The logic of paradox
TL;DR: A new way of handling the logical paradoxes is suggested, where instead of trying to dissolve them, or explain what has gone wrong, the authors should accept them and learn to come to live with them.
Abstract: The purpose of the present paper is to suggest a new way of handling the logical paradoxes. Instead of trying to dissolve them, or explain what has gone wrong, we should accept them and learn to come to live with them. This is argued in Sections I and II. For obvious reasons this will require the abandonment, or at least modification, of 'classical' logic. A way to do this is suggested in Section III. Sections IV and V discuss some implications of this approach to paradoxes.
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787 citations
Book•
01 Jan 2000
TL;DR: This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural' and systematically survey the new results and the significant impact that this class oflogics has had on a wide range of fields.
Abstract: This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading.
456 citations
Book•
01 Dec 1998TL;DR: This chapter discusses the need to derive reasonable inferences without deriving the trivial inferences that follow the ex falso quodlibet proof rule that holds in classical logic.
Abstract: In practical reasoning, it is common to have “too much” informationabout some situation. In other words, it is common for there to be classically inconsistent information in a practical reasoning database [Besnard et al., 1995]. The diversity of logics proposed for aspects of practical reasoning indicates the complexity of this form of reasoning. However, central to practical reasoning seems to be the need to reason with inconsistent information without the logic being trivialized [Gabbay and Hunter, 1991; Finkelstein et al., 1994]. This is the need to derive reasonable inferences without deriving the trivial inferences that follow the ex falso quodlibet proof rule that holds in classical logic.
454 citations
01 Jan 2007
TL;DR: The Logicas da Inconsistencia Formal (LIFs) as mentioned in this paper form a classe of logicas paraconsistentes particularmente expressivas, in which a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to.
Abstract: Segundo a pressuposicao de consistencia classica, as contradicoes tem um cara[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocinio sensato pode entao ter lugar. Uma logica e paraconsistente se ela rejeita uma tal pressuposicao, e aceita ao inves que algumas teorias inconsistentes conquanto nao-triviais facam perfeito sentido. A? Logicas da Inconsistencia Formal, LIFs, formam uma classe de logicas paraconsistentes particularmente expressivas nas quais a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to. Como consequencia, as LIFs sao capazes de recapturar o raciocinio consistente pelo acrescimo de assuncoes de consistencia apropriadas. Assim, por exemplo, enquanto regras classicas tais como o silogismo disjuntivo (de A e {nao-,4)-ou-13, infira B) estao fadadas a falhar numa logica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presuncao de que estamos raciocinando em um ambiente consistente (neste caso, pelo acrescimo de (consistente-.A) como uma hipotese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustracoes destas logicas e de suas propriedades, mostrando que tais logicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocinio consistente dentro de tais sistemas inconsistentes sao tambem ilustradas Em cada caso, interpretacoes em termos de semânticas polivalentes, de traducoes possiveis ou modais sao fornecidas, e os problemas relacionados a provisao de contrapartidas algebricas para tais logicas sao examinados. Uma abordagem formal abstra[cjta e proposta para todas as definicoes relacionadas e uma extensa investigacao e feita sobre os principios logicos e as propriedades positivas e negativas da negacao
Abstract
348 citations
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TL;DR: A minimal tableaus as a satisfactory proof theory for LP/ sub m/ is provided and the soundness and completeness theorems of the tableaux with respect to the semantics of LP and LP/sub m/ are proved.
Abstract: G. Priest introduced nonmonotonicity into a paraconsistent logic, so-called logic of paradox LP, that yields a solution to the weakness of paraconsistent logic. The resulting logic (of minimal parades) LP/sub m/ is nonmonotonic in the sense that inconsistency is minimal. The problem of proof theory of logic LP/sub m/ left open because the base logic LP is paraconsistent so that syntactic formulations of nonmonotonic logic are not available for LP/sub m/, though LP/sub m/ is well characterized by minimal semantics. In this paper, we provide a minimal tableaus as a satisfactory proof theory for LP/sub m/. We first present a signed tableaux for LP. Then minimal tableaux for LP/sub m/ is obtained by revising signed tableaux for LP to fit LP/sub m/ in which the branches of non-minimally-inconsistent models of the tableaux are eliminated. The soundness and completeness theorems of the tableaux with respect to the semantics of LP and LP/sub m/ are proved, respectively.
330 citations
Cites background from "The logic of paradox"
...In 1979 [12] Priest has presented under the name LP (Logic of Paradox) a paraconsistent logic having the same truth-tables for negation, conjunction and disjunction as the three-valued logics L3 of Łukasiewicz [11] and K3 of Kleene [9]:...
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References
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TL;DR: In this article, the notion of material adequacy of a definition of truth is defined as follows: if all these equivalences follow from it, then the definition is true if, and only if, p.X is true.
Abstract: X is true if, and only if, p. [...] we shall call a definition of truth “adequate” if all these equivalences follow from it. [...] The definition of truth which was outlined above [...] implies all equivalences of the form (T). In this connnection it is important to notice that the conditions for the material adequacy of the definition determine uniquely the extension of the term “true.” Therefore, every definition of truth which is materially adequate would necessarily be equivalent to that actually constructed. The semantic conception of truth gives us, so to speak, no possibility of choice between variaous non-equivalent definitions of this notion..
940 citations
TL;DR: In this article, the notion of material adequacy of a definition of truth is defined as follows: if all these equivalences follow from it, then the definition is true if, and only if, p.X is true.
Abstract: X is true if, and only if, p. [...] we shall call a definition of truth “adequate” if all these equivalences follow from it. [...] The definition of truth which was outlined above [...] implies all equivalences of the form (T). In this connnection it is important to notice that the conditions for the material adequacy of the definition determine uniquely the extension of the term “true.” Therefore, every definition of truth which is materially adequate would necessarily be equivalent to that actually constructed. The semantic conception of truth gives us, so to speak, no possibility of choice between variaous non-equivalent definitions of this notion..
761 citations
Book•
01 Jan 1934
TL;DR: In this article, a vorliegende Buch soll einer eingehenden Orientierung uber den gegenwartigen Stoff der HILBERTschen Beweistheorie dienen.
Abstract: Der vorliegende Band schliesst die Darstellung der Beweistheorie ab, die ich vor einigen Jahren zusammen mit P. BERNAYS begann. Auf meinen Wunsch hat P. BERNAYS wieder die Abfassung des Textes uber nommen. Ich danke ihm fur die Sorgfalt und Treue, mit der er meine Gedanken wiedergegeben hat, an deren Entwicklung er in jahrelanger Zusammenarbeit aufs starkste beteiligt war. Ohne seine Mithilfe ware die Vollendung dieses Buches unmoglich gewesen. Den Herren W. ACKERMANN, G. GENTZEN, A. SCHMIDT, H. SCHOLZ danke ich fur ihre freundliche Mitwirkung bei den Korrekturen. Gottingen, im Marz 1939 HILBERT Zur Einfuhrung Das vorliegende Buch soll einer eingehenden Orientierung uber den gegenwartigen Stoff der HILBERTschen Beweistheorie dienen. Wenn gleich das bisher hier Erreichte gemessen an den Zielen der Theorie sehr bescheiden ist, so liegt doch ein reichlicher Stoff an pragnanten Ergebnissen, an Gesichtspunkten und Beweisgedanken vor, die zur Kenntnis zu bringen als lohnend erscheint. Fur die inhaltliche Gestaltung dieses zweiten Bandes waren durch den Zweck des Buches zwei Hauptthemata vorgezeichnet. - Es handelte sich einmal darum, die hauptsachlichen, an das e-Symbol sich knupfenden beweistheoretischen Ansatze HILBERTS und ihre Durchfuhrung zur ein gehenden Darstellung zu bringen
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