Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different

Hazen, Allen; Pelletier, Francis Jeffry

doi:10.1007/s11225-014-9564-1

Cited by 19 publications

(11 citation statements)

References 33 publications

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“…In a recent paper, Hazen and Pelletier (2014) compared Jaśkowski's (1934) and Gentzen's (1934) versions of natural deduction. In Section 2.3 of their paper, Hazen and Pelletier consider Price's (1961) natural deduction rules for the Sheffer stroke (NAND).…”

Section: Introductionmentioning

confidence: 99%

Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

Zach

2015

J Philos Logic

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Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke (NAND) and Peirce's arrow (NOR). The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of the connectives in question.

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Section: Introductionmentioning

confidence: 99%

Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

Zach

2015

J Philos Logic

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“…This can be seen by comparing Gentzen's ND-systems with those of Jaśkowski [13,14], where the latter are not so strictly divided into (I/E)-rules. For a detailed discussion of those differences (viewed only from the traditional view of formalising mathematics), see [9].…”

Section: Discussionmentioning

confidence: 99%

Does the Implication Elimination Rule Need a Minor Premise?

Francez

2017

LLP

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Abstract. The paper introduces NJg , a variant of Gentzen's NJ natural deduction system, in which the implication elimination rule has no minor premise. The NJ g -systems extends traditional ND-systems with a new kind of action in derivations, assumption incorporation, a kind of dual to the assumption discharge action. As a result, the implication (I/E)-rules are invertible and, almost by definition, harmonious and stable, a major condition imposed by proof-theoretic semantics on ND-systems to qualify as meaning-conferring. There is also a proof-term assignment to NJ gderivations, materialising the Curry-Howard correspondence for this system.

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“…In 1, I indicated that negation rules can be added without any problems. The translations extend to negation rules in a 24 For more on the trade offs involved in cascade and tree systems, see Restall (2014) and Hazen & Pelletier (2014). 25 I have presented the (•E) rule that I did, rather than a general elimination-style rule to keep the correspondence with cascade systems.…”

Section: X C a Amentioning

confidence: 99%

Translations Between Linear and Tree Natural Deduction Systems for Relevant Logics

Standefer

2021

The Review of Symbolic Logic

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Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

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Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different

Cited by 19 publications

References 33 publications

Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

Does the Implication Elimination Rule Need a Minor Premise?

Translations Between Linear and Tree Natural Deduction Systems for Relevant Logics

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