Natural deduction with general elimination rules

Plato, Jan von

doi:10.1007/s001530100091

Cited by 87 publications

(74 citation statements)

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“…These can be motivated in terms of Prawitz's inversion principle 1 : "the conclusion obtained by an elimination does not state anything more than what must have already been obtained if the major premiss of the elimination was inferred by an introduction" [25, p. 246]. Normality is now the simple idea [39] that the major premiss of each elimination step should be an assumption; see also [13,36].…”

Section: Background On General Elimination Rulesmentioning

confidence: 99%

“…López-Escobar [13] distinguishes between the premiss A of ⊃G E as a "minor" premiss and that of C (assuming B) as a "transfer" premiss. 3 One thus has a calculus of rules in natural deduction style for Int; such calculi, and their normalisation results, have been studied by von Plato [39], by López-Escobar [13] and by Tennant [36]. With the definition (given above) that a deduction is normal iff the major premiss of every elimination step is an assumption, the main results are:…”

Section: Cmentioning

confidence: 99%

“…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%

“…This raises difficulties for the construction of the appropriate GE-rules: Prawitz [26] got it wrong (corrected in [27]), SchroederHeister [30] gave an answer in the form of a system of rules of higher level, allowing discharge not just of assumptions but of rules (which may themselves discharge …)-but, although much cited, use of this system seems to be modest. As already discussed, an alternative was mentioned (disparagingly) in [3] and (independently) adopted more widely by others [13,36,39], the "flattened" GE-rule for ⊃ being…”

Section: Premiss Of I-rule Discharges Some Assumptionsmentioning

confidence: 99%

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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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Section: Background On General Elimination Rulesmentioning

confidence: 99%

Section: Cmentioning

confidence: 99%

“…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%

Section: Premiss Of I-rule Discharges Some Assumptionsmentioning

confidence: 99%

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Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

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“…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. Von Plato [44] has shown that such a solution allows much simpler proof of normalization theorem for natural deduction system.…”

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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Translations from natural deduction to sequent calculus

Plato

2003

Mathematical Logic Qtrly

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Gentzen's "Untersuchungen" [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination.

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Natural deduction with general elimination rules

Cited by 87 publications

References 5 publications

Some Remarks on Proof-Theoretic Semantics

Some Remarks on Proof-Theoretic Semantics

A Survey of Nonstandard Sequent Calculi

Translations from natural deduction to sequent calculus

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