Power and Weakness of the Modal Display Calculus

Kracht, Marcus

doi:10.1007/978-94-017-2798-3_7

Cited by 75 publications

(160 citation statements)

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“…A primitive axiom is an axiom of the form A → B where both A and B are built using propositional variables, ∧, ∨, ♦, and . Kracht [13] shows that any extension of tense logic with primitive axioms has a display calculus which enjoys cut elimination. He shows that any such axiom can be turned into a left structural rule.…”

Section: Lemma 2 (Admissibility Of Weakening) Supposementioning

confidence: 99%

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

Goré

Postniece

Tiu

2009

Lecture Notes in Computer Science

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Abstract. We consider two sequent calculi for tense logic in which the syntactic judgements are nested sequents, i.e., a tree of traditional onesided sequents built from multisets of formulae. Our first calculus SKt is a variant of Kashima's calculus for Kt, which can also be seen as a display calculus, and uses "shallow" inference whereby inference rules are only applied to the top-level nodes in the nested structures. The rules of SKt include certain structural rules, called "display postulates", which are used to bring a node to the top level and thus in effect allow inference rules to be applied to an arbitrary node in a nested sequent. The cut elimination proof for SKt uses a proof substitution technique similar to that used in cut elimination for display logics. We then consider another, more natural, calculus DKt which contains no structural rules (and no display postulates), but which uses deep-inference to apply inference rules directly at any node in a nested sequent. This calculus corresponds to Kashima's S2Kt, but with all structural rules absorbed into logical rules. We show that SKt and DKt are equivalent, that is, any cut-free proof of SKt can be transformed into a cut-free proof of DKt, and vice versa. We consider two extensions of tense logic, Kt.S4 and S5, and show that this equivalence between shallow-and deep-sequent systems also holds. Since deep-sequent systems contain no structural rules, proof search in the calculi is easier than in the shallow calculi. We outline such a procedure for the deep-sequent system DKt and its S4 extension.

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Section: Lemma 2 (Admissibility Of Weakening) Supposementioning

confidence: 99%

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

Goré

Postniece

Tiu

2009

Lecture Notes in Computer Science

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“…In [33], Kracht establishes a correspondence between special rules and primitive formulas in the setting of tense modal logic, which will be generalized in Section 5.1 below.…”

Section: Remarkmentioning

confidence: 99%

“…In [33], Kracht states without proof that any analytic structural rules in the language of classical tense logic Kt is equivalent to some special structural rule. Kracht's claim has been proved with model-theoretic techniques in [9], [42].…”

Section: Remarkmentioning

confidence: 99%

“…In [33,Theorem 16], Kracht showed that primitive formulas of basic normal/tense modal logic on a classical propositional base can be equivalently transformed into (a set of) special structural rules satisfying the defining conditions of proper display calculi (cf. Subsection 2.2).…”

Section: Primitive Inequalities and Special Rulesmentioning

confidence: 99%

“…[39,6,10,32,7,36,34,38,35]). Perhaps the first paper in this line of research is [33], which addresses these questions in the setting of display calculi for basic normal modal and tense logic. Interestingly, in [33], the connections between Sahlqvist theory and display calculi started to be observed, but have not been systematically explored there nor (to the knowledge of the authors) in subsequent papers in the same research line.…”

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confidence: 99%

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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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Proofs and Expressiveness in Alethic Modal Logic

Rijke¹,

Wansing²

2006

A Companion to Philosophical Logic

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Power and Weakness of the Modal Display Calculus

Cited by 75 publications

References 7 publications

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

Unified correspondence as a proof-theoretic tool

Proofs and Expressiveness in Alethic Modal Logic

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