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Axiom schemes for m-valued propositions calculi

Published online by Cambridge University Press:  12 March 2014

J. B. Rosser  and
A. R. Turquette
J. B. Rosser
Affiliation:
Cornell University
A. R. Turquette
Affiliation:
Cornell University
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Extract

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 10 , Issue 3 , September 1945 , pp. 61 - 82
Copyright
Copyright © Association for Symbolic Logic 1945

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References

1 Łukasiewicz, Jan, O logice trójwartościowej, Ruch filozoficzny, vol. 5 (1920), pp. 169–171.Google Scholar

2 Łukasiewicz, Jan and Tarski, Alfred, Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 30–50.Google Scholar

3 Wajsberg, M., Aksjomalyzacja trójwartościowego rachunku zdań, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 24 (1931), pp. 259–262.Google Scholar

4 Post, Emil L., Introduction to a general theory of elementary propositions, American journal of mathematics, vol. 43 (1921), pp. 163–185.CrossRefGoogle Scholar

5 Post's own term is “complete,” but since we shall find it necessary to distinguish various types of completeness, we add the adjective “functional” to Post's original term and shall mean by “functional completeness” what Post means by “completeness.”

6 Slupecki, Jerzy, Der volle dreiwertige Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 29 (1936), pp. 9–11.Google Scholar

7 Op. cit.

8 Throughout the present paper when Łukasiewicz-Tarski calculi are being considered, we will adopt the type of symbolism used by these Polish logicians.

9 Carl Hempel states definitions for some formulas which have the same truth-value properties as our Jκ(P) in his, Ein System verallgemeinerter Negationen (See Travaux du IXe Congrès International de Philosophie, vol. 6 (1937), pp. 26-32). We believe thatoural-ternative definitions are not without interest.

10 Throughout the present paper theorems will be numbered in the following manner: The κth theorem of the s th section will have the number s.k.

11 See p. 61 of the present paper.

12 Op. cit.

13 The notion of “strong completeness” originated with Emil Post (op. cit., p.177). Since Post does not distinguish various types of completeness, however, the notion is introduced through what he calls a “closed system” and Post does not use the qualified term “strong completeness.”

14 For example, see Post's definition of consistency, op. cit., p. 177.

15 See pp. 65-66 of the present paper.

16 For the distinction between axiom schemes and postulates see von Neumann, J., Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46.CrossRefGoogle Scholar

17 See p. 74 of the present paper.

18 See p. 70 of the present paper.

19 Op. cit.

20 See pp. 73-74 of the present paper.

21 See p. 74 of the present paper.

22 See theorem 2.8, p. 67 of the present paper.

23 We will now abandon our use of the Polish logical symbolism. Our justification for this is twofold: (1) The Polish logical symbolism was used in section 2 since we were there concerned with the formalization of m-valued prepositional calculi based on the connectives C, N, and T. Most of the literature dealing with these calculi makes consistent use of Polish logical symbolism. Hence, any change of symbolism in working with these systems would only enhance the burdens of comparative study. (2) In the present section we shall be concerned with m-valued propositional calculi in general and not with particular calculi based on definite connectives. In a forthcoming paper the present authors will extend this general treatment of m-valued calculi to cover the theory of quantification. In this extension the ordinary Peano-Russell symbolism proves to be most useful. For consistency of symbolism, then, it seems advisable to introduce the Peano-Russell symbolism at the beginning of the general treatment in the present paper.

24 By virtue of assumption (2) we may define a product and the sums Sa(Pι, … , Pn), and Sκ in terms of ˙ and v in a strictly analogous manner to the way in which these were defined in section 2 in terms of Κ and A. See pp. 66, 69, and 72 of the present paper.

25 See p. 71 of the present paper.

26 See pp. 72-73 of the present paper.

27 See the definition of plausibility, p. 68.

28 Both formalizations and functions satisfying standard conditions are of particular importance in the study of m-valued quantification theory.

29 See pp. 74-76 of the present paper.

30 See pp. 68-73 of the present paper.

31 Note, too, that IPQ is definable in an analogous way for s = 1 and will satisfy standard conditions in this case.

32 This is especially true in working with the theory of quantification for m-valued calculi.

33 Note that, like IPQ, is definable analogously for s = 1, and will satisfy standard conditions in this case.

34 See p. 80 of the present paper.

35 See p. 81 of the present paper.

36 See p. 62 of the present paper.