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Axiom schemes for m-valued functional calculi of first order. Part II1

Published online by Cambridge University Press:  12 March 2014

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

In part I of the present paper axiom schemes and rules of inference were defined for m-valued functional calculi of first order with s(m > s > 1) designated truth-values. A proof of plausibility was given, and it was shown that it is not difficult to extend to m-valued functional calculi of first order certain concepts that are closely analogous to the ordinary 2-valued notions of “consistency with respect to an operator” and “absolute consistency.” The purpose of the present paper is to show that the concept of “deductive completeness” may be extended to m-valued functional calculi of first order. For this purpose we define “analytic formula” for the m-valued case and show that if a formula is analytic, then it is provable in our formalization of m-valued functional calculi of first order.

In proving deductive completeness for the m-valued case, it is possible to use a method which is analogous to that used by Gödel in establishing the completeness of 2-valued functional calculi of first order. However, in the present paper we will indicate only very briefly how the Gödel procedure may be extended to the m-valued case. Our chief concern will be the problem of extending to our formalization of m-valued functional calculi the more elegant proof of deductive completeness for the 2-valued case which has recently been developed by Leon Henkin.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 16 , Issue 1 , March 1951 , pp. 22 - 34
Copyright
Copyright © Association for Symbolic Logic 1951

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Footnotes

1

See this Journal, vol. 13, no. 4, pp. 177–192. In the present paper, all references to axioms and rules of inference will be to Al through A15, R1 and R2 of part I of this paper.

References

2 Ibid., pp. 190–192.

3 For an abstract of a paper which indicates how the Gödel proof could be extended to the m-valued case, see the present authors' he Gödel completeness of m-valued functional calcuof first order, this Journal, vol. 14, no. 1, p. 74Google Scholar. For expositions of Gödel's completeness proof, see Hilbert, and Ackermann, , Grundzüge der theoretischen Logik, 2nd revised edition, pp. 76–81Google Scholar, and Church, Alonzo, Introduction to mathematical logic, part I, pp. 66–82Google Scholar. Attention should be called to the fact that each of these expositions contains an error which may be corrected as suggested in the review X 19 in this Journal.

4 See Henkin, , The completeness of the first-order functional calculus, this Journal, vol. 14, no. 3, pp. 159–166Google Scholar.

5 See the authors, Axiom schemes for m-valued prepositional calculi, this Journal, vol. 10, no. 3, pp. 61–82Google Scholar.

6 Ibid., pp. 78–80.

7 For our definition of the yields sign “¢” see part I of the present paper, op. cit., p. 190.

8 For a definition of “standard conditions” see ibid., footnote 37, p. 188.

9 For example, see Alonzo Church, op. cit., pp. 45–46.

10 For a typical 2-vaIued functional calculus of first order, see Alonzo Church's F1, op. cit., p. 40.

11 Ibid.

12 For the definition of Nj,i(P) see part I of the present paper, op. cit., p. 188.

13 Ibid., p. 185.

14 Ibid., p. 182.

15 It is clear from the definition of (P ≡ Q) that if it always takes a designated truthvalue, then P will be weakly equivalent to Q in the sense denned in part I of the present paper, op. cit., p. 178.

16 Our use of “analytic formula” corresponds to such 2-valued concepts as “identische Formel” and “valid formula.” For examples of these latter concepts see Hilbert and Ackermann, op. cit., pp. 76–81, or Alonzo Church, op. cit., pp. 73–82.

17 For a definition of an associated truth-value for P over a universe of a single individual, see part I of the present paper, op. cit., pp. 186–187. One effect of our definition of a generalized associated truth-value is to allow us to deal with a universe of more than one individual.

18 For example, see Church's condition 3(3), op. cit., p. 80.

19 See “Motivation,” part I of the present paper, op. cit., pp. 182–184.

20 Ibid., pp. 182–186.

21 For the definition of “closure” see Alonzo Church, op. cit., p. 81.

22 If Λ is not a finite set, then it is clear that no essential changes need be made in the definition of the yields sign.

23 This could be done by using the standard Gödel method of assigning number names to the well-formed formulas of a calculus. See Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar.

24 For example, we could select the set {A1}. This is clearly a consistent set, since otherwise we would have P⊃ P ⊢ Q Q · ∼ Q and ⊢ Q · ∼ Q, which contradicts the plausibility of our formalization.

25 In their natural order.

26 Since is used as an individual constant, F* contains no free individual variables and T1 may be applied.

27 It is not difficult to show in a manner analogous to the 2-valued case that if ⊢ F then F is analytic.

28 For a detailed proof of this statement, one may proceed in a manner closely analogous to the proof of Lemma 1 in part I of the present paper, op. cit., pp. 189–190.

29 See Alonzo Church, op. cit., pp. 59–61.

30 See S19 of the present paper.

31 See Alonzo Church, op. cit.