Liar Paradox

Miller, Bradwardine and the Truth

by Stephen Read

Discusiones Filosoficas (Colombia) 12(18), 229-35

In his article "Verdades antiguas y modernas" (in the same issue, pp. 207-27), David Miller criticised... more In his article "Verdades antiguas y modernas" (in the same issue, pp. 207-27), David Miller criticised Thomas Bradwardine’s theory of truth and signification and my defence of Bradwardine’s application of it to the semantic paradoxes. Much of Miller’s criticism is sympathetic and helpful in gaining a better understanding of the relationship between Bradwardine’s proposed solution to the paradoxes and Alfred Tarski’s. But some of Miller’s criticisms betray a misunderstanding of crucial aspects of Bradwardine’s account of truth and signification.

Jump Liars and Jourdain's Card via the Relativized T-scheme

by Ming Xiong

Studia Logica, Vol. 91 (2), pp. 239-271, December 2009.

A relativized version of Tarski’s T-scheme is introduced as a new principle of the truth predicate. Under the... more 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.

Chrysippus Confronts the Liar: The Case for Stoic Cassationism

by Michael Papazian

The Stoic philosopher Chrysippus wrote extensively on the liar paradox, but unfortunately the extant testimony on his... more The Stoic philosopher Chrysippus wrote extensively on the liar paradox, but unfortunately the extant testimony on his response to the paradox is meager and mainly hostile. Modern scholars, beginning with Alexander Rüstow in the first decade of the twentieth century, have attempted to reconstruct Chrysippus’ solution. Rüstow argued that Chrysippus advanced a cassationist solution, that is, one in which sentences such as ‘I am speaking falsely’ do not express propositions. Two more recent scholars, Walter Cavini and Mario Mignucci, have rejected Rüstow's thesis that Chrysippus used a cassationist approach. Each has proposed his own thesis about Chrysippus’ solution. I argue that Rüstow's view is fundamentally correct, and that the cassationist thesis gains greater plausibility when viewed in light of a passage in Sextus Empiricus’ Adversus mathematicos that the previous commentators have ignored, and when understood within the broader context of Stoic logical theory and philosophy of language. I close with a brief remark on the significance of Chrysippus’ work for the modern debate on the semantic paradoxes.

Reaching Transparent Truth

by Pablo Cobreros

Joint work with Paul Egré, Dave Ripley and Robert van Rooij

This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T<A>... more This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T<A> and A are everywhere intersubstitutable, where all T-biconditionals hold, and where truth can be made compositional. A key feature of our framework, called STT (for Strict-Tolerant Truth), is that it supports a nontransitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STT allows for arise in paradoxical cases

Can a many‐valued language functionally represent its own semantics?

by Jeffrey Ketland

Analysis 2003

A proof of the (strengthened) Liar formula in a semantical extension of Peano Arithmetic

by Jeffrey Ketland

Analysis 2000

On what truth is about

by Ignacio Ojea

Dialetheism

by Francesco Berto

Co-authored with Graham Priest.

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout... more 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...

How to Sell a Contradiction: The Logic and Metaphysics of Inconsistency

by Francesco Berto

There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the... more 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.

My own truth, relative truth and the pathologies of self-reference

by Alexandre Billon

Preprintversion, forthcoming
The Realism-Antirealism Debate in the Age of Alternative Logics

Series: Logic, Epistemology, and the Unity of Science, Vol. 23

Rahman, Shahid; Primiero, Giuseppe; Marion, Mathieu (Eds.)

1st Edition., 2011, XXII, 366 p.

*

Semantic pathologies of self-reference include the Liar (`this sentence
is false'), the Truth-Teller (`this... more Semantic pathologies of self-reference include the Liar (`this sentence
is false'), the Truth-Teller (`this sentence is true') and the Open Pair
(`the neighbouring sentence is false' `the neighbouring sentence is false'). Although
they seem like perfectly meaningful declarative sentences, truth value
assignment to their uses seems either inconsistent (the Liar) or arbitrary (the
Truth-Teller and the Open-Pair). These pathologies thus call for a resolution.
I propose such a resolution in terms of relative-truth: the truth value
of a pathological sentence use varies with the context of its assessment. It
always has a determinate truth value, but this truth value is relative to the
context of its assessment. I start by considering truth-tellers, that is, sentences which say of themselves that they are true.
I make the case that truth value of a given truth-teller use must in general
depend on the context of its assessment, and that one can indeed change its truth value at will. I then show how the notion
of assessment-sensitive truth can help us provide solutions to other semantic
paradoxes such as the Liar and the Open Pair and that those solutions
are immune to revenge problems. I conclude by situating my proposal among
the main approaches to the semantic paradoxes, and by drawing a very broad
moral about pathological self-reference and intentionality.

A Simple Solution to the Hardest Logic Puzzle Ever

by Landon Rabern

A Simple Solution to the Hardest Logic Puzzle Ever

by Landon Rabern

Free Assumptions and the Liar Paradox

by Patrick Greenough

published in American Philosophical Quarterly 2001.

A new solution to the liar paradox is developed using the insight that it is illegitimate to even suppose (let alone... more 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 liar is that not all assumptions are for free. One merit of this proposal is that it is free from the revenge problem.

Truthmaker Gaps and the No-No Paradox

by Patrick Greenough

forthcoming in Philosophy and Phenomenological Research, 2011.

Consider the following sentences:

The neighbouring sentence is not true.
The neighbouring sentence is... more Consider the following sentences:

The neighbouring sentence is not true.
The neighbouring sentence is not true.

Call these the no-no sentences. Symmetry considerations dictate that the no-no sentences must both possess the same truth-value. Suppose they are both true. Given Tarski’s truth-schema—if a sentence S says that p then S is true iff p—and given what they say, they are both not true. Contradiction! Conclude: they are not both true. Suppose they are both false. Given Tarski’s falsity-schema—if a sentence S says that p then S is false iff not-p—and given what they say, they are both true, and so not false. Contradiction! Conclude: they are not both false. Thus, despite their symmetry, the no-no sentences must differ in truth-value. Such is the no-no paradox.[1] Sorensen (2001, 2005a, 2005b) has argued that: (1) The no-no paradox is not a version of the liar but rather a cousin of the truth-teller paradox. (2) Even so, the no-no paradox is more paradoxical than the truth-teller. (3) The no-no and truth-teller sentences have groundless truthvalues—they are bivalent but give rise to “truthmaker gaps”. (4) It is metaphysically impossible to know these truth-values. (5) A truthmaker gap response to the no-no paradox provides reason to accept a version of epistemicism. In this paper it is shown that a truthmaker gap solution to the no-no and truth-teller paradoxes runs afoul of the dunno-dunno paradox, the strengthened no-no paradox, and the strengthened truth-teller paradox. In consequence, the no-no paradox is best seen as a form of the liar paradox. As such, it cannot provide a case for epistemicism.