On Refutation Rules

“…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%

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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“…The complementary systems contain, essentially, two subsystems: one for deriving the asserted propositions, and another -for deriving the rejected propositions. For instance, in [49,50,59] the authors consider two closure operators: the regular one Cn that gives the theorems, and the complementary one Cn − , that gives anti-theorems (see also [45,Section 5.2]).…”

Section: 42mentioning

confidence: 99%