The first sentence in this essay is a lie. There is something odd about saying so, as has been known since ancient times. To see why, remember that all lies are untrue. Is the first sentence true? If it is, then it is a lie, and so it is not true. Conversely, suppose that it is not true. As we (viz., the authors) have said it, presumably with the intention of you believing it when it is not true, it is (...) a lie. But then it is true! (shrink)

The Liar paradox arises when we consider a sentence that says of itself that it is not true. If such self-referential sentences exist? and examples like?This sentence is not true? certainly suggest this?, then our logic and standard notion of truth allow to infer a contradiction: The Liar sentence is true and not true. What has gone wrong? Must we revise our notion of truth and our logic? Or can we dispel the common conviction that there are (...) such self-referential sentences? The present study explores the second path.0After comparing the Liar reasoning in formal and informal logic and showing that there are no Gödelian Liar sentences, the study moves on from the semantics of self-reference to the metaphysics of expressions and proposes a novel solution to the Liar paradox: Meaningful expressions are distinct from their syntactic bases and exist only relative to contexts. Detailed semantico-metaphysical arguments show that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the circular reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals. (shrink)

In this paper we concentrate on the nature of the liar paradox asa cognitive entity; a consistently testable configuration of properties. We elaborate further on a quantum mechanical model (Aerts, Broekaert and Smets, 1999) that has been proposed to analyze the dynamics involved, and we focus on the interpretation and concomitant philosophical picture. Some conclusions we draw from our model favor an effective realistic interpretation of cognitive reality.

The liar paradox is still an open philosophical problem. Most contemporary answers to the paradox target the logical principles underlying the reasoning from the liar sentence to the paradoxical conclusion that the liar sentence is both true and false. In contrast to these answers, Buddhist epistemology offers resources to devise a distinctively epistemological approach to the liar paradox. In this paper, I mobilise these resources and argue that the liar sentence is what (...) Buddhist epistemologists call a contradiction with one’s own words. I situate my argument in the works of Dignāga and Dharmakīrti and show how Buddhist epistemology answers the paradox. (shrink)

A semantic solution to the liar paradox (“This statement is not true”) is presented in this article. Since the liar paradox seems to evince a contradiction, the principle of non-contradiction is preliminarily discussed, in order to determine whether dismissing this principle may be reason enough to stop considering the liar paradox a problem. No conclusive outcome with respect to the value of this principle is aspired to here, so that the inquiry is not concluded (...) at this point and the option to explore an alternative, semantic, solution remains open. This proposed solution is focused on what the liar paradox expresses and what it fails to express. (shrink)

A historically popular response to the liar paradox (“this sentence is false”) is to say that the liar sentence is meaningless (or semantically defective, or malfunctions, or…). Unfortunately, like all other supposed solutions to the liar, this approach faces a revenge challenge. Consider the revenge liar sentence, “this sentence is either meaningless or false”. If it is true, then it is either meaningless or false, so not true. And if it is not true, then it (...) can’t be either meaningless or false, so it must be true. Either way, we are back in a paradox. This paper provides a detailed and exhaustive discussion of the options for responding to revenge on behalf of “meaningless” theories. Though I attempt to discuss all of the options fairly, I will ultimately opt for one specific response and discuss some of its challenges. Various technical and logical matters will be discussed throughout the paper, but my focus will be philosophical, throughout. My overall conclusion is that the “meaningless” strategy is at least as well off in the face of revenge as any other approach to the liar and related paradoxes. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a model of language and conceptual knowledge within the Predictive Processing framework. I suggest that the paradox arises as (...) a failure of synchronization between two ways of generating the liar situation in two different PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Although most scholars trace the Liar Paradox to Plato’s contemporary, Eubulides, the paper argues that Plato builds something very like the Liar Paradox into the very structure of his dialogues with significant consequences for understanding his views. After a preliminary exposition of the liar paradox it is argued that Plato builds this paradox into the formulation of many of his central doctrines, including the “Divided Line” and the “Allegory of the Cave” and the (...) “Ladder of Love”. Thus, Plato may have been the first to formulate the view that Graham Priest calls dialetheism, roughly, the view that some contradictions are, in an illuminating way, inescapable and true. The paper argues that Plato builds this Liar paradox into the formulation of his signature views because he holds that the attempt by finite human beings to theorize about transcendent realities results in the simultaneous necessity, and impossibility, of transgressing the limits of language—leading to the paradoxes. Finally, it is argued that the existence of these paradoxes in these Platonic doctrines is the direct result of an intrinsic hermeneutical circle in Plato’s aforementioned signature views. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a cognitive-computational model of language and conceptual knowledge within the Predictive Processing framework. I suggest that the paradox arises (...) as a failure of synchronization between two ways of generating the liar situation in two different PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a model of language and conceptual knowledge within thePredictive Processing(PP) framework. I suggest that the paradox arises as a (...) failure of synchronization between two ways of generating the liar situation in two different (idealized) PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates thePrinciple of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Here is the liar paradox. We have a sentence, (L), which somehow says of itself that it is false. Suppose (L) is true. Then things are as (L) says they are. (For it would appear to be a mere platitude that if a sentence is true, then things are as the sentence says they are.) (L) says that (L) is false. So, (L) is false. Since the supposition that (L) is true leads to contradiction, we can assert that (...) (L) is false. But since this is just what (L) says, (L) is then true. (For it would appear to be a mere platitude that if things are as a given sentence says they are, the sentence is true.) So (L) is true. So (L) is both true and false. Contradiction. (shrink)

ABSTRACTTwo rival scholars from Shiraz in Persia, Dawānī and Dashtakī engaged in a bitter and extended dispute over a range of metaphysical and logical issues. One of these was the liar paradox. Their debate on this point marked the most extensive scrutiny of the paradox in Arabic until that time. Dashtakī’s solution was to deny that the statement ‘What I say is false’ is true or false, on the ground that there is one statement and one application (...) of the falsity predicate. Given that – ex hypothesi – there is no other statement, there is no basis for a reiteration of the truth or falsity predicate and describing the statement itself as true or false. Dawānī’s solution was to deny that ‘What I say is false’ is a statement at all, and he argued that it is instead akin to a performative utterance such as ‘I hereby sell you this’. (shrink)

In Beyond the Limits of Thought [2002], Graham Priest argues that logical and semantic paradoxes have the same underlying structure (which he calls the Inclosure Schema ). He also argues that, in conjunction with the Principle of Uniform Solution (same kind of paradox, same kind of solution), this is sufficient to 'sink virtually all orthodox solutions to the paradoxes', because the orthodox solutions to the paradoxes are not uniform. I argue that Priest fails to provide a non-question-begging method to (...) 'sink virtually all orthodox solutions', and that the Inclosure Schema cannot be the structure that underlies the Liar paradox. Moreover, Ramsey was right in thinking that logical and semantic paradoxes are paradoxes of different kinds. (shrink)

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.

The liar paradox in its simplest form is the following argument. Consider the sentence 'this sentence is false'; call that the "liar sentence". Suppose the liar sentence is true. Then, since it says it is false, the liar sentence is false. So our supposition that it is true was mistaken, and the liar sentence must be false. But that's precisely what the liar sentence says, so it is true after all. The liar (...) sentence is, therefore, both true and false---an absurd result. ;Hans Herzberger has argued that several solutions to the Grelling paradox exhibit a common pattern: the very devices we introduce in order to avoid one version of the paradox are integral components of new versions of the paradox. Herzberger suggests that this is typical of semantic paradox in general. Chapters 1, 2, and 3 of this dissertation examine several possible solutions to the liar---especially appeals to language levels and to contexts---and attempt to locate these possible solutions in Herzberger's pattern. ;There is some reason to think that this pattern is inescapable---that is, that the liar paradox is ultimately unsolvable---so the remaining two chapters of this dissertation explore what follows from the claim that it is unsolvable. Chapter 4 develops an argument from the claim that the liar is unsolvable to the claim that we cannot understand ourselves. Chapter 5, then, attempts to make sense of this circumstance by invoking a Nietzschean view of thinking and truth---perspectivalism, which holds that all thinking concerns interpretation and that there are no uninterpreted facts. Given some perspectivalist assumptions, we can see how it is inevitable that we will be unable to understand ourselves---and how we can nevetherless attain a certain limited kind of understanding. Perspectivalism also recommends a certain intellectual humility: if everything is interpretation, then no view of mine can be a comprehensive and accurate view of the way the world really is. This by itself is not a surprising claim; the surprising claim is that the liar paradox is the basis of it. (shrink)

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “xis true” and satisfying the “dequotation schema”for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

Next SectionCharlie Pelling presents an impropriety paradox for the truth account of assertion. After solving his paradox I show that it is a version of the liar paradox. I then show that for any account of truth there is a strengthened liar-like paradox, and that for any solution to the strengthened liar paradox, there is a parallel solution to each of these “new” paradoxes.

There is no need to prove the significance to philosophy of the famous liar paradox. Attempts to solve it have not ceased over the course of many centuries, from hoary antiquity to our own day. The failure of attempts undertaken within the bounds of everyday, natural languages has led many philosophers, particularly of the positivist trend, to pessimistic conclusions with respect to the potentials of such languages. This is very clearly evident, for example, in the statement of A. (...) Tarski, who writes: "The liar paradox made its first appearance in our discussions as a kind of evil force possessed of great destructive energy. It compels us to reject all attempts to explicate the notion of truth for natural languages and to confine ourselves to the formalized languages of scientific reasoning." Tarski's viewpoint has been repeatedly criticized. We should prefer not to cast doubt on the significance of that highly valuable criticism. However, we cannot fail to observe that Tarski's notion retains its force so long as its basic argument - the insolubility of the liar paradox within the confines of natural languages - remains standing. (shrink)

The medieval Islamic solutions to the liar paradox can be categorized into three different families. According to the solutions of the first family, the liar sentences are not well-formed truth-apt sentences. The solutions of the second family are based on a violation of the classical principles of logic (e.g. the principle of non-contradiction). Finally, the solutions of the third family render the liar sentences as simply false without any contradiction. In the Islamic tradition, almost all the (...) well-known solutions of the third family are inspired by the solution proposed by At_īr al-Dīn al-Abharī (d. 1265). Providing a logical analysis of his discussion of the liar paradox, I show that his solution is based on a conception of truth according to which every sentence signifies, usually among other things, its own truth. This makes Abharī’s solution of the same spirit as certain solutions that were later developed in the Latin tradition, in particular by John Buridan (d. 1358) and Albert of Saxony (d. 1390). (shrink)

adr al-Dīn al-Dashtakī (d. 1498) has proposed a solution to the liar paradox according to which the liar sentence is a self-referential sentence in which the predicate ‘false’ is iterated. Discussing the conditions for the truth-aptness of the sentences with nested and iterated instances of the predicates ‘true’ and/or ‘false’, Dashtakī argued that the liar sentence is not truth-apt at all. In the tradition of Arabic logic, the central elements of Dashtakī's solution—the self-referentiality of the (...) class='Hi'>liar sentence and the implicit iteration of the predicate ‘false’—were initially highlighted in two earlier solutions proposed by Naīr al-Dīn al-ūsī (d. 1274) and Shams al-Dīn al-Samarqandī (d. 1322), respectively. Here I investigate all three solutions and show that Dashtakī's solution can be taken as a synthesis of the other two. None of these solutions seems to be convincing at the end of the day. Nevertheless, all of them include significant logical and philosophical insights. In particular, although Dashtakī's solution is not itself compelling, it is only a few steps away from a promising solution. The appendix to this paper includes translations of the relevant passages. (shrink)

Systems of paraconsistent logics violate the law of explosion: from contradictory premises not every formula follows. One of the philosophical options for interpreting the contradictions allowed as premises in these cases was put forward recently by Carnielli and Rodrigues, with their epistemic approach to paraconsistent logics. In a nutshell, the plan consists in interpreting the contradictions in epistemic terms, as indicating the presence of non-conclusive evidence for both a proposition and its negation. Truth, in this approach, is consistent and is (...) dealt with by classical logic. In this paper we discuss the fate of the Liar paradox in this picture. While this is a paradox about truth, it cannot be accommodated by the classical part of the approach, due to trivialization problems. On the other hand, the paraconsistent part does not seem fit as well, due to the fact that its intended reading is in terms of non-conclusive evidence, not truth. We discuss the difficulties involved in each case and argue that none of the options seems to accommodate the paradox in a satisfactory manner. (shrink)

In his Lectures on the History of Philosophy, Hegel develops a subtle analysis of Megarian paradoxes: the Liar, the Veiled Man and the Sorites. In this paper, we focus on Hegel's interpretation of...

Al-Taftāzānī introduces the Liar Paradox, in a commentary on al-Rāzī, in a short passage that is part of a polemic against the ethical rationalism of the Muʿtazila. In this essay, we consider his remarks and their place in the history of the Liar Paradox in Arabic Logic. In the passage, al-Taftāzānī introduces Liar Cycles into the tradition, gives the paradox a puzzling name—the fallacy of the “irrational root” —which became standard, and suggests a connection (...) between the paradox and what it tells us about truth and falsehood, and arguments for divine voluntarism and what they tell us about the nature of goodness and badness. On this last point, we also discuss a passage from al-Rāzī, which suggests similar connections. (shrink)

The story goes that Epimenides, a Cretan, used to claim that all Cretans are always liars. Whether he knew it or not, this claim is odd. It is easy to see it is odd by asking if it is true or false. If it is true, then all Cretans, including Epimenides, are always liars, in which case what he said must be false. Thus, if what he says is true, it is false. Conversely, suppose what Epimenides said is false. Then (...) some Cretan at some time speaks truly. This might not tell us anything about Epimenides. But if, to make the story simple, he were the only Cretan ever to speak, and this was the only thing he ever said, then indeed, he would have to speak truly. And we would then have shown that if what he said was false, it must be true. (shrink)

A new solution to the liar paradox is developed using the insight that it is illegitimate to even suppose (let alone assert) that a liar sentence has a truth-status (true or not) on the grounds that supposing this sentence to be true/not-true essentially defeats the telos of supposition in a readily identifiable way. On that basis, the paradox is blocked by restricting the Rule of Assumptions in Gentzen-style presentations of the sequent-calculus. The lesson of the (...) class='Hi'>liar is that not all assumptions are for free. One merit of this proposal is that it is free from the revenge problem. (shrink)

The medieval Islamic solutions to the liar paradox can be categorized into three different families. According to the solutions of the first family, the liar sentences are not well-formed truth-apt...

Jean Buridan has offered a solution to the Liar Paradox, i.e. to the problem of assigning a truth-value to the sentence ‘What I am saying is false’. It has been argued that either this solution is ad hoc since it would only apply to self-referencing sentences [Read, S. 2002. ‘The Liar Paradox from John Buridan back to Thomas Bradwardine’, Vivarium, 40 , 189–218] or else it weakens his theory of truth, making his ‘a logic without truth’ (...) [Klima, G. 2008. ‘Logic without truth: Buridan on the Liar’, in S. Rahman, T. Tulenheimo and E. Genot, Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, Berlin: Springer, 87–112 ; Dutilh Novaes, C. 2011. ‘Lessons on truth from mediaeval solutions to the Liar Paradox’, The Philosophical Quarterly, 61 , 58–78]. Against , I will argue that Buridan's solution by means of truth by supposition does not involve new principles. Self-referential sentences force us to handle supposition more carefully, which does not warrant the accusation of adhoccery. I will also argue, against (2), that it is exaggerated to assert that this solution leads to a ‘weakened’ theory of truth, since it is consistent with other passages of the Sophismata, which only gives necessary conditions for the truth of affirmative propositions, but sufficient conditions for falsity. (shrink)

Do logical paradoxes, like Eubulides’s Liar Paradox (the claim that the sentence “I am now lying” is true if and only if it is false), have any “existential” significance or are they mere brain puzzles for the mathematically minded? The paper argues that Randall Jarrell’s poem, “Eighth Air Force”, contains a poetic use of Eubulides’ Liar Paradox, spoken by Pontius Pilate’s wife in her statements about the “murder” of Jesus, in order to capture, symbolically, the inherent (...) universal duplicity (inauthenticity) of human life, specifically, the fact that human life, even in its true statements about fundamental moral-existential issues, is an inseparable blend of “truth” and “lies.”. (shrink)

This article presents an empirical examination of the consequences of the virtual entailment principle proposed by Jean Buridan to resolve the Liar paradox. This principle states that every sentence in natural language implicitly asserts its own truth. Adopting this principle means that the Liar sentence is not paradoxical but false, because its content is contradictory to what is virtually implied. As a result, humans should perceive the Liar sentence the same way as any other false sentence. (...) This solution to the Liar paradox received criticism for making ad hoc claims about the natural language. However, thanks to modern advancements in psychophysiology, it became possible to empirically investigate if the human brain really perceives the Liar sentence like a false sentence. We designed and conducted an experiment to examine brain activity in response to true sentences, false sentences and self-referential sentences. Our results provide support for the Buridan’s hypothesis and show that the Liar sentence is processed by the human brain identically to false sentences, while the Truthteller sentence is perceived identically to true sentences. This agrees with predictions derived from the virtual entailment principle and supports the idea that humans think with the logic of truth—a logic for which the truth is a designated value of its adequate semantics. (shrink)

We describe the earliest occurrences of the Liar Paradox in the Arabic tradition. e early Mutakallimūn claim the Liar Sentence is both true and false; they also associate the Liar with problems concerning plural subjects, which is somewhat puzzling. Abharī (1200-1265) ascribes an unsatisfiable truth condition to the Liar Sentence—as he puts it, its being true is the conjunction of its being true and false—and so concludes that the sentence is not true. Tūsī (1201-1274) argues (...) that self-referential sentences, like the Liar, are not truth-apt, and defends this claim by appealing to a correspondence theory of truth. Translations of the texts are provided as an appendix. (shrink)

In recent years, speech-act theory has mooted the possibility that one utterance can signify a number of different things. This pluralist conception of signification lies at the heart of Thomas Bradwardine’s solution to the insolubles, logical puzzles such as the semantic paradoxes, presented in Oxford in the early 1320s. His leading assumption was that signification is closed under consequence, that is, that a proposition signifies everything which follows from what it signifies. Then any proposition signifying its own falsity, he showed, (...) also signifies its own truth and so, since it signifies things which cannot both obtain, it is simply false. Bradwardine himself, and his contemporaries, did not elaborate this pluralist theory, or say much in its defence. It can be shown to accord closely, however, with the prevailing conception of logical consequence in England in the fourteenth century. Recent pluralist theories of signification, such as Grice’s, also endorse Bradwardine’s closure postulate as a plausible constraint on signification, and so his analysis of the semantic paradoxes is seen to be both well-grounded and plausible. (shrink)

In this paper we explain our pretense account of truth-talk and apply it in a diagnosis and treatment of the Liar Paradox. We begin by assuming that some form of deflationism is the correct approach to the topic of truth. We then briefly motivate the idea that all T-deflationists should endorse a fictionalist view of truth-talk, and, after distinguishing pretense-involving fictionalism (PIF) from error- theoretic fictionalism (ETF), explain the merits of the former over the latter. After presenting the (...) basic framework of our PIF account of truth-talk, we demonstrate a few advantages it offers over T-deflationist accounts that do not explicitly acknowledge pretense at work in the discourse. In turning to the Liar Paradox, we explain how the quasi-anaphoric functioning that our account attributes to truth-talk provides a diagnosis of the Liar Paradox (and other instances of semantic pathology) as having no content—in the sense of not specifying any of what we call M-conditions. At the same time, however, we vindicate the intuition that we can understand liar sentences, thereby avoiding one standard objection to “meaningless strategy” responses to the Liar Paradox. With this diagnosis in place, we then, by way of treatment, introduce a new predicate, ‘semantically defective’, and show how the explanation we give for its application allows for a consistent, yet revenge-immune, (dis)solution of the Liar Paradox, and semantic pathology generally. (shrink)

The ancient puzzle of the Liar was shown by Tarski to be a genuine paradox or antinomy. I show, analogously, that certain puzzles of contemporary game theory are genuinely paradoxical, i.e., certain very plausible principles of rationality, which are in fact presupposed by game theorists, are inconsistent as naively formulated. ;I use Godel theory to construct three versions of this new paradox, in which the role of 'true' in the Liar paradox is played, respectively, by (...) 'provable', 'self-evident', and 'justifiable'. I also construct in modal operator logic a paradox involving reflexive empirical reasoning. Unlike the paradox of the Liar, the paradox of reflexive reasoning does not depend on self-reference. ;I consider various solutions to the Liar paradox and evaluate how well these solutions cope with the paradox of reflexive reasoning. I then formalize the solution to the paradoxes which I favor: the indexical-hierarchical approach, first sketched out by Charles Parsons and Tyler Burge. In this solution, occurrences of the predicate 'true' in sentence-tokens are contextually relativized to levels of a hierarchy. Drawing also on some brief remarks of Bertrand Russell and Charles Parsons, I develop an account of the kind of schematic generality needed for this theory to be statable. ;Finally, I demonstrate that the principles shown to be paradoxical are in fact presupposed by contemporary game theorists in their reliance on the notion of common knowledge or, more precisely, mutual belief. I create novel analyses of and corresponding solutions to several recalcitrant puzzles within game theory, including the "chain-store paradox". (shrink)