Translations from natural deduction to sequent calculus

Plato, Jan von

doi:10.1002/malq.200310047

Cited by 12 publications

(4 citation statements)

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“…In recent years, there is a lot of interest in what became known as general elimination rules (G E-rules), that are structured very much as the (∨E)-rule (see, for example, von Plato, 2001Plato, , 2003SchroederHeister, 1984;Tennant, 2002). The format of such a rule (for a generic operator ' * ') is…”

Section: General-elimination Rulesmentioning

confidence: 99%

On the Notion of Canonical Derivations From Open Assumptions and Its Role in Proof-Theoretic Semantics

Francez

2015

The Review of Symbolic Logic

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The paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of localcompleteness of a natural-deduction proof-system, underlying its harmony.

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Section: General-elimination Rulesmentioning

confidence: 99%

On the Notion of Canonical Derivations From Open Assumptions and Its Role in Proof-Theoretic Semantics

Francez

2015

The Review of Symbolic Logic

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“…In all article which studied the connections (I) and (II) from the Introduction (see for example Zucker (1974), Pottinger (1977), Minc (1996), von Plato (2003, Troelstra & Schwichtenberg (2000), Negri & von Plato (2001), von Plato (2011), and Dyckhoff (2015)), the connections between an N-system and a G-system (from Troelstra & Schwichtenberg (2000)) were studied. Namely, N-systems from Troelstra & Schwichtenberg (2000) have characteristics of natural deduction: its rules are in fact introduction and elimination rules from natural deduction.…”

Section: 1mentioning

confidence: 99%

AN ANALYSIS OF THE RULES OF GENTZEN’SNJANDLJ

Borisavljevic¹

2018

The Review of Symbolic Logic

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“…The translation T operates by transforming introduction rules into right rules and elimination rules into left ones (for details see von Plato, , section 5). In the case of ∧, for the introduction rule we have

\frac{\begin{matrix} Γ^{'} & Γ^{″} \\ ⋮ & ⋮ \\ A_{1} & A_{2} \end{matrix}}{A_{1} \land A_{2}} \land -intro^{⇝} T \frac{\begin{matrix} ⋮ & ⋮ \\ Γ ⊢ A_{1} & Γ ⊢ A_{2} \end{matrix}}{Γ ⊢ A_{1} \land A_{2}} \land_{R}

where Γ is equal to Γ′ ∪ Γ″ .…”

Section: Two Inferentialist Criteriamentioning

confidence: 99%

Are Uniqueness and Deducibility of Identicals the Same?

Naibo

Petrolo

2014

Theoria

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A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of analysis is driven by exploiting the Curry-Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity.1 Note also that Belnap requires the deducibility relation to satisfy a set of structural properties that contains the cut rule and the weakening rule(s); however, unlike Hacking, he does not explicitly ask for the admissibility of these rules (cf. Belnap, 1962, p. 132).3 The difference between uniqueness and strict uniqueness (Došen and Schroeder-Heister, 1985, p. 166) is not taken into account here: in a sequent calculus setting such a difference is difficult to identify in a very neat way. 4 Belnap's original definition of UNI is slightly different from the one we presented here. However, in presence of transitivity of the deducibility relation -i.e., in presence of the cut rule -our own definition entails Belnap's one. The latter will be discussed in more details in section 5.2.

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Translations from natural deduction to sequent calculus

Cited by 12 publications

References 10 publications

On the Notion of Canonical Derivations From Open Assumptions and Its Role in Proof-Theoretic Semantics

On the Notion of Canonical Derivations From Open Assumptions and Its Role in Proof-Theoretic Semantics

AN ANALYSIS OF THE RULES OF GENTZEN’SNJANDLJ

Are Uniqueness and Deducibility of Identicals the Same?

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