Il paradosso del mentitore

In this paper, the paradox of the liar is studied in the framework of (computational) verb logic. Unlike other research on liar's paradox, which were based on classical logical or fuzzy logic framework, the study of liar's paradox in verb logic emphasizes the contribution of verbs to the perception of TRUTH in verb statements or sentences. A new interpretation of the paradox of the liar under verb logic is presented. Then the conditions under which verb liar's paradoxes occur are presented based on BE-transformations. Based on different paradoxical functions, the concepts of strong and weak verb paradoxes are presented. The main conclusion is that liar's paradoxes in verb logic are dynamical processes with time-varying degrees of being paradox (paradoxical value) and are sensitive to different contexts; namely, different BE-transformations. Another conclusion is that in natural language systems, weak verb paradoxes can be used intuitively correctly due to the uncertainties in brain dynamics and can be useful for expressing human emotions. © 2001 John Wiley & Sons, Inc.

Hartry Field's book Saving Truth from Paradox is without question among the best works on truth and the liar paradox in the analytic tradition; it should become the standard reference on the liar paradox for years to come. Field offers lucid, technically accurate but accessible discussions of most of the approaches to the liar paradox that are currently being debated in the literature. He also defends his favoured approach, which requires a change from classical to paracomplete logic. After a brief flirtation with dialetheism around the turn of the century, he now offers a novel, powerful and technically dazzling way of dealing with the liar paradox to accompany his influential version of disquotationalism.1 Together they provide a unified view of the nature and logic of truth.2 Field's solution to the liar, together with his fair and charitable discussion of the alternatives, make this book required reading by anyone remotely interested in issues associated with truth, philosophical logic and philosophy of language.

The Liar Paradox has been widely discussed from the ancient times and preserved its importance in contemporary philosophy of logic and mathematics. At the beginning of the 20th century, F.P. Ramsey asserted that the Liar Paradox is different from pure logical paradoxes such as Russell’s paradox. The Liar Paradox is connected with language and can be considered a semantic paradox. Ramsey's point of view has become widespread in the logic of the 20th century. The author of the article questions this view. It is argued that the Liar Paradox cannot be unequivocally attributed to the semantic paradoxes and therefore Ramsey's point of view should be revised.

In this pair of essays, I revisit the logical paradoxes. In the present essay I discuss the most famous of the so-called semantical paradoxes, the paradox of the Liar, the sentence that says of itself that it is not true, and in the essay that follows (Paradox Revisited II) I shall consider whether we should really accept a view once expressed by Godel, the view that the paradoxes of set theory are ones that we can see through, can definitely and satisfactorily resolve, even if (as he conceded) the same cannot be said for the semantical paradoxes. The Liar Paradox The best presentation I know of the Liar Paradox is Charles Parsons', and in the end the view I shall defend is, I believe, an elaboration of his. In “The Liar Paradox,” a paper I have thought about for almost twenty years, the paradox is stated in different ways. One of these ways is in terms of three alternatives: either a sentence expresses a true proposition, or it expresses a false proposition, or it does not express a proposition at all. A second way mentioned in that paper is the one I followed in my presentation of the Liar paradox in Realism with a Human Face , in which talk of propositions is avoided, and I mostly employ that way here in order to facilitate comparison with Tarski's work.

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.

Dialeteias são contradições verdadeiras. Dialeteísmo é a visão de que há dialeteias e dialeteístas são aqueles que defendem tal visão. Uma das principais motivações para o dialeteísmo encontra-se nos paradoxos semânticos, como o paradoxo do Mentiroso. A sentença do mentiroso, na abordagem dialeteísta, instancia uma dialeteia. O problema é: como aceitar contradições verdadeiras? Como dialeteísmo é a visão que algumas, mas não todas, contradições são verdadeiras, o dialeteísmo demanda um tratamento paraconsistente. Porém, não é qualquer sistema paraconsistente que pode ser aplicado ao dialeteísmo. A interpretação do símbolo de negação deve ser um operador formador de contradição (ofc) com o sentido relevante para a aplicação dialeteísta. Especificamente, na versão de dialeteísmo que discutiremos nesse artigo, defendida por Graham Priest, temos que, para que o dialeteísmo faça sentido, a negação da lógica subjacente deve atender a dois requisitos: (i) ser um ofc e (ii) não ser explosiva. A Lógica do Paradoxo (LP) tem sido apontada como a lógica adequada ao dialeteísmo. Veremos que, a fim de lidar com contradições sem trivialidade, dialeteístas “esticam” a verdade de modo que a verdade possa incluir, em alguns casos, a falsidade também. Nesse caso, haveria sentenças verdadeiras e falsas, chamadas de aglutinações de valores de verdade (truth-value gluts). Em LP, aglutinações de valores de verdade são fundamentais para garantir os requisitos (i) e (ii). Todavia, vamos argumentar que tal procedimento de esticar a verdade, permitindo aglutinações, garante a paraconsistência ao custo de distorcer a interpretação da noção de contradição envolvida no dialeteísmo. Especificamente, tal distorção enfraquece a interpretação da negação comprometendo o sentido de contradição relevante para o dialeteísmo. Vamos argumentar que as próprias restrições dialeteístas a uma compreensão adequada do Mentiroso mostram que as condições (i) e (ii) são incompatíveis e que, com isso, o projeto dialeteísta enfrenta consideráveis obstáculos.

The study of language as a means of communication, storage and transmission of human experience has led to the formation of two different philosophical and methodological strategies in scientific cognition: one, based on the theory of sets, considers language from the point of view of its formal structure; the other, based on the theory of systems, interprets language as a carrier of information, as a sign information system. It is proved that these strategies of language research should be characterized not so much as competing, but as mutually complementing each other, helping to reveal the specifics of the linguistic phenomenon. The main thesis of the study is reduced to the statement that an utterance, before it becomes an integral component of social knowledge, must undergo a “filtration” procedure according to two different selection criteria. The first criterion is contextual, when understandable information is extracted from the content of the sentence. The second – logical – involves the “filtering” of true judgments based on formal requirements accepted in science. While the formal-logical strategy of language research is described in some detail in the scientific literature, a systematic approach to its study needs more thorough elaboration and practical coping. An example of the mentioned double filtering of a sentence, carried out on a logical and informational basis, can be the liar's paradox. Despite the fact that many experts perceive the liar's statement “I’m lying” as a logical paradox, however, it does not affect our understanding of natural language and its use for cognitive purposes. As a result of the conducted research, the interconnection of the processes of formalization and contextualization as two key philosophical and methodological approaches to the study of language is revealed.

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

There can be no doubt that the existence of paradoxes has stimulated vigorous and highly productive activity in philosophy and in logic. Take two famous examples: The Liar Paradox, which arises from a sentence such asL This statement is false.and the Russell Paradox which arises from the sentenceR The class of all classes which are not members of themselvesis a member of itself.The first of these has been a source of anguish for over 2,000 years. The second has engaged the serious attention of logicians for over three quarters of a century. Investigation of paradoxes of this sort has spawned whole new fields of study, such as technical semantics and axiomatic set theory.It has been claimed that legal reasoning is infected with paradoxes and that these paradoxes are similar in structure to those, like the two we have cited, which are of interest to the logician. If this claim were true one of two consequences would follow. Either the jurisprudent would face what would in all likelihood be a protracted struggle with these legal paradoxes resulting, perhaps, in significant additions to legal theory, or else, if these paradoxes were sufficiently similar to those of the logician, he might try to utilise the logician's results to solve his own legal puzzles.The first alternative, though attractive to a theoretician, may appear rather dismal to those engaged in the business of law. Whereas reflection on the logical paradoxes can lead to only more refined abstractions—the philosopher's meat and drink—legal theory is rather intimately connected with practical affairs.

Internal Quantum Measurements and the Growth of Information

The Liar Paradox and Many-Valued Logic

This article presents a close reading of Montaigne's "De Democritus et Heraclitus" (Essais I,50) and explores the notion of judgment that Montaigne develops in this chapter as well as the rhetoric used in expressing this notion. I argue that several rhetorical structures used by Montaigne in this essay – structures such as paradox, inversion of subject and object, and polysemic language that invites multiple interpretations – play an important role in expressing Montaigne's critique of judgment. As he notes in his "Apologie de Raimond Sebond," sceptical views of judgment are difficult to express: they are subject to logical paradoxes similar to the liar's paradox. The rhetoric of "De Democritus et Heraclitus" can be understood as an answer to this logical difficulty. At the heart of my reading is a reinterpretation of a maxim well known to readers of the Essais: "Tout mouvement nous descouvre." This reinterpretation grew out of a study of the polysemous words used in this maxim, of Montaigne's claims about judgment in other essays, and of a careful inspection of the way that it appears in the Exemplaire de Bordeaux.

Logics of Formal Inconsistency (LFIs) are usually considered a philosophically neutral logic with respect to paraconsistency, in the sense that they provide a good basis to think in terms of dialetheias, i.e. true contradictions, or in terms of conflicting information, a notion weaker than truth. In this article, I will show how this claim of neutrality fails in the face of the Liar Paradox: in classical logic, the sentence A: “A is false.” implies a contradiction. By adopting a paraconsistent logic, we avoid that contradictions lead to triviality. LFIs, however, add a consistency operator that recovers classical logic to propositions to which it applies. In this context, the Liar Paradox arises once again by a type of Liar’s Revenge: now, we can construct the sentence B: “B is only false.”, and that gives us triviality once again. With this result, we conclude that LFIs are not suitable logics to deal with semantic paradoxes

The fact that Gödel's famous incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of the community of logicians. Indeed, almost every more or less formal treatment of the theorem makes a reference to this connection. Gödel himself remarked in the paper announcing his celebrated result (cf. [7]):The analogy between this result and Richard's antinomy leaps to the eye;there is also a close relationship with the ‘liar’ antinomy, since … we are… confronted with a proposition which asserts its own unprovability.In the light of the fact that the existence of this connection is commonplace it is all the more surprising that very little can be learnt about its exact nature except perhaps that it is some kind of similarity or analogy. There is, however, a lot more to it than that. Indeed, as we shall try to show below, the general ideas underlying the three central theorems concerning internal limitations of formal deductive systems can be taken as different ways to resolve the Liar paradox. More precisely, it will turn out that an abstract formal variant of the Liar paradox, which can almost straightforwardly inferred from its original ordinary language version, is a possible common generalization of (both the syntactic and semantic versions of) Gödel's incompleteness theorem, the theorem of Tarski on the undefinability of truth, and that of Church concerning the undecidability of provability.

Wittgenstein’s Deflationism - In this paper, I will aim at clarifying up to what extent the later Wittgenstein is a deflationist. I will claim that he is a (moderate) ontological but not a metaphysical deflationist. Up to a certain point, his moderate ontological deflationism amounts to a grammatical ontological deflationism. In its turn, this deflationism can be equated with a form of idealism. Moreover, once such a deflationism is combined with his metaphysical inflationism, i.e., grammatical essentialism, Wittgenstein turns out to be an internal realist. Finally, with respect to Glock’s tripartition between objectual, linguistic and existential deflationism, this way of putting things will enable me to negatively answer the question of whether Wittgenstein is either an objectual or a linguistic deflationist. This leaves the possibility open that he is an existential deflationist.