Expanding the Realm of Systematic Proof Theory

Ciabattoni, Agata; Straßburger, Lutz; Terui, Kazushige

doi:10.1007/978-3-642-04027-6_14

Cited by 19 publications

(22 citation statements)

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“…Above, we use the notation ∧ and ∨ to reflect that in a classical calculus, the connectives conjunction and disjunction satisfy the respective properties. Our recipe abstracts and reformulates for display calculi the procedure in [5,6], defined for (hyper)sequent calculi and substructural logics. To transform axioms into structural rules we use: (1) the invertible logical rules of C and (2) the display calculus formulation, below, of the so-called Ackermann's lemma that allows a formula in a rule to switch sides of the sequent moving from conclusion to premises.…”

Section: Definition 4 (Amenable Calculus)mentioning

confidence: 99%

“…[5,6,16,14,15,11] introduce methods to extract rules out of suitable Hilbert axioms. More precisely [5,6] generate sequent and hypersequent rules, [11] nested sequent rules, [15] sequent rules for certain modal axioms, and [16] labelled rules; finally [14] transforms suitable modal and tense axioms (called primitive tense axioms) into structural rules for the display calculus. [14] also provides a characterisation as it is shown that each such rule added to the base system is equivalent to the extension of the logic by primitive tense axioms.…”

Section: Introductionmentioning

confidence: 99%

“…All the above results start with a specific logic and introduce calculi for (some of) its axiomatic extensions, e.g., Full Lambek calculus with exchange FLe for [5,6], or the tense logic Kt for [14,11]. This paper proposes instead a recipe that utilises the more common Hilbert axioms to construct analytic calculi 1 .…”

Section: Introductionmentioning

confidence: 99%

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Structural Extensions of Display Calculi: A General Recipe

Ciabattoni

Ramanayake

2013

Logic, Language, Information, and Computation

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Abstract. We present a systematic procedure for constructing cut-free display calculi for axiomatic extensions of a logic via structural rule extensions. The sufficient conditions for the procedure are given in terms of (purely syntactic) abstract properties of the display calculus and thus the method applies to large classes of calculi and logics. As a case study, we present cut-free calculi for extensions of well-known logics including Bi-intuitionistic and tense logic.

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Section: Definition 4 (Amenable Calculus)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Structural Extensions of Display Calculi: A General Recipe

Ciabattoni

Ramanayake

2013

Logic, Language, Information, and Computation

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“…In recent years, these questions have been intensely investigated in the context of various proof-theoretic formalisms (cf. [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.…”

Section: Tools Of Unified Correspondence Theorymentioning

confidence: 99%

“…9 ALBA is the acronym of Ackermann Lemma Based Algorithm. 10 Throughout the paper, order-types will be typically associated with arrays of variables p := (p 1 , . .…”

Section: Definitionmentioning

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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Hypersequent and Labelled Calculi for Intermediate Logics

Ciabattoni

Maffezioli

Spendier

2013

Lecture Notes in Computer Science

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Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cut-free calculi for non-classical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rules. We show that these methods are closely related and we extend them to capture larger classes of intermediate logics.

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Expanding the Realm of Systematic Proof Theory

Cited by 19 publications

References 18 publications

Structural Extensions of Display Calculi: A General Recipe

Structural Extensions of Display Calculi: A General Recipe

Unified correspondence as a proof-theoretic tool

Hypersequent and Labelled Calculi for Intermediate Logics

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