Liar Paradox Research Articles

Dialetheism and the countermodel problem

AbstractAccording to some dialetheists, we ought to reject the distinction between object and meta‐languages. Given that dialetheists advocate truth‐value gluts within their object‐language, whether in order to solve the liar paradox or for some other reason, this rejection of the object‐/meta‐language distinction comes with the commitment to use a glutty metatheory. While it has been pointed out that a glutty metatheory brings with it expressive deficiencies, we highlight here another complication arising from the use of a glutty metatheory, this time evidential in nature. According to this countermodel problem, while the thoroughgoing dialetheist who embraces a glutty metatheory can justify their acceptance of a rule of inference's invalidity using countermodels, to justify their renunciation of an unwanted rule they actually require the means to warrant their rejection of the rule's validity—which cannot be supplied by countermodels based on a standard dialetheic semantics. We end by sketching out a possible solution for the thoroughgoing dialetheist using a bilaterialist semantics.

Defusing a Paradox to a Hypodox

One way of resolving a paradox is to defuse it to a hypodox. This way is relatively unknown though. The goal of this paper is to explain this way with varied examples. The hypodoxes are themselves a broad class: both the Truth-teller and the 21st birthday of someone born on 29th February can be construed as hypodoxes. The most familiar kind of relation between paradoxes and hypodoxes is exemplified by the relation between the Liar and the Truth-teller. This article concerns a second kind where a paradox is defused to a hypodox by restricting or rejecting some granted principles. The Liar paradox has this second kind of relation to a Liar hypodox, which will be introduced. In some cases, defusing a paradox to a hypodox is only a partial resolution, as the hypodox itself may then need resolving. Even so, such a partial resolution decomposes a complex problem into more easily understood problems. Moreover, I compare the result of defusing a paradox to a hypodox with the results of resolving paradoxes in other ways. I give four examples. The first is mainly pedagogic, concerning a birthday. The second is a lightweight legal case, presenting a parking voucher paradox. The third is a formal system in which a Liar and Liar-like sentences are hypodoxical. The fourth is a philosophical critique of ways of solving Bertrand’s chord paradox.

The many faces of the Liar Paradox

The Liar Paradox is a classic argument that creates a contradiction by reflection on a sentence that attributes falsity to itself: ‘this sentence is false’. In our paper we will discuss the ways in which the Liar sentence (and its paradoxical argument) can be represented in first-order logic. The key to the representation is to use first-order logic to model a self-referential language. We will also discuss several related sentences, like the Liar cycles, the empirical versions of the Liar and the Truth teller sentences.

On the structural properties of paradoxes: the distinction between formal language and natural language that comes with the use of the liar paradox

This study delves into the intriguing realm of paradoxes that have long fascinated philosophers and logicians throughout history. It begins by discussing the nature and purpose of paradoxes, ranging from their role in entertainment to their capacity to reveal flaws within logical systems. This work emphasizes the challenge paradoxes pose to the completeness of systems and the subsequent development of axiomatic systems that aim to eliminate paradoxes. Rather than providing definitive solutions to paradoxes, the primary aim of this study is to defend the idea that systems containing paradoxes can coexist with completeness. The focus is on categorizing paradoxes, with special attention given to the group known as liar paradoxes, including Russell’s famous variation. The text demonstrates how these paradoxes are intrinsically linked to the principle of non-contradiction and argues that the truth and contradiction of propositions are both the cause and consequence of these paradoxes, presenting a dilemma within the system. The work introduces the concept of Aristotelian Sets (A-Sets) and Empty Sets as potential solutions to these paradoxes. It explores the idea that these sets, when carefully defined, can provide a meaningful representation of individual substances and predicates without violating the principle of non-contradiction. By proposing the inclusion of non-existents within a naive set theory and introducing A-Sets, this work seeks to contribute to the ongoing discourse surrounding paradoxes and their resolution. Ultimately, this study offers a fresh perspective on the handling of paradoxes, emphasizing the importance of reevaluating the foundations of formal and natural languages in the pursuit of a more comprehensive understanding of logic and philosophy.

Saving logic from paradox via nonclassical recapture

The Liar paradox arguably shows that a coherent and self-applicable notion of truth is governed by nonclassical logic. It then seems natural to conclude that classical logic is inadequate for defining a truth theory. In this article, we argue that this is not the case. In the spirit of Reinhardt (Math Logic Formal Syst 94:227, 1985; J Philos Logic 15:219–251, 1986), and in analogy with Hilbert’s program for the foundation of classical mathematics, we will articulate an instrumentalist justification for the use classical logic: it will be argued that classical reasoning is a useful but dispensable instrument, which can yield philosophically adequate truth theories.

The Liar Paradox and “Meaningless” Revenge

The Liar Paradox and “Meaningless” Revenge

Inferential Deflationism

Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. We implement the deflationist insight in a bilateral theory by taking the meaning of the truth predicate to be explained by its inferential relation to assertion. We combine this account of the meaning of the truth predicate with a new diagnosis of the liar paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth, which solves the liar paradox in a principled manner. We end by showing that our theory and simple extensions thereof have the resources to axiomatize the internal logic of several supervaluational hierarchies, including Cantini’s. This solves open problems of Halbach (2011) and Horsten (2011).

The import of hypodoxes for the Liar and Russell’s paradoxes

Is the set of all self-membered sets, S, a member of itself? In naive set theory, this is Russell’s hypodox. By the Laws of Excluded Middle and Non-contradiction, S is a member of itself xor it is not, but no principle of classical logic or naive set theory determines which. (Herein, ‘xor’ extends English with a specifically exclusive disjunction.) As a hypodox, the Truth-teller is a sentence that says of itself simply that it is true; by the above mentioned principles, the Truth-teller is true xor not true, but no principle of classical logic or naive truth determines which. Many concepts that have paradoxes have related hypodoxes, although the focus here is on these two. I argue that the Truth-teller does not rely on a principle of truth, in particular it does not rely on the T-schema. I also argue that Russell’s hypodox does not rely on Comprehension. Yet I relate Russell’s paradox and hypodox based on involution functions. The Liar and the Truth-teller are also related by means of involutions. Relations based on these involution functions can be represented in duality squares. Although hypodoxes are not paradoxes, these involution functions and duality relations warrant some commonality in a correct solution to a paradox and its dual hypodox. Using this and some other premises, I provide novel arguments supporting non-classical solutions for the Liar paradox and the significance of Russell’s hypodox for solutions to Russell’s paradox.

Is truth inconsistent?

AbstractA popular and enduring approach to the liar paradox takes the concept of truth to be inconsistent. Very roughly, truth is an inconsistent concept if the central principles of this concept (taken together) entail a contradiction, where one of these central principles is Tarski's T‐schema for truth: a sentence S is true if and only if p, (where S says that p). This article targets a version of Inconsistentism which: retains classical logic and bivalence; takes the truth‐predicate “is true” to pick out a property (and determine a non‐empty extension relative to a given world); and holds that liar sentences exhibit a certain kind of indeterminacy in truth‐value. Call such a view Modest Inconsistentism since it is somewhat more conservative in its outlook than various other forms of Inconsistentism. Such a modest view has its attractions: we retain the thesis that the liar sentence is meaningful; we get to respect the claims that there are truths and that there is a property of truth; we get to keep classical logic and bivalence; and, prima facie, no strengthened liar paradox is in the offing. The main aim in this paper is to show that Modest Inconsistentism, despite its initial attractions, is in deep trouble—because it does, after all, give rise to a strengthened liar paradox. We shall also see that there are related kinds of theory which are also subject to the same worry.

The fixed points of belief and knowledge

Abstract Self-referential sentences have troubled our understanding of language for centuries. The most famous self-referential sentence is probably the Liar, a sentence that says of itself that it is false. The Liar Paradox has encouraged many philosophers to establish theories of truth that manage to give a proper account of the truth predicate in a formal language. Kripke’s Fixed Point Theory from 1975 is one famous example of such a formal theory of truth that aims at giving a plausible notion of truth by allowing truth value gaps. However, not only the concept of truth gives rise to paradoxes. A syntactical treatment of epistemic notions like belief and knowledge leads to contradictions that very much resemble the Liar Paradox. Therefore, it seems to be fruitful to apply the established theories of truth to epistemic concepts. In this paper, I will present one such attempt of solving the epistemic paradoxes: I adapt Kripke’s Fixed Point Theory and interpret truth, knowledge and belief within the framework of a partial logic. Thereby I find not only the fixed point of truth but also the fixed points of knowledge and belief. In this fixed point, the predicates of truth, belief and knowledge find their definite interpretation and the paradoxes are avoided.

POSSIBLE SOLUTION TO GÖDEL'S INCOMPLETENESS THEOREM AND GÖDEL'S SECOND THEOREM

Plan
 
 Statement of the problem
 Find information to solve the problem
 Clarification of information to solve the problem
 Formulation of the lemma to solve the problem
 Search for a principle to solve the problem
 Proving the lemma that every mathematical system needs an observer
 Proving the lemma that every mathematical system needs an observer, whose existence only he can know, because whose existence cannot be proved
 
 8.Mathematical record of problem solving
 
 Confirmation of the consistency and completeness of the formal system for one observer.
 Necessary and sufficient conditions for the formation of a consistent system for society (group of observers)
 Solving the liar paradox as a byproduct of solving the problem
 Using observer’s view on The Ship of Theseus
 The unexpected hanging paradox
 The sorites paradox
 The philosophical basis of the theorem proof
 Some reasonable conclusions from this work that can be applied in other scientific
 Conclusions of solving the problem
 My sincere thanks for the provided knowledge / information sources

A FIFTEENTH-CENTURY OTTOMAN SOLUTION TO THE LIAR PARADOX BY ḪAṬĪBZĀDE MUḤYIDDĪN

AbstractThis paper deals with a solution to the infamous liar paradox, usually known in the Arabic literature as Maġlaṭat al-ǧaḏr al-aṣamm. The solution is raised by a fifteenth-century Ottoman treatise that is attributed, among others, to Ḫaṭībzāde Muḥyiddīn Efendī. The paper also compares it with the solution by the contemporary Persian philosopher, Ǧalāl al-Dīn al-Dawānī. The short treatise devoted to the paradox is one of the few works by Ottomans on the subject and it comprehensively addresses the paradox in its two forms. An analysis of the solution offered by the treatise to the paradox, the paper aims to bring Ottoman discussions of the liar to the attention of contemporary scholarship and contribute to filling the obvious gap in the literature on the paradox in Islamic thought.

Dashtakī's Solution to the Liar Paradox: A Synthesis of the Earlier Solutions Proposed by Ṭūsī and Samarqandī

Ṣadr 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 liar sentence and the implicit iteration of the predicate ‘false’—were initially highlighted in two earlier solutions proposed by Naṣīr al-Dīn al-Ṭūsī (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.

Trouble no more: how non-truth-functionality makes the alethic indeterminacy solution to the Liar Paradox viable

ABSTRACT Jay Newhard [2021. “Alethic Undecidability and Alethic Indeterminacy.” Synthese 199: 2563–2574. doi:10.1007/s11229-020-02900-z] proposes a novel solution to the Liar Paradox, which he calls the alethic indeterminacy solution to the Liar Paradox. Bradley Armour-Garb [2021. “Troubles for Alethic Indeterminacy.” Inquiry. doi:10.1080/0020174X.2021.1932166] raises a pair of objections to the alethic indeterminacy solution. Both objections are based upon the alethic indeterminacy solution’s alleged commitment that the truth conditions for a Liar Sentence are indeterminate, and therefore not true. In this paper, this alleged commitment is shown to be mistaken. The alethic indeterminacy solution is compatible with maintaining that the truth conditions for a Liar Sentence are a non-truth-functional biconditional which is true, and not indeterminate. Two independent accounts of non-truth-functional biconditionals are defended. The first account is based on the claim that indicative conditionals express a non-truth-functional dependency relation between consequent and antecedent. The second account is based upon connected sentences having interdependent truth conditions, which can occur with any polyadic connective. These accounts are fully general, not ad hoc measures. Each account solves both of Armour-Garb’s objections. With both objections met, the alethic indeterminacy solution is shown to be a viable solution to the Liar Paradox worth further development.

Determinism in hidden variable interpretation and Turings halting problem

Heisenberg deduced the famous uncertainty principle which shows that there exist several conjugated quantities that can never be measured precisely at the same time. Further, Copenhagen interpretation gives a new perspective about quantum mechanics, which claims that particles do not have properties like position or momentum until people measure it. Hence, the famous EPR paradox was proposed to question the realism and locality of Quantum mechanics, which leads to the Hidden Variable explanation. However, this theorem was proved wrong by John Bell in 1964 with Bell Inequality. In addition, Hidden Variable Interpretation was developed by De Broglie and Bohm, they came up with the Bohmian Mechanics, which can be considered as Non-local Hidden Variable theorem. This interpretation gives physical meaning to waves. Unlike Copenhagen Interpretation, this theorem claims that particles do have a determined position. Some people may argue that it can be proved contradicted by using the Turing method (self-reference). Because if the algorithm represented by physics law described Hidden Mechanics to determine particles state, then the future is determined and predictable. Therefore, another algorithm can be established based on that, which will lead to Liar Paradox. This article will briefly introduce the uncertainty principle and some interpretations about quantum mechanics. Furthermore, this article will combine some ideas in Alan Turings Halting Problem to the universe of non-local hidden variables as a thought experiment, which involve self-reference, to give an interesting result of locality and determinism.

Buddhist Epistemology and the Liar Paradox

ABSTRACT 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.

The Liar Paradox: A Case of Mistaken Truth Attribution

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.

Truth and Falsity in Buridan’s Bridge

This paper revisits Buridan’s Bridge paradox (Sophismata, chapter 8, Sophism 17), itself close kin to the Liar paradox, a version of which also appears in Bradwardine’s Insolubilia. Prompted by the occurrence of the paradox in Cervantes’s Don Quixote, I discuss and compare four distinct solutions to the problem, namely Bradwardine’s “just false” conception, Buridan’s “contingently true/false” theory, Cervantes’s “both true and false” view, and then the “neither true simpliciter nor false simpliciter” account proposed more recently by Jacquette. All have in common to accept that the Bridge expresses a truth-apt proposition, but only the latter three endorse the transparency of truth. Against some previous commentaries I first show that Buridan’s solution is fully compliant with an account of the paradox within classical logic. I then argue that Cervantes’s insights, as well as Jacquette’s treatment, are both supportive of a dialetheist account, and Jacquette’s in particular of the strict-tolerant account of truth. I defend dialetheist intuitions (whether in LP or ST guise) against two objections: one concerning the future, the other concerning the alleged simplicity of the Bridge compared to the Liar.

Self-Reference and Diagonalisation

This poem is an exercise on self-reference and diagonalisation in mathematics featuring Turing’s proof of the undecidability of the halting problem, Cantor’s cardinality argument, the Burali-Forti paradox, and Epimenides' liar paradox.

An Illegal Assembly of One

In Singapore, the law holds that one person may constitute an illegal assembly. This makes each person, individually and at all times, latently assembled if not actually so. But where exactly does the permissible, non-assembled one end and the unlawful, gathered one begin? How and when does one become more than one, that is, some? For here an excess of one is not many, but rather an indeterminate some. Of what does this someness consist? This essay draws on Foucault and Lacan’s discussion of the liar paradox and set theory’s concept of the “not-all” via Bateson and Kordela to make a few observations about the political subject’s constitution under illiberal democracy.