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A criterion of fullness of many-valued systems of propositional logic

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References

  1. The reader will find a definition of many-valued systems of the propositional logic, or shortly of many-valued logics in § 10 of the paperZnaczenie analizy lgicznej dla poznania (The significance of logical analysis for the knowledge) by J. Łukasiewicz, in Przegląd Filozoficzny, Vol. 37, Warszawa 1934.

  2. The proof of this theorem was given in my still unpublished workPelny trójwartościowy rachunek zdań (The full three-valued propositional calculus). [That paper has been published afterwards in: Annales Universitatis Mariac Curie-Skłodowska, Sectio F, Vol. I, 1946, pp. 193–209.—Editor.] In that work the above property of the binary functors of the three-valued logic was proved. However, the generalization of this theorem on any finite-many-valued logic does not pose any difficulties.

  3. The theorem can be found in the work:Sur les fonctions definies dans les ensembles finis quelconques par Sophie Piccard (Neuchâtel), in Fundamenta Mathematicae, Vol. 24, Warszawa 1935, pp. 298–301.

  4. The reader will find a definition of the concept of an interpretative table in the work by Jan ŁukasiewiczElementy logiki matematycznej, Warszawa 1929. [English translation:Elements of Mathematical Logic, Oxford 1963, Pergamon Press.—Editor.]

  5. In this paper we shall use the parantheses-free notation of J. Łukasiewicz. Its explanation can be found in theElements of Mathematical Logic (cf. note 4.)

  6. This part of our proof follows analogical proofs of J. Łukasiewicz.

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Authors
  1. Jerzy Słupecki
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Editor's note. This paper appeared originally in Polish under the titleKryterium pelności wielowarteściowych systemów logiki zdań in: Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 32 (1939), pp. 102–109.

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Słupecki, J. A criterion of fullness of many-valued systems of propositional logic. Stud Logica 30, 153–157 (1972). https://doi.org/10.1007/BF02120845

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