Substructural Logics and Residuated Lattices — an Introduction

Ono, Hiroakira

doi:10.1007/978-94-017-3598-8_8

Cited by 68 publications

(63 citation statements)

References 16 publications

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“…Relevance logic for example consists in adding (c) and (e) to the distributive Lambek calculus, adding only (c) defines the positive MALL + fragment of linear logic ( [Res02]), whilst the combination of (lw), (rw) and (e) defines affine logic. These structural rules correspond to (in)equations in the theory of residuated lattices (see [Res02,Ono03,GJKO07]); that is, in the language of L RL -algebras. Let us go through them in order.…”

Section: Additional Frame Conditionsmentioning

confidence: 99%

“…Many additional features could be added to the quotient of L(L RL + L ∂ RL ) under FC1-6 and FC ∂ 1-6, most notably one could define the 'negation' operations ∼ a ≜ a -⊗ J and ¬a ≜ J ⊗-a and use them to connect the behaviour of the two halves of the signature. We refer the reader to [Res02], [Ono03] and [GJKO07] for such considerations. To conclude we return to our heap model and show that it is a model for the logic we have just defined.…”

Section: Additional Frame Conditionsmentioning

confidence: 99%

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Coalgebraic completeness-via-canonicity for distributive substructural logics

Dahlqvist

Pym

2017

Journal of Logical and Algebraic Methods in Programming

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We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.

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Section: Additional Frame Conditionsmentioning

confidence: 99%

Section: Additional Frame Conditionsmentioning

confidence: 99%

Coalgebraic completeness-via-canonicity for distributive substructural logics

Dahlqvist

Pym

2017

Journal of Logical and Algebraic Methods in Programming

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“…We denote the variety of residuated bounded-lattices by bRL. Residuated bounded-lattices have been considered in the context of logic, see [15,16], because of the natural interpretation of the bounds as absolute truth and falsehood.…”

Section: Residuated Bounded-latticesmentioning

confidence: 99%

“…Residuated lattices provide algebraic semantics for substructural logics, logics that, when presented in a sequent calculus system, lack some of the three structural rules of exchange, contraction and weakening, see [12,15,16]. On the other hand, residuated lattices generalize well-studied algebraic structures, including latticeordered groups, Brouwerian algebras and (generalized) MV-algebras.…”

Section: Introductionmentioning

confidence: 99%

Minimal varieties of residuated lattices

Galatos

2005

Algebra univers.

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Abstract. In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated boundedlattice.

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“…2.1, 2.2, 2.3 below, that will be used in what follows, are well known in the lattice theory literature (e.g. [33]); in fact they are easy to establish by the reader herself. Hence we give them without proof.…”

Section: Introductionmentioning

confidence: 99%

Graded consequence: an institution theoretic study

Diaconescu

2014

Soft Comput

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We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multisignature one, in the spirit of institution theory. We consider graded consequence at three different conceptual levels: entailment, semantic, and closure operators, and explore several interpretations between them. We also study logical connectors and quantifiers both at the entailment and semantic level, compactness and soundness properties.

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Substructural Logics and Residuated Lattices — an Introduction

Cited by 68 publications

References 16 publications

Coalgebraic completeness-via-canonicity for distributive substructural logics

Coalgebraic completeness-via-canonicity for distributive substructural logics

Minimal varieties of residuated lattices

Graded consequence: an institution theoretic study

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