Display logic

Belnap, Nuel D.

doi:10.1007/bf00284976

Cited by 212 publications

(219 citation statements)

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“…This property of the structural rules is similar to Belnap's condition (C6) for cut elimination for display logics [2]. A primitive axiom is an axiom of the form A → B where both A and B are built using propositional variables, ∧, ∨, ♦, and .…”

Section: Lemma 2 (Admissibility Of Weakening) Supposementioning

confidence: 83%

“…Belnap's Display Logic [2] (we prefer the term display calculi) is an extremely general proof-theoretical framework with the property that any sequent containing a particular formula occurrence A can be transformed into another sequent in which the occurrence of A is either the whole of the antecedent or the whole of the succedent, using only a subset of the rules called the display postulates. The occurrence of A is then said to be displayed.…”

Section: Introductionmentioning

confidence: 99%

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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: 83%

Section: Introductionmentioning

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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“…We allow sequences, multisets or sets of formulae but exclude from further considerations other kinds of structures. Hence we do consider neither display calculi invented by Belnap [6] (see Ciabattoni, Ramanayake, Wansing [10] in this issue) nor labelled sequent calculi of different sorts 5 . In general all these systems briefly referred to in this section, and do not taken into consideration in further sections, may be called generalised sequent calculi (GSC).…”

Section: Special Issue: Gentzen's and Jaśkowski's Heritage 80 Years Omentioning

confidence: 99%

“…One can for example distinguish between systems which use intuitionistic sequents with at most one formula in the succedent of a sequent (many-one) and systems using classical sequents which admit more formulae (many-many) 6 . Other solutions, e.g.…”

Section: Special Issue: Gentzen's and Jaśkowski's Heritage 80 Years Omentioning

confidence: 99%

“…calculi belonging either to the first type but with nonprogressive rules, or belonging to the remaining two types. 6 The names refer to Gentzen's original solution where a calculus LJ with sequents of the first sort was defined for intuitionistic logic, whereas LK with sequents of the second sort was developed for classical logic. Of course it is to some extent arbitrary since one can develop SC for intuitionistic logigic with many-many sequents (see e.g.…”

Section: Special Issue: Gentzen's and Jaśkowski's Heritage 80 Years Omentioning

confidence: 99%

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A Survey of Nonstandard Sequent Calculi

Indrzejczak

2014

Stud Logica

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Abstract. The paper is a brief survey of some sequent calculi (SC) which do not follow strictly the shape of sequent calculus introduced by Gentzen. We propose the following rough classification of all SC: Systems which are based on some deviations from the ordinary notion of a sequent are called generalised; remaining ones are called ordinary. Among the latter we distinguish three types according to the proportion between the number of primitive sequents and rules. In particular, in one of these types, called Gentzen's type, we have a subtype of standard SC due to Gentzen. Hence by nonstandard ones we mean all these ordinary SC where other kinds of rules are applied than those admitted in standard Gentzen's sequent calculi. We describe briefly some of the most interesting or important nonstandard SC belonging to the three abovementioned types.Keywords: Sequent Calculi, Natural Deduction, Proof Methods. IntroductionThe notion of sequent calculus (SC) is commonly, and rightly, connected with the name of Gerhard Gentzen. Quite automatically, we think of calculi with structural and logical rules introducing constants either to antecedent or to succedent of a sequent. However, one should be aware that nowadays this term may be applied not only to such calculi like original Gentzen's LJ, LK or their variants, but also to a variety of calculi which use sequents in a significantly different way than they are used in Gentzen's like calculi. Some of them are even older than Gentzen's LK (or LJ) -like Hertz's calculus -but most were invented after Gentzen, mainly for providing more flexible or natural systems for actual proof search than original Gentzen's calculus constructed rather for theoretical purposes. In this paper we survey some of the most important or interesting proposals. The paper is based on the 10th chapter of Indrzejczak [31].Let us define an ordinary sequent calculus (SC) as a finite collection of (schemata) of (primitive) sequent rules of the form: Sequents on the left of / are premises of the rule, whereas S n+1 is its conclusion. Note that n, i.e. the number of premises, may be 0 -in this case we will say on (primitive) axiomatic sequent; in case n > 0 we will simply say on rules. Note that every SC have a nonempty set of primitive sequents and a nonempty set of rules 1 .So far, the only restriction we put on the notion of a sequent calculus is that rules have unique conclusions. The next restriction is connected with the very notion of a sequent. We define it in a standard way as an ordered pair Γ ⇒ Δ, where Γ is the antecedent and Δ the succedent of a sequent. Hence we exclude from our considerations such calculi which operate on sequents having more arguments 2 or being structures of different character 3 or using more types of sequents in one system 4 .One should also specify what kind of structures are denoted by arguments of a sequent. We allow sequences, multisets or sets of formulae but exclude from further considerations other kinds of structures. Hence we do consider neither display calculi ...

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Proofs and Expressiveness in Alethic Modal Logic

Rijke¹,

Wansing²

2006

A Companion to Philosophical Logic

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Display logic

Cited by 212 publications

References 11 publications

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

Taming Displayed Tense Logics Using Nested Sequents with Deep Inference

A Survey of Nonstandard Sequent Calculi

Proofs and Expressiveness in Alethic Modal Logic

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