The theory of rejected propositions. II

Słupecki, Jerzy; Bryll, Grzegorz; Wybraniec-Skardowska, Urszula

doi:10.1007/bf02120839

Cited by 28 publications

(21 citation statements)

References 7 publications

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“…10 This builds upon analysis of the relationship between intuitionistic and co-intuitionistic logic as discussed in [4,5,7,12,16,35,37,39,52]; and systems of proof and refutation discussed in [41,42,43,44,45,53,54]. Unlike the construction suggested here, the typical relationship between intuitionistic and co-intuitionistic logic in the literature is not without significant 6 Note that this feature is not forced on intuitionistic sequents, though restrictions on rules or dependency relations are required to ensure that only intuitionistic derivations are valid in sequents with multiple conclusions, see [9] for discussion.…”

Section: Bi-intuitionismmentioning

confidence: 99%

“…45 This has the consequence that certain "formulas" are barred. For example, ¬ I α − and ¬ C α + are not wf f 's, and so neither is ¬ I ¬ C α +,− .…”

Section: Harnessing Collapse For Dialogue Between Proofs and Refutationsmentioning

confidence: 99%

“…As mentioned above, we are considering the co-intuitionistic logic L C (hence co-Heyting algebra) as a refutation structure, following the direction of [41,44,45]. 24 A valid sequent "α β" of a logic is ordinarily interpreted as saying that, under the assumption that there exists a proof of α, there exists a proof of β also.…”

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confidence: 99%

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Structuring Co-constructive Logic for Proofs and Refutations

Trafford¹

2016

Log. Univers.

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Abstract. This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following [49]. It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such that classical logic (interpreted in a Boolean topos) is simulated where proofs and refutations are conclusive.

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Section: Bi-intuitionismmentioning

confidence: 99%

“…45 This has the consequence that certain "formulas" are barred. For example, ¬ I α − and ¬ C α + are not wf f 's, and so neither is ¬ I ¬ C α +,− .…”

Section: Harnessing Collapse For Dialogue Between Proofs and Refutationsmentioning

confidence: 99%

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Structuring Co-constructive Logic for Proofs and Refutations

Trafford¹

2016

Log. Univers.

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“…We will present the axiom system of the original so-called Tarski's enriched deductive system theory denoted by T + and the axioms of its equivalent modification T ++ . Tarski's ideas were elaborated by Pogorzelski and Słupecki [24], Słupecki et al [19,20], Wójcicki [4], Pogorzelski and Wojtylak [6], and the current author [22,23], among others.…”

Section: Deductive Systems Theories Based On Classical Sentential Calmentioning

confidence: 99%

“…In Sections 2-4, we present the axiom systems of these three theories: T + , T´and T 1 (The theories T + and T´were presented or sketched earlier in [19][20][21]; see also axiom systems in [7,22]. The axioms of the theory T 1 were discussed in [23].).…”

Section: Introductionmentioning

confidence: 99%

On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency

Wybraniec-Skardowska

2016

Axioms

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Abstract:In this paper, two axiomatic theories T´and T 1 are constructed, which are dual to Tarski's theory T + (1930) of deductive systems based on classical propositional calculus. While in Tarski's theory T + the primitive notion is the classical consequence function (entailment) Cn + , in the dual theory T´it is replaced by the notion of Słupecki's rejection consequence Cn´and in the dual theory T 1 it is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T + , T´, and T 1 are equivalent.

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Accepting/rejecting propositions from accepted/rejected propositions: A unifying overview

Caferra¹,

Peltier²

2008

Int. J. Intell. Syst.

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Looking at inference as a way of transforming information so as to make it more easily usable (or interpretable) allows to consider accepted and rejected propositions as equally relevant and naturally gives a bipolar view of reasoning. The four possibilities of transforming information from accepted or rejected propositions into accepted or rejected ones are analyzed and examples illustrating them are given. This analysis is not only interesting per se but can also be useful in increasing capabilities of existing theorem provers. A unified framework based on former work by the authors is extended by incorporating the idea of theory-anti-subsumption related to Plotkin's generalization. Working on some technical details of this framework should allow automated reasoning tools to deal with different ways of connecting accepted and rejected propositions. C 2008 Wiley Periodicals, Inc.

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The theory of rejected propositions. II

Cited by 28 publications

References 7 publications

Structuring Co-constructive Logic for Proofs and Refutations

Structuring Co-constructive Logic for Proofs and Refutations

On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency

Accepting/rejecting propositions from accepted/rejected propositions: A unifying overview

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