Structural Proof Theory

Negri, Sara; Plato, Jan von; Ranta, Aarne

doi:10.1017/cbo9780511527340

Cited by 394 publications

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“…Weak Normalisation (WN): every deduction can be replaced by a normal deduction of the same conclusion from the same assumptions [13,20,36,39]. 2.…”

Section: Cmentioning

confidence: 99%

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

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This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential "constant" • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on "general elimination rules" and harmony, with a retraction of one of the main ideas of that work, i.e. the use of "flattened" general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

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“…Weak Normalisation (WN): every deduction can be replaced by a normal deduction of the same conclusion from the same assumptions [13,20,36,39]. 2.…”

Section: Cmentioning

confidence: 99%

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

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“…Consider rule Cut in λ Gtz and rule Subst in λ Nat . First, we observe, as Negri and von Plato in [22], that the right cut-formula of Cut, but not the right substitution formula in Subst, may be the conclusion of a sequence of left introductions. Second, and here comes the novelty, we may also observe that the left substitution formula in Subst, but not left cut-formula in Cut, may be the conclusion of a sequence of elimination rules.…”

Section: Unificationmentioning

confidence: 75%

“…One of the main tasks of structural proof theory [22] is to investigate whether these differences between sequent calculus and natural deduction (and the concomitant relative (dis)advantages and different applications of the systems) are absolute or just apparent. This task has been carried out over the last 70 years [15,25,28,24,21,27], and the outcome is that the differences are most of the time just apparent.…”

Section: Introductionmentioning

confidence: 99%

The λ-Calculus and the Unity of Structural Proof Theory

Santo

2009

Theory Comput Syst

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Abstract. In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cutelimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a "coercion calculus", in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a "multiary" calculus, because "applicative terms" may exhibit a list of several arguments. But the combination of "multiarity" and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.

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“…Of course it is to some extent arbitrary since one can develop SC for intuitionistic logigic with many-many sequents (see e.g. Negri, von Plato [42]), and as we will see in the next sections, most of formalizations of classical logic use many-one sequents. 2.6 Dosen's structural SC [13].…”

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

confidence: 99%

“…It is worth mentioning that interesting application of such generalised elimination rules but in standard natural deduction is provided by Negri and von Plato [42]. Let us note that in this context every elimination rule is a proof construction rule since we do not break every compound formula directly but by means of subproofs initiated with respective subformulae.…”

Section: Leblanc's Systemmentioning

confidence: 99%

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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Structural Proof Theory

Cited by 394 publications

References 0 publications

Some Remarks on Proof-Theoretic Semantics

Some Remarks on Proof-Theoretic Semantics

The λ-Calculus and the Unity of Structural Proof Theory

A Survey of Nonstandard Sequent Calculi

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