A note on tableaux of logic of paradox

1994 ◽  
pp. 296-307 ◽  
Author(s):  
Zuoquan Lin ◽  
Wei Li
Keyword(s):  
Logic Of Paradox

scholarly journals COMPLETENESS VIA CORRESPONDENCE FOR EXTENSIONS OF THE LOGIC OF PARADOX

2012 ◽  
Vol 5 (4) ◽  
pp. 720-730 ◽  
Author(s):  
BARTELD KOOI ◽  
ALLARD TAMMINGA
Keyword(s):  
Correspondence Analysis ◽  
Natural Deduction ◽  
Correspondence Theory ◽  
Truth Table ◽  
Deduction System ◽  
Natural Deduction System ◽  
Logic Of Paradox

AbstractTaking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for LP. Third, we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics.

Subprevarieties versus extensions. Application to the logic of paradox

2000 ◽  
Vol 65 (2) ◽  
pp. 756-766 ◽  
Author(s):  
Alexej P. Pynko
Keyword(s):  
General Result ◽  
Classical Logic ◽  
Modus Ponens ◽  
Galois Connection ◽  
Normal Extension ◽  
Material Implication ◽  
Element Chain ◽  
Class Of Matrices ◽  
The One ◽  
Logic Of Paradox

AbstractIn the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

Extensions of paraconsistent weak Kleene logic

2020 ◽  
Author(s):  
Francesco Paoli ◽  
Michele Pra Baldi
Keyword(s):  
General Result ◽  
Classical Logic ◽  
Order Ideal ◽  
Consequence Relation ◽  
Reduced Models ◽  
Logic Of Paradox

Abstract Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

scholarly journals A Note on Freedom from Detachment in the Logic of Paradox

2013 ◽  
Vol 54 (1) ◽  
pp. 15-20 ◽  
Author(s):  
Jc Beall ◽  
Thomas Forster ◽  
Jeremy Seligman
Keyword(s):  
Logic Of Paradox

Logic of paradox revisited

1984 ◽  
Vol 13 (2) ◽  
pp. 153-179 ◽  
Author(s):  
Graham Priest
Keyword(s):  
Logic Of Paradox

Cajetan of Thiene on the Logic of Paradox

2016 ◽  
Vol 13 (1) ◽  
pp. 71-95
Author(s):  
Miroslav Hanke ◽  
Keyword(s):  
Logic Of Paradox

scholarly journals The “Ontological Difference” Again. A Dialetheic Perspective on Heidegger’s Mainstay

2018 ◽  
Vol 1 (1) ◽  
pp. 143-154
Author(s):  
Francesco Gandellini
Keyword(s):  
Crucial Role ◽  
Paraconsistent Logic ◽  
Main Argument ◽  
Ontological Difference ◽  
The Third ◽  
Classical Notion ◽  
Late Works ◽  
Logic Of Paradox ◽  
Previous Paragraph

Abstract This paper intends to offer a new assessment of the “Ontological Difference” (OD), one of Martin Heidegger’s mainstays, in the light of the metaphysical view called “dialetheism”. In the first paragraph I briefly summarize the main argument of Heidegger’s contradiction of Being, where OD is present as a premise. In the second paragraph I introduce dialetheism, indicate two kinds of dialetheic solutions to the paradox and explain why they face comeback troubles from OD. The third paragraph is devoted to a review of Heidegger’s uses of OD and underlines the crucial role of negation in it. In the fourth paragraph I investigate the philosopher’s account of negation and show similarities with the account provided by the paraconsistent logic called “Logic of Paradox”. The fifth paragraph puts forward two possible readings of OD, the first based on the classical notion of negation and the second on the notion of negation pointed out in the previous paragraph. The second reading is proved suitable for dialetheists and in accordance with the exegesis of some textual passages from Heidegger’s late works.

scholarly journals Dialetheists’ Lies About the Liar

2018 ◽  
Vol 22 (1) ◽  
pp. 59-85 ◽  
Author(s):  
Jonas R. B. Arenhart ◽  
Ederson S. Melo
Keyword(s):  
Natural Language ◽  
Expressive Power ◽  
Formal Approach ◽  
The Liar ◽  
Logic Of Paradox ◽  
Logical Treatment ◽  
Negation Sign ◽  
Graham Priest ◽  
Contradictory Conclusion

Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, or else because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Paradox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, internal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the resources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. As a result, the formal account suffers from severe limitations, which make it unable to represent the contradiction obtained in the conclusion of each of the paradoxes.

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