Basic Relevant Theories for Combinators at Levels I and II

Pal, Koushik; Meyer, Robert K.

doi:10.26686/ajl.v3i0.1770

Cited by 3 publications

(2 citation statements)

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“…The canonical application of the bubbling lemma for B ∧ is in providing nice filter models of the λ-calculus (with intersection types) and combinatory logic. In the relevant logic literature, Meyer and a number of co-authors over the years (see [16,14,28]) used this fact to build nice ternary relation models of combinatory logic. 13…”

Section: Some Sufficient Conditions For Channel Compositionmentioning

confidence: 99%

Information Flow in Logics in the Vicinity of BB

Tedder¹

2021

AJL

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Situation theory, and channel theory in particular, have been used to provide motivational accounts of the ternary relation semantics of relevant, substructural, and various non-classical logics. Among the constraints imposed by channel-theory, we must posit a certain existence criterion for situations which result from the composites of multiple channels (this is used in modeling information flow). In associative non-classical logics, it is relatively easy to show that a certain such condition is met, but the problem is trickier in non-associative logics. Following Tedder (2017), where it was shown that the conjunction-conditional fragment of the logic B admits the existence of composite channels, I present a generalised ver- sion of the previous argument, appropriate to logics with disjunction, in the neighbourhood ternary relation semantic framework. I close by suggesting that the logic BB+(^I), which falls between Lavers' system BB+ and B+ , satisfies the conditions for the general argument to go through (and prove that it satisfies all but one of those conditions).

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Section: Some Sufficient Conditions For Channel Compositionmentioning

confidence: 99%

Information Flow in Logics in the Vicinity of BB

Tedder¹

2021

AJL

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“…The theory of (weak) equality is more challenging; see[3] for an account that succeeds based on the filters of ITD (or the theories of the logic B∧T), and[8],[14], and[16] for discussions of the difficulties applying that method when the type theory is extended to include disjunction, or union, types (or to the logic B+T).…”

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confidence: 99%

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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The Better Bubbling Lemma

Meyer

2007

Electronic Notes in Theoretical Computer Science

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Basic Relevant Theories for Combinators at Levels I and II

Cited by 3 publications

References 12 publications

Information Flow in Logics in the Vicinity of BB

Information Flow in Logics in the Vicinity of BB

Combinatory Logic and the Semantics of Substructural Logics

The Better Bubbling Lemma

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