The semantics of entailment ? III

Routley, Richard; Meyer, Robert K.

doi:10.1007/bf00650498

Cited by 122 publications

(48 citation statements)

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“…For what concerns us in this article, we will only introduce a fragment of this general framework. The semantics of this fragment is based on the ternary relation of the frame semantics for relevant logic originally introduced by Routley & Meyer [60,61,59,62]. Another semantics proposed independently by Urquhart [65,63,64] at about the same time will be discussed at the end of this section.…”

Section: A Substructural Languagementioning

confidence: 99%

“…The wellknown semantics for substructural logics is based on a ternary relation introduced by Routley & Meyer for relevance logic in the 1970's [60,61,59,62]. However, the introduction of this ternary relation was originally motivated by technical reasons, and it turns out that providing a non-circular and conceptually grounded interpretation of this relation remains problematic [18].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Epistemic Logic as a Substructural Logic

Aucher

2014

Outstanding Contributions to Logic

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Dynamic Epistemic Logic (DEL) is an influential logical framework for reasoning about the dynamics of beliefs and knowledge. It has been related to older and more established logical frameworks. Despite these connections, DEL remains, arguably, a rather isolated logic in the vast realm of non-classical logics and modal logics. This is problematic if logic is to be viewed ultimately as a unified and unifying field and if we want to avoid that DEL goes on "riding off madly in all directions" (a metaphor used by van Benthem about logic in general). In this article, we show that DEL can be redefined naturally and meaningfully as a two-sorted substructural logic. In fact, it is even one of the most primitive substructural logics since it does not preserve any of the structural rules. Moreover, the ternary semantics of DEL and its dynamic interpretation provides a conceptual foundation for the Routley & Meyer's semantics of substructural logics.

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Section: A Substructural Languagementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic Epistemic Logic as a Substructural Logic

Aucher

2014

Outstanding Contributions to Logic

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“…The earliest systems were proposed by Orlov in 1928 (Došen 1992), and by Moh (1950), Church (1951), and Ackermann (1956) in the 1950s. Semantics were introduced and developed only much later, in the 1970s; see (Routley and Routley 1972), (Routley and Meyer 1972a), (Routley and Meyer 1972b), (Urquhart 1972), , (Fine 1974), , (Anderson and Belnap 1975), (Routley, Plumwood, Meyer, and Brady 1982), (Anderson, Belnap, and Dunn 1992), and (Brady 2003).…”

Section: Introductionmentioning

confidence: 99%

Relevance Logic and the Calculus of Relations

Maddux

2010

Rev. symb. logic

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Abstract. Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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“…With Sylvan (né Routley) 1 , Meyer proposed in [19] and [18] a minimal positive relevant logic B + . As they conceived it, B + had a role to play for relevant logics analogous to that played by the system K among normal modal logics with a Kripke-style "possible worlds" semantics in the style of [16].…”

Section: Introductionmentioning

confidence: 99%

“…For λx.xx is one of the terms that has no type on Curry's scheme. Still, on the "correspondence theory" implicit in [19], with the ternary relation R to explicate → on our relational "worlds semantics", the validity of the formula ((p → q) ∧ p) → q enforces and is enforced by the total ternary reflexivity postulate Rwww. Rightly viewed, that semantical postulate is just a way of saying that λx.xx (a.k.a.…”

Section: Introductionmentioning

confidence: 99%

The Semantics of Entailment Omega

Dezani‐Ciancaglini¹,

Meyer²,

Motohama³

2002

Notre Dame J. Formal Logic

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ABSTRACT. This paper discusses the relation between the minimal positive relevant logic B + and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B ∧ T of B + , which saves implication → and conjunction ∧ but drops disjunction ∨. The filter models of the λ-calculus (and its intimate partner Combinatory Logic CL) of the first author and her co-authors then become theory models of these calculi. (The logician's Theory is the algebraist's Filter.) The coincidence extends to a dual interpretation of key particles -the subtype ≤ translates to provable →, type intersection ∩ to conjunction ∧, function space → to implication and whole domain ω to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union ∪ should correspond to the logical disjunction ∨ of B + . But the simulation of functional application by a fusion (or modus ponens product) operation • on theories leaves the key Bubbling lemma of work on ITD unprovable for the ∨-prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of λ and CL.

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The semantics of entailment ? III

Cited by 122 publications

References 8 publications

Dynamic Epistemic Logic as a Substructural Logic

Dynamic Epistemic Logic as a Substructural Logic

Relevance Logic and the Calculus of Relations

The Semantics of Entailment Omega

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