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Bibliography
The definition of many-valued systems of calculus of propositions (or—more briefly—manyvalued logics) can be found in section 10 of prof.J. Łukasiewicz's paperZnaczenie analizy logicznej dla poznania, Przegląd Filozoficzny, vol. XXXVII, Warszawa 1934.
The rules of notation without parentheses of meaningful propositions can be found in mimeographed notes: JanŁukasiewicz,Elementy logiki matematycznej. Skrypt autoryzowany opracował M. Presburger. Nakładem Komisji Wydawniczej Koła Matematyczno-Fizycznego Słuchaczów Uniwersytetu Warszawskiego, Warszawa 1929. Reader can also find there definition of the notion of meaningful proposition of calculus of propositions.
Reader can find definition of notion of matrix in:J. Łukasiewicz andA. Tarski,Badania nad rachunkiem zdań, Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego, Vol. XXIII, 1930, Wydział III, § 1, Definition 3, p. 4.
Our deliberations will concern also two-valued logic, so further we shall use the term “many-valued logic” in a wider sense which includes two-valued calculus of propositions, too.
Reader can find this theorem in the paper:Sur les fonctions définies dans les ensembles finis quelconques, par SophiePiccard (Neuchâtel), Fundamenta Mathematicae, vol. XXIV, Warszawa 1935, pp. 298–301.
The proof of this theorem will follow the pattern of the proof of completeness of two-valued calculus of propositions which is in prof.J. Łukasiewicz's Elementy logiki matematycznej (particulars concerning these mimeographed lectures are given in the reference 2). In that paper the reader can also find exact definitions of series of notions, e.g. the ones of independent proposition, inferential equivalence of propositions, complete system and others; the notions will be used in our proof.
Bounds for the values which can be taken by the superscripti in this and further, analogous cases will not play an essential rôle later in our proof; therefore we shall not go into details of questions connected with it. The fact which is important for us is finiteness of these bounds, and that can be checked in every case with no difficulties.
Prof. J. Łukasiewicz regards this proposition to be the shortest axiom of two-valued implicative logic. Cf.J. Łukasiewicz,W obronie logistyki, Studia Gnesnensia 15, Poznań 1937.
We use reflexiveness of relation of inferential equivalence here.
Let us remark that propositions appearing on the right side of either equivalence (b) or (c) of the lemma IV are meaningful propositions of orders not exceeding the orders of propositions which appear on the left side of respective equivalence.
The theorem generalizes results of prof. S. Leśniewski concerning two-valued logic. The results were not published.
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Editor's note. This paper appeared originally in Polish under the titleDowód aksjomatyzowalności pelnych systemów wielowartościowych rachunku zdań in Comptes Rendus des Seances de la Sociéte des Sciences et des Lettres de Varsovie XXXII, Année 1939, Classe III, Fasc. 1–3, pp. 110–128.
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Słupecki, J. Proof of axiomatizability of full many-valued systems of calculus of propositions. Stud Logica 29, 155–168 (1971). https://doi.org/10.1007/BF02121872
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DOI: https://doi.org/10.1007/BF02121872