Kneale’s Natural Deductions as a Notational Variant of Beth’s Tableaus


Gentzen’s singular sequential system of first-order logic was an alternative notation for his system of natural deductions. His multiple sequential system was his symmetric generalization that was more appropriate to classical logic. Beth’s tableaus system was a system that was derived directly from the semantic analysis of connectives and quantifiers. It was soon realized that the Beth’s system and the Gentzen’s multiple system were only notational variants of each other. Kneale’s system of multiple natural deductions was a generalization of Gentzen’s system of natural deductions. We prove that Kneale’s natural deductions are also a notational variant of Beth’s tableaus.

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Correspondence to Zvonimir Šikić.

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Dedicated to the late Kosta Došen who asked me to write it.

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The paper was awarded the Georgius Beningus Logic Prize (Croatia) and will be presented at the universal logic contest at UNILOG 2022 in Crete.

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Šikić, Z. Kneale’s Natural Deductions as a Notational Variant of Beth’s Tableaus. Log. Univers. (2021).

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  • Beth’s tableaus
  • Gentzen’s sequential system
  • Kneale’s natural deductions

Mathematics Subject Classification

  • 03B10