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1

Inconsistency-adaptive modal logics. On how to cope with modal inconsistency

100%
Lycke H.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 1-2
31–61
EN
In this paper, I will characterize a new class of inconsistency-adaptive logics, namely inconsistency-adaptive modal logics. These logics cope with inconsistencies in a modal context. More specifically, when faced with inconsistencies, inconsistency-adaptive modal logics avoid explosion, but still allow the derivation of sufficient consequences to adequately explicate the part of human reasoning they are intended for.
2

Minimising disjunctive information

100%
Wong P.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 1-2
159–192
EN
In [5, 6], Belnap proposed a number of amendments to Rescher’s strategy for reasoning with maximal consistent subsets. More recently in [18], Horty explicitly endorsed Belnap’s amendment to address a related problem in handling inconsistent instructions and commands. In this paper, we’ll examine Belnap’s amendment and point out that Belnap’s suggestion in the use of conjunctive containment is open to the very objection he raised. We’ll propose a way out. The strategy turns on the use of First Degree Entailment in combination with Quine’s notion of prime implicate.
3
Content available remote

A Generalisation of a Refutation-related Method in Paraconsistent Logics

100%
Trybus A.
Logic and Logical Philosophy
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2018
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vol. 27
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issue 2
235-261
EN
This article describes a refutation method of proving maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experimental setting that confirms the theoretical results.
4

Co-constructive Logics for Proofs and Refutations

88%
Trafford J.
Studia Humana
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2014
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vol. 3
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issue 4
22-40
EN
This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.
5
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The Case of Dialetheism

75%
Tkaczyk M.
Logic and Logical Philosophy
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2016
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vol. 25
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issue 2
203-224
EN
The concept of dialetheia and the claim of dialetheism has been examined and compared to such related concept as contradiction, antinomy, consistency and paraconsistency. Dialetheia is a true contradiction and dialetheism is the claim that there exists at least one dialetheia. It has been observed that dialetheism is equivalent to the negation of the traditional principle of contradiction. Hence, dialetheism itself is no new idea in whatsoever. The novelty of dialetheism consists in the arguments delivered for its case. Key justification the partisans deliver for dialetheism has been examined and evaluated: antinomies, an alleged Gödel’s paradox, and existence of limits of thought. The structure of those arguments has been analyzed. It has been claimed that they share one and the same simple structure which may be called reverse paradox. The vital content dialetheists add to the traditional paradoxes is only the thesis of reliability of the vernacular prima facie knowledge. Three objections have been raised against the justification of dialetheism: firstly, it has been claimed that exactly the same argument supports principle of contradiction, secondly, it has been questioned whether the arguments preserve their value when logic is subject to revision, and thirdly, it has been claimed that the underlying logic of dialetheism is classical.
6

Gerland’s Dialectica and Paraconsistency

75%
Loffredo D’Ottaviano I. M., Gomes E. L.
Edukacja Filozoficzna
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2020
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issue 70
143-170
EN
An analysis of positions for and against the principle of ex falso sequitur quodlibet is essential to the history of a paraconsistent approach in scholastic logic and in Western thought. In this paper we analyze the role that the Dialectica of Gerland of Besançon played in initiating the discussion about the ex falso in the 12th century, and we interpret his position as contrary to the acceptance of the principle. We consider Gerland one of the earliest authors to prepare the path and examine properly the role of the ex falso sequitur quodlibet principle, making it central in the philosophical context of the time. We adopt the thesis of Józef Maria Bocheński, according to which the formal aspects of logical theory are essential, decisive, and indispensable to a good historiography of logic.
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