Ternary relations and relevant semantics

Meyer, Robert K.

doi:10.1016/j.apal.2003.11.015

Cited by 8 publications

(4 citation statements)

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“…Quite recently, after an early start [43], a few publications concentrated on further developing a proper semantics for relevant logics [5,37,41]. There, a possible worlds approach is also defined, but with the aid of a ternary accessibility relation between worlds.…”

Section: Relevance Logicsmentioning

confidence: 99%

Constrained Consequence

2011

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There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss-or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reasoning demands (beyond α |= β) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence •α with β |= •α. Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies.Mathematics Subject Classification (2010). Primary 03B20; Secondary 03B22, 03B45, 03B47, 03B53, 03B60, 03A05.

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Section: Relevance Logicsmentioning

confidence: 99%

Constrained Consequence

2011

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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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“…Meyer (2004) added in addition the axiom → (A → ) to the •-free axiomatization of the positive fragment of B. The reason, presumably, being that the •-free fragment would otherwise be non-conservatively extended by •.3 If the structure to be replaced is clear from context, then I will on occasion drop the slashnotation to increase readability.Australasian Journal of Logic (18:2) 2021, Article no.…”

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confidence: 99%

From Hilbert proofs to consecutions and back

Øgaard

2021

AJL

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Restall set forth a "consecution" calculus in his An Introduction to Substructural Logics. This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules.

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Ternary relations and relevant semantics

Cited by 8 publications

References 25 publications

Constrained Consequence

Constrained Consequence

Combinatory Logic and the Semantics of Substructural Logics

From Hilbert proofs to consecutions and back

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