The Semantics of Entailment Omega

Dezani‐Ciancaglini, Mariangiola; Meyer, Robert K.; Motohama, Yoko

doi:10.1305/ndjfl/1074290712

Cited by 13 publications

(20 citation statements)

References 11 publications

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“…It is included here, and in the type assignment system TAS in Section 2.2, so that TAS will be an extension of the Intersection Type Discipline (ITD) of [3], [8], etc. T is the ω of ITD, where it does have a significant role to play.…”

Section: Axiomaticsmentioning

confidence: 99%

“…This is an adaptation of the Intersection Type Discipline, ITD, of [3], [7] and [8]. Aside from a somewhat different style of presentation, the present system differs from ITD only in allowing for disjunction, or union, types and also fusion types.…”

Section: Typed Combinatory Logicmentioning

confidence: 99%

“…It makes no special assumptions about these, however, except as the rules of ITD are extended to apply to the new vocabulary. 8 We follow Hindley and Seldin [12], Ch. 14, and its TA C in the presentation of this system, though the types assigned by TAS outrun those of TA C because of the additional kinds of types, and so TAS, like ITD, has some assignment rules that have no counterparts in TA C .…”

Section: Typed Combinatory Logicmentioning

confidence: 99%

“…As noted in footnote 1, T plays the role of ω in ITD of[3] and[8]. It is not required for the results to follow here, though it is needed for those of[3], etc 8.…”

mentioning

confidence: 99%

“…It is not required for the results to follow here, though it is needed for those of[3], etc 8. By allowing for disjunction types, TAS corresponds to the ITD∨ described in[8] (cf. also[2] and the assignment system induced by the type theory Ξ).…”

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confidence: 99%

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Combinatory Logic and the Semantics of Substructural Logics

Goble

2007

Stud Logica

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The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B•T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself.In this paper I explore how the concepts and structures of combinatory logic can be used to define a very broad class of substructural logics, both prooftheoretically and semantically. In particular, I establish a general result that, for every combinator, X, there is a corresponding logic LX defined by adding the set of types assigned to X (by an appropriate type assignment system TAS) as axioms to the logic B+T, or rather its extension B•T, that is the basic positive relevant or substructural logic familiar from the work of Routley and Meyer [17], [18], etc., and moreover this logic LX is characterized by the class of relational frames of the Routley-Meyer semantics for relevant logics that meet a first-order condition that corresponds in a very direct way to the combinator X itself.To demonstrate this result, even to articulate it precisely, requires laying some groundwork about the basic logic B•T and some elementary combinatory logic. Accordingly, in Section 1 below, I describe B•T and the relational semantics that applies to it and all its extensions to be considered later. Then in Section 2, I present some rudiments of combinatory logic, the theory of weak reduction, both untyped and typed. These two sections cover standard material, and are provided chiefly to establish a language for the results to follow and so that this paper will, by and large, be self-contained. Section 3 draws the two frameworks together, discovering in the relational semantics counterparts of the key elements of combinatory logic. This pro-

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Section: Axiomaticsmentioning

confidence: 99%

Section: Typed Combinatory Logicmentioning

confidence: 99%

Section: Typed Combinatory Logicmentioning

confidence: 99%

“…As noted in footnote 1, T plays the role of ω in ITD of[3] and[8]. It is not required for the results to follow here, though it is needed for those of[3], etc 8.…”

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confidence: 99%

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confidence: 99%

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