel does credit Spinoza with the discovery of this most fundamental insight, he believes Spinoza failed to appreciate the importance of his discovery. The issue of negation and the possibility of self-negation stand at the very center of the philosophical dialogue between the systems of Spinoza and Hegel, and in this paper I will attempt to provide a preliminary explication of this foundational debate between the two systems. In the first part of the paper I will argue that the “determination is negation” formula has been understood in at least three distinct senses among the German Idealists, and as a result many of the participants in the discussion of this formula were actually talking past each other. The clarification of the three distinct senses of the formula will lead, in the second part of the paper, to a more precise evaluation of the fundamental debate between Spinoza and Hegel (and the German Idealists in general) regarding the possibility (or even necessity) of self-negation. In this part I will evaluate the validity of each interpretation of the determination formula, and motivate the positions of the various participants in the debate. (shrink)

forthcoming in Philosophy and Phenomenological Research.

Neil Tennant and Joseph Salerno have recently attempted to rigorously formalize Michael Dummett's argument for logical revision. Surprisingly, both conclude that Dummett commits elementary logical errors, and hence fails to offer an argument that is even prima facie valid. After explicating the arguments Salerno and Tennant attribute to Dummett, I show how broader attention to Dummett's writings on the theory of meaning allows one to discern, and formalize, a valid argument for logical revision. Then, after correctly providing a rigorous statement (...) of the argument, I am able to delineate four possible anti-Dummettian responses. Following recent work by Stewart Shapiro and Crispin Wright, I conclude that progress in the anti-realist's dialectic requires greater clarity about the key modal notions used in Dummett's proof. (shrink)

Graham Priest's book In Contradiction (Dordrecht: Martinus Nijhoff, 1987) is a bold and well argued for defense of the existence of true contradictions. Priest's case for true contradictions -- or «dialetheias», as he calls them -- is by no means the only one in contemporary analytical philosophy, let alone in philosophy tout court . In some sense, other defenses of the existence of true contradictions are less philosophically «heterodox» than his is, since, unlike Priest's orientation, other approaches are closer to (...) prevailing ideas in mainstream («Quinean») analytical philosophy, whereas Priest's leanings are strongly anti realist, and not distant from the logical empiricism of the thirties. (shrink)

One might ask of two or more texts—what can be inferred from them, taken together? If the texts happen to contradict each other in some respect, then the unadorned answer of standard logic is EVERYTHING. But it seems to be a given that we often successfully reason with inconsistent information from multiple sources. The purpose of this paper is to attempt to develop an adequate approach to accounting for this given.

A dialetheia is a sentence, A, such that both it and its negation, A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and (...) false. (shrink)

Dialetheism is the view that some contradictions are true. This is a view which runs against orthodoxy in logic and metaphysics since Aristotle, and has implications for many of the core notions of philosophy. Doubt Truth to Be a Liar explores these implications for truth, rationality, negation, and the nature of logic, and develops further the defense of dialetheism first mounted in Priest's In Contradiction, a second edition of which is also available.

In Contradiction advocates and defends the view that there are true contradictions (dialetheism), a view that flies in the face of orthodoxy in Western philosophy since Aristotle. The book has been at the center of the controversies surrounding dialetheism ever since its first publication in 1987. This second edition of the book substantially expands upon the original in various ways, and also contains the author's reflections on developments over the last two decades. Further aspects of dialetheism are discussed in the (...) companion volume, Doubt Truth to be a Liar, also published by Oxford University Press in 2006. (shrink)

The Law of Non-Contradiction - that no contradiction can be true - has been a seemingly unassailable dogma since the work of Aristotle, in Book G of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the 'law', considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into (...) a venerable principle of logic, one that raises questions at the very centre of logic itself. The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a 'dialetheic' resolution of that paradox. The editors contribute an introduction which surveys the issues and serves to frame the debate, and a useful bibliography offering a guide to further reading. This volume will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them. (shrink)

Proponents of “truth-value glut” responses to the paradoxes of self-reference, such as Priest [6, 7] argue that “truth-value gap” analyses of the paradoxes fall foul of the strengthened liar paradox: “this sentence is not true.” If we pay attention to the role of assertion and denial and the behaviour of negation in both “gap” and “glut” analyses, we see that the situation with these approaches has a pleasing symmetry: gap approaches take some denials to fail to be expressible by negation, (...) and glut approaches take some negations to not express denials. But in the light of this symmetry, considerations against a gap view point to parallel considerations against a glut view. Those who find some reason to prefer one view over another (and this is almost everyone) must find some reason to break this symmetry. (shrink)

The purpose of this essay is to shed some light on a certain type of sentence, which I call a borderline contradiction. A borderline contradiction is a sentence of the form F a ∧ ¬F a, for some vague predicate F and some borderline case a of F , or a sentence equivalent to such a sentence. For example, if Jackie is a borderline case of ‘rich’, then ‘Jackie is rich and Jackie isn’t rich’ is a borderline contradiction. Many theories (...) of vague language have entailments about borderline contradictions; correctly describing the behavior of borderline contradictions is one of the many tasks facing anyone offering a theory of vague language. Here, I first briefly review claims made by various theorists about these borderline contradictions, attempting to draw out some predictions about the behavior of ordinary speakers. Second, I present an experiment intended to gather relevant data about the behavior of ordinary speakers. Finally, I discuss the experimental results in light of several different theories of vagueness, to see what explanations are available. My conclusions are necessarily tentative; I do not attempt to use the present experiment to demonstrate that any single theory is incontrovertibly true. Rather, I try to sketch the auxiliary hypotheses that would need to be conjoined to several extant theories of vague language to predict the present result, and offer some considerations regarding the plausibility of these various hypotheses. In the end, I conclude that two of the theories I consider are better-positioned to account for the observed data than are the others. But the field of logically-informed research on people’s actual responses to vague predicates is young; surely as more data come in we will learn a great deal more about which (if any) of these theories best accounts for the behavior of ordinary speakers. (shrink)

According to metaphysical realism, there may be features of reality which we cannot conceive. If this thesis of cognitive closure is inconsistent, then, pace dialetheism, metaphysical realism proves incoherent. Recently, Graham Priest has revived Berkeley's idealist argument meant to show that cognitive closure is inconsistent. If cogent, this argument poses a threat to metaphysical realism. I argue that while Priest's reconstruction of Berkeley's argument may be seen to be paradoxical on one interpretation of ‘conceive’, that interpretation is not the intended (...) one. On the intended interpretation, the argument fails and hence leaves metaphysical realism unassailed. (shrink)

The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in the world. For example, (...) there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of non-contradiction faces; the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of non-contradiction understood as a metaphysical principle. (shrink)

The purpose of this paper is to examine the status of logic from a metaphysical point of view – what is logic grounded in and what is its relationship with metaphysics. There are three general lines that we can take. 1) Logic and metaphysics are not continuous, neither discipline has no bearing on the other one. This seems to be a rather popular approach, at least implicitly, as philosophers often skip the question altogether and go about their business, be it (...) logic or metaphysics. However, it is not a particularly plausible view and it is very hard to maintain consistently, as we will see. 2) Logic is prior to metaphysics and has metaphysical implications. The extreme example of this kind of approach is the Dummettian one, according to which metaphysical questions are reducible to the question of which logic to adopt. 3) Metaphysics is prior to logic, and your logic should be compatible with your metaphysics. This approach suggests an answer to the question of what logic is grounded in, namely, metaphysics. Here I will defend the third option. (shrink)