An Introduction to Substructural Logics

Restall, Greg

doi:10.4324/9780203252642

Cited by 225 publications

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“…The class of T RL -coalgebras satisfying the second frame condition is described in Theorem 19. It is intuitive and indeed corresponds to the usual relational semantics of distributive substructural logics (see, e.g., [Res02]) or separation logic/BI. However, proving it 'from first principles' as we do here is much more intricate than might be expected and, indeed, much more so than is clear in [DP15].…”

Section: Describing T Rl -Coalgebras Validating Fc1-6mentioning

confidence: 99%

“…These rules are the exchange rule (e), the contraction rule (c), the left weakening rule (lw), and the right weakening rule (rw). 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.…”

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: Describing T Rl -Coalgebras Validating Fc1-6mentioning

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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“…Display calculi have been successful in giving adequate proof-theoretic semantic accounts of logics-such as certain modal and substructural logics [30], and more recently also Dynamic Epistemic Logic [25] and PDL [24]-which have notoriously been difficult to treat with other approaches. Here we mainly report and elaborate on the work of Belnap [2], Wansing [51], Goré [30,29], and Restall [43].…”

Section: Preliminaries On Display Calculimentioning

confidence: 99%

“…C 2 : Shape-alikeness of parameters. This condition is based on the relation of congruence between parameters (i.e., non-active parts) in inferences; the congruence relation is an equivalence relation which is meant to identify the different occurrences of the same formula or substructure along the branches of a derivation [2, Section 4], [43,Definition 6.5]. Condition C 2 requires that congruent parameters be occurrences of the same structure.…”

Section: Proper Display Calculi and Canonical Cut Eliminationmentioning

confidence: 99%

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

158

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The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap.These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into 'analytic' structural rules of display calculi. In this context, a rule is 'analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem.In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvisttype classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices).We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics. Keywords: Display calculi, unified correspondence, distributive lattice expansions, properly displayable logics. Math. Subject Class. 03B45, 06D50, 06D10, 03G10, 06E15. , and mu-calculus [11,12]. The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [40,15], or of the phenomenon of pseudocorrespondence [19]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [27]. Finally, the insights of unified correspondence theory have made it possible to determine the extent to which the Sahlqvist theory of classes of normal DLEs can be reduced to the Sahlqvist theory of normal Boolean expansions, by means of Gödel-type translations [20]. These and other results have given rise to a theory called unified correspondence [13]. Contents Tools of unified correspondence theory.The most important technical tools in unified correspondence are: (a) a very general syntactic definition of the class of Sahlqvist formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; (b) the algorithm ALBA, which effectively computes first-order correspondents of input term-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which, like the Sahlqvist class, can be defi...

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“…In the canonical model, the points are non-trivial prime theories [4]. In other words, they are sets of sentences closed under consequence (if A B holds and A ∈ X then B ∈ X) and conjunction (if A, B ∈ X then A ∧ B ∈ X), respecting the constants ( ∈ X and ⊥ ∈ X) and disjunction (if A ∨ B ∈ X then A ∈ X or B ∈ X).…”

Section: Lemma 23 the Canonical Model For A Logic Containing ∼(A∧b) mentioning

confidence: 99%

Defining double negation elimination

Restall

2000

Logic Journal of IGPL

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In his paper "Generalised Ortho Negation" [2] J. Michael Dunn mentions a claim of mine to the effect that there is no condition on 'perp frames' equivalent to the holding of double negation elimination ∼∼A A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. 1 Compatibility FramesDunn's work on general models for negation has been a significant advance in our understanding of negation in non-classical logics [1,2]. These models generalise Kripke models for intuitionistic logic and Routley-Meyer models for relevant implication. I will recount the essential details of these models for negation here before we look at the behaviour of inference patterns such as double negation elimination. Definition 1.1 A frame is a triple P, C, consisting of a set P of points, and two binary relations C and on P such that• is a partial order on P . That is, is reflexive, transitive and antisymmetric on P .• C is antitone in both places. That is, for any x and y in P , if xCy, x x and y y then x Cy .The relation between points may be interpreted as one of information inclusion. A point x is informationally included in y if everything warranted by x (or encoded by x or made true by x or included in x or however else information is thought to be related to points) is also warranted by y. This is a transitive, reflexive relation on points. If the information content of a point is all you care about (as it is here) then antisymmetry is also an appropriate condition for inclusion.C is the relation of compatibility between points. 2 A point x is compatible with y if and only if all claims excluded by x are not warranted by y. It is clear that this merits the antitonicity condition -if x is compatible with y, then x (excluding no more than x) must be compatible with y (warranting no more than y).It is important to realise what "excluding" amounts to here. A point x excludes a piece of information when it rules it out. A point might neither warrant nor exclude 1 Thanks to anonymous referees of the journal for comments which helped improve the presentation of this paper. 2 I use 'C' for a relation of compatibility instead of '⊥' for the relation of incompatibility used by Dunn, simply because I use '⊥' as the false proposition which entails all others. The difference is merely notational. 853L.

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An Introduction to Substructural Logics

Cited by 225 publications

References 0 publications

Coalgebraic completeness-via-canonicity for distributive substructural logics

Coalgebraic completeness-via-canonicity for distributive substructural logics

Unified correspondence as a proof-theoretic tool

Defining double negation elimination

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