Chapter 5 LOGIC WITHOUT TRUTH Buridan on the Liar

The article presents a critique of logical foundations for formal realism. Formal realism is not satisfied with the ban on groundlessness by Hans Herzrberger and the ban on self-references by Bertrand Russell and Alfred Tarski which were taken by logicians as a means of effectively preventing alethic paradoxes. According formal realism, a source for arise alethic paradoxes is so-called “essentially negativity”. All of the examples of those paradoxes (like sentences of the Liar family) are directly linked with cases of using negative alethic predicate (‘to be false’). However, that approach to prevent paradoxes completely ignores any example of epistemic paradoxes, in particular the so-called “Buridan’s sophisms” (like sentences of the Truthteller family or the No-No family). In cases with Buridan’s sophisms (in contrary to Liar-like sentences), there are gluts of consistent pairs of truth-values for their sentences. The study of these sophisms allows to recognizing the actual difference between two different types of truth-value assignment – the unique truth-values (‘to be true’ or ‘to be false’) and the criteria truth-values (‘to have the same truth-value’ or ‘to have the opposite truth-value’). In cases with assignment of criteria truth-values we should talk about characteristics for logical relations between truth-values of sentences which constitute alethic and epistemic paradoxes. Thus, in cases with Buridan’s sophisms, sentences of the Truthteller family (‘S1: S2 is true’ and ‘S2: S1 is true’) indicate that any of them has the same truth-value as another (no matter what exactly it is), and sentences of the No-No family (‘S1: S2 is false’ and ‘S2: S1 is false’) have reported that each of them has the opposite truth-value to another. The facility for using of criteria truth-values is not limited to a group of epistemic paradoxes only. For example, in cases of athletic paradoxes (like sentences of the Liar family), criteria truth-values also allow to find a solution with consistent truth-values not only for sentences of the classic Liar (‘S1: S2 is false’ and ‘S2: S1 is true’), but for sentences to the infinite Liar (‘S1: For all k>1, Sk is untrue’, ‘S2: For all k>2, Sk is untrue’, ‘S3: For all k>3, Sk is untrue’, etc.).

Notre Dame Journal of Formal Logic, 1978

Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar.

Some fourteenth-century treatises on paradoxes of the liar family offer a promising starting-point for the formulation of full-fledged theories of truth with systematic relevance in their own right. In particular, Bradwardine's thesis that sentences typically say more than one thing gives rise to a quantificational approach to truth, and Buridan's theory of truth based on the notion of suppositio allows for remarkable metaphysical parsimony. Bradwardine's and Buridan's theories both have theoretical advantages, but fail to provide a satisfactory account of truth because both are committed to the thesis, fatal for both, that every sentence signifies/implies its own truth. I close with remarks on Greg Restall's recent model-theoretic formalization of Bradwardine's theory of truth.

Vivarium, 2002

Croatian Journal of Philosophy 23 (67): 1-31. 2023., 2020

This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed solutions ("the revenge of the Liar"). This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through an appropriate semantic shift allows us to express the failure in a classical two-valued language. Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fi xed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides an argumentation, which has not been present in contemporary debates so far, for the choice of LIF-PSK3 and its classical closure as the right model for the truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out.

History and Philosophy of Logic, 2007

This paper uses the resources of illocutionary logic to provide a new understanding of the Liar Paradox. In the system of illocutionary logic of the paper, denials are irreducible counterparts of assertions; denial does not in every case amount to the same as the assertion of the negation of the statement that is denied. Both a Liar statement, (a) Statement (a) is not true, and the statement which it negates can correctly be denied; neither can correctly be asserted. A Liar statement, more precisely, an attempted Liar statement, fails to fulfill conditions essential to statements, but no linguistic rules are violated by the attempt. Ordinary language, our ordinary practice of using language, is not inconsistent or incoherent because of the Liar. We are committed to deny Liars, but not to accept or assert them. This understanding of the Liar Paradox and its sources cannot be fully accommodated in a conventional logical system, which fails to mark the distinction between sentences/statements and illocutionary acts of accepting, rejecting, and supposing statements.

It is proposed that "This sentence is not true" may be true under some interpretations and false under others. The revenge challenge given by "This sentence is false in at least one interpretation" is handled through the observation that a sentence is arguable true under all interpretations if and only if it is provable. This makes the revenge challenge equivalent to the one given by "This sentence is not provable." Apart from arguing a link with Gödel's first incompleteness theorem we also argue that the way a Liar sentence may be true under one interpretation and false under another is related to the way a "many-valued function" in mathematics may have more than one value. We set up a mathematical framework where sentences may be given many different interpretations. Examples are also given of how Liar sentences may be given many interpretations in practice depending on the intentions of speakers and how they are understood, and our approach to dealing with Liar sentences is compared to those of others.

It seems that the most common strategy to solve the liar paradox is to argue that liar sentences are meaningless and, consequently, truth-valueless. The other main option that has grown in recent years is the dialetheist view that treats liar sentences as meaningful, truth-apt and true. In this paper I will offer a new approach that does not belong in either camp. I hope to show that liar sentences can be interpreted as meaningful, truth-apt and false, but without engendering any contradiction. This seemingly impossible task can be accomplished once the semantic structure of the liar sentence is unpacked by a quantified analysis. The paper will be divided in two sections. In the first section, I present the independent reasons that motivate the quantificational strategy and how it works in the liar sentence. In the second section, I explain how this quantificational analysis allows us to explain the truth teller sentence and a counter-example advanced against truthmaker maximalism, and deal with some potential objections.

2005

The Liar sentence is a singularly important piece of philosophical evidence. It is an instrument for investigating the metaphysics of expressing truths and falsehoods. And an instrument too for investigating the varieties of conflict that can give rise to paradox. It shall serve as perhaps the most important clue to the shape of human judgment, as well as to the nature of the dependence of judgment upon language use.