Liar Paradox Research Articles

Truth and Paradox: Solving the Riddles

1. Two Versions of the Liar Paradox 2. On the Origin of Truth Values 3. What is Truth, and What is a Theory of Truth? 4. A Language That Can Express Its Own Truth Theory 5. The Norms of Assertion and Denial 6. Solving the Inferential Liar Antinomy 7. Reasoning about Permissible Sentences 8. The Permissibility Paradox 9. The Metaphysics of Truth Bibliography

What Theories of Truth Should be Like (but Cannot be)

AbstractThis article outlines what a formal theory of truth should be like, at least at first glance. As not all of the stated constraints can be satisfied at the same time, in view of notorious semantic paradoxes such as the Liar paradox, we consider the maximal consistent combinations of these desiderata and compare their relative advantages and disadvantages.

An illocutionary logical explanation of the liar paradox

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.

Platitudes against Paradox

We present a strategy to dissolve semantic paradoxes which proceeds from an explanation of why paradoxical sentences or their definitions are semantically defective. This explanation is compatible with the acceptability of impredicative definitions, self-referential sentences and semantically closed languages and leaves the status of the so-called truth-teller sentence unaffected. It is based on platitudes which encode innocuous constraints on successful definition and successful expression of propositional content. We show that the construction of liar paradoxes and of certain versions of Curry’s paradox rests on presuppositions that violate these innocuous constraints. Other versions of Curry’s paradox are shown not to be paradoxical at all once their presuppositions are made explicit. Part of what we say rehearses a proposal originally made by Laurence Goldstein in 1985. Like Goldstein we dispose of certain paradoxes by rejecting some of the premises from which they must be taken to proceed. However, we disagree with his more recent view that the premises to be rejected are neither true nor false.

Hyper-Contradictions, Generalized Truth Values and Logics of Truth and Falsehood

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

Symmetry and Paradox

The ‘no’–‘no’ paradox (so-called by Sorensen) consists of a pair of propositions each of which says of the other that it is false. It is not immediately paradoxical, since it has a solution in which one proposition is true, the other false. However, that is itself paradoxical, since there is no clear ground for determining which is which. The two propositions should have the same truth-value. The paper shows how a proposal by the medieval thinker Thomas Bradwardine solves not only the Liar paradox, but also symmetric paradoxes like the ‘no’–‘no’, the descending ‘no’–‘no’, and the Truth-teller paradoxes.

Don Quixote and the Liar's Paradox

"I am lying." It might be said that this statement is, implicitly, the single real truth of the fiction-forming author, and its complications are both the emblem and bane of all novels. In his epic <em>Don Quixote</em>, Miguel de Cervantes exploits the problem of making lies real by intricately waving together such disparate elements as autobiography and high fantasy within the unreliably unreliable voices of several subjective storytellers. And his conclusion, it seems, is Don Quixote himself, a man who combats the protean strangeness of the world by replacing it with a deliberate reality of his own. At the cusp of the European Renaissance, Cervantes' Don Quixote makes the choice which will become a necessity for post-Cartesian thinkers, namely, a personal determination, an authorship of "the real" in a world where inconsistencies and subjectivities constantly threaten to chain the truth-seeking man, like Prometheus, to the confusion of the liar's paradox.

True, false and paranormal

explain how, if at all, wecan truly characterize – specify the ‘semantic status of’ – allsentences of our language (in our language).These projects reflect the core appearances that give rise to the Liar para-dox (and its ilk). Semantic paradoxes arise, at least in part, from the ap-pearance that we can ‘exhaustively characterize’ all sentences of our lan-guage in terms of ‘semantically significant’ predicates, and truly do asmuch in our language.

True, false and paranormal

explain how, if at all, wecan truly characterize – specify the ‘semantic status of’ – allsentences of our language (in our language).These projects reflect the core appearances that give rise to the Liar para-dox (and its ilk). Semantic paradoxes arise, at least in part, from the ap-pearance that we can ‘exhaustively characterize’ all sentences of our lan-guage in terms of ‘semantically significant’ predicates, and truly do asmuch in our language.

Future Law: Prepunishment and the Causal Theory of Verdicts

Is the King’s Messenger being treated unjustly? Why not prepunish people for their future crimes? True, it is generally easier to have knowledge about past crimes than future crimes. But is unpredictability the only problem with prepunishment? My answer is a causal theory of verdicts that condemns all actual prepunishment but for reasons that are temporally neutral. These reasons echo the semantic concerns of logicians who study the liar paradox.

Deflationism and the Autonomy of Truth

In Thought and World, Chris Hill offers us the most sophisticated version of minimalism to date. Hill's starting point is Horwich's minimal theory of truth (MT), according to which propositions - or thoughts in Hill's preferred terminology - are the truth-bearers, and the axioms of the theory are the thoughts expressed by sentences like The thought that aardvarks amble is true if and only if aardvarks amble. But Hill's theory is superior to Horwich's in a number of crucial respects. For example, while Horwich gives up on a finite formulation of MT, taking it that an infinity of axioms is needed, Hill's basic theory (simple substitutionalism or SS) is composed of just one axiom. And unlike MT, SS can accommodate certain generalizations about truth (for example, the generalization that only thoughts are true, and general propositions about the truth-conditional properties of logical connectives.) Moreover, Hill provides a detailed deflationary theory not only of truth, but also of related semantic notions such as reference and concept-denotation, notions that have received relatively little attention in the literature on deflationism; he takes correspondence intuitions much more seriously than most deflationists, and is at pains to show how those intuitions may be accommodated within his framework; he provides the most developed deflationary account available of indexical concepts and indexical thoughts; and unlike many deflationists, he is not content to merely set aside problems concerning bi valence and the Liar paradox. Those of us interested in the proper treatment of our semantic concepts, whether we are deflationists or not, are in Hill's debt.

Semantic Regularity and the Liar Paradox

Semantic Regularity and the Liar Paradox

Sense and the Computation of Reference

The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of this relation must be given in the form of a logic program. The resulting system does not only throw light on the properties of sense and their relation to computation, but also shows circular behaviour if some ingredients of the Liar Paradox are added. The connection is natural, as algorithms can be inherently circular and the Liar is explained as expressing one of those. Many ideas in the present paper are closely related to those in Moschovakis (1994), but receive a considerably lighter formalization.

MicroReviews by the Book Review Editor: The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time: Marcel Danesi

MicroReviews by the Book Review Editor: <i>The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time</i>: Marcel Danesi

Review: Sense, Reference, and Philosophy

Sense, Reference, and Philosophy develops the far-reaching consequences for philosophy of adopting non-Fregean intensionalism, showing that long-standing problems in the philosophy of language, and indeed other areas, that appeared intractable can now be solved. Katz proceeds to examine some of those problems in this new light, including the problem of names, natural kind terms, the Liar Paradox, the distinction between logical and extra-logical vocabulary, and the Raven paradox. In each case, a non-Fregean intentionalism provides a philosophically more satisfying solution.

Ramseying liars

Despite the volume of discussion on the Liar Paradox recently, there is one stream of largely British thought on the matter which is hardly represented in the wider literature. This paper points out salient aspects of the history of this tradition, from its origin in forms of propositional quantification found in Ramsey, through to more precise symbolisations which have emerged more recently. But its purpose is to exposit, with respect to a number of contested cases, the ensuing results. Thus it goes on to apply the analysis to several other well known paradoxes, including one rarely discussed, which reveals more fully the consequent consistency and completeness of natural language.

The Generalised Liar Paradox: A Quantum Model and Interpretation

The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere. Basically, the intrinsic contextuality of the truth-value of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. This includes (i) the truth-falsehood state of the m-Liar Paradox can be represented by an embedded 2m-dimensional quantum vector in a (2m)^m dimensional complex Hilbert space, with cognitive interactions corresponding to projections, (ii) the construction of a continuous 'time' dynamics is possible: typical truth and falsehood value oscillations are described by Schrodinger evolution, (iii) Kirchoff and von Neumann axioms are satisfied by introduction of 'truth-value by inference' projectors, (iv) time invariance of unmeasured state.

Minimalism, the Generalization Problem and the Liar

In defense of the minimalist conception of truth, Paul Horwich(2001) has recently argued that our acceptance of the instances of the schema,`the proposition that p is true if and only if p', suffices to explain our acceptanceof truth generalizations, that is, of general claims formulated using the truth predicate.In this paper, I consider the strategy Horwich develops for explaining our acceptance of truth generalizations. As I show, while perhaps workable on its own, the strategy is in conflictwith his response to the liar paradox. Something must give. I consider and reject variousalternatives and emendations to the strategy. In order to resolve the conflict,I propose an alternative approach to the liar, one that supports Horwich's strategywhile leaving minimalism maximally uncompromised.

An Elementary Resolution of the Liar Paradox

An Elementary Resolution of the Liar Paradox

An Elementary Resolution of the Liar Paradox

An Elementary Resolution of the Liar Paradox