Structuring Co-constructive Logic for Proofs and Refutations

Trafford, James

doi:10.1007/s11787-016-0138-z

Cited by 3 publications

(4 citation statements)

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“…For, first, bilateral systems include the following rule: (+¬I) −α; +(¬α) And in passing, let's remark that bilateral systems will also include the converse rule. 33 How can (+¬I) be accepted without entailing anti-rejectivism? It seems perfectly acceptable for a Bivalentist as:…”

Section: Bilateralism and Bivalentismmentioning

confidence: 99%

“…In this respect, the "⇒" is a metalinguistic symbol of ensuing action after a preceding speech act. 33 In AR 4 , this is: (+¬I) a 1 (p) = 0; a 1 (¬p) = a 2 (p) = 1. 34 In QAS, this is: a 1 (¬α) = a 2 (α) = 1; a 1 (α) = 0.…”

Section: Reduction and Rejectivismmentioning

confidence: 99%

“…See also Section 2.1, formula (3) Cf. Rumfitt or Ripley, who mentioned something similar whilst the latter admits "paracoherence"; see also[33] for a detailed discussion of coherence and paracoherence in the context of constructive logics.…”

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confidence: 99%

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Is ‘No’ a Force-Indicator? Yes, Sooner or Later!

Schang

Trafford²

2017

Log. Univers.

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This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question-Answer Semantics (QAS). We develop a generalised account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression 'no'is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators. We finish by highlighting the benefits of our approach with regard to the categoricity of logics, and also a conditional that intends to solve the Frege-Geach Problem.

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Section: Bilateralism and Bivalentismmentioning

confidence: 99%

Section: Reduction and Rejectivismmentioning

confidence: 99%

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Is ‘No’ a Force-Indicator? Yes, Sooner or Later!

Schang

Trafford²

2017

Log. Univers.

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“…Trafford [27] defines an interpretation of co-intuitionistic logic into a topos-theoretic model to represent both proofs, in an elementary topoi, and refutations, in a complement topoi. He then shows that classical logic can be simulated in his model.…”

Section: Related and Future Workmentioning

confidence: 99%

A Cointuitionistic Adjoint Logic

Eades¹,

Bellin²

2017

Preprint

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Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to include the duals of each logical connective. One leading question with respect to BINT is, what does BINT look like across the three arcs -logic, typed λ-calculi, and category theory -of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard [11] showed that combining these two categories into the same category results in it degenerating to a poset. However, this degeneration does not occur when both logics are linear. We propose that IL and coIL need to be separated, and then mixed in a controlled way using the modalities from linear logic. This separation can be ultimately achieved by an adjoint formalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionistic logic. They are then related via two adjunctions. The adjunction between IL and ILL is known as a Linear/Non-linear model (LNL model) of ILL, and is due to Benton [4]. However, the dual to LNL models which would amount to the adjunction between coILL and coIL has yet to appear in the literature. In this paper we fill this gap by studying the dual to LNL models which we call dual LNL models. We conduct a similar analysis to that of Benton for dual LNL models by showing that dual LNL models correspond to dual linear categories, the dual to Bierman's [5] linear categories proposed by Bellin [3]. Following this we give the definition of bi-LNL models by combining our dual LNL models with Benton's LNL models to obtain a categorical model of bi-intuitionistic logic, but we leave its analysis and corresponding logic to a future paper. Finally, we give a corresponding sequent calculus, natural deduction, and term assignment for dual LNL models.

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Rules in Dialogue

Trafford¹

2016

Studies in Applied Philosophy, Epistemology and Rational Ethics

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Structuring Co-constructive Logic for Proofs and Refutations

Cited by 3 publications

References 40 publications

Is ‘No’ a Force-Indicator? Yes, Sooner or Later!

Is ‘No’ a Force-Indicator? Yes, Sooner or Later!

A Cointuitionistic Adjoint Logic

Rules in Dialogue

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