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The Sheffer functions of 3-valued logic

Published online by Cambridge University Press:  12 March 2014

Norman M. Martin
Norman M. Martin*
Affiliation:
Willow Run Research Center, Ypsilanti, Michigan, U.S.A.
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Extract

In previous papers, Post, Webb, Götlind and the present author have described some Sheffer functions (in Swift's terminology, “independent binary generators”) in m-valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in m-valued logic and isolate all of the Sheffer functions of 3-valued logic.

Before we proceed we will define some terms which we will find convenient. A set of functions in m-valued logic is functionally complete, if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of m-valued logic. A function is functionally complete, if its unit set is functionally complete. A Sheffer function is a two-place functionally complete function. If i and j are truth values (1 i, j ≤ m), we will say i ~ j (D), if D is a decomposition of the truth values 1, …, m into 2 or more disjoint non-empty classes and i and j are elements of the same class. A binary function f(p, q) satisfies the substitution law for a decomposition D, if for any truth values h, i, j, k, whenever h ~ j (D) and i~k(D), then f(h, i) ~ f(j, k) (D). The function f(p,q) satisfies the co-substitution law for D, if for any truth values h, i, j, k, whenever f(h, i) ~ f(j, k) (D), then h ~ j (D) or i ~ k (D). We will say f(p, q) has the proper substitution property, if there is a decomposition of the truth values into less than m classes for which it satisfies the substitution law.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 1 , March 1954 , pp. 45 - 51
Copyright
Copyright © Association for Symbolic Logic 1954

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References

REFERENCES

[1]Götlind, Erik, Some Sheffer functions in n-valued logic, Portugaliae mathematicae, vol. 11 (1952), pp. 141–149.Google Scholar
[2]Kalicki, Jan, A test for the equality of truth-tables, this Journal, vol. 17 (1952), pp. 161–163.Google Scholar
[3]Łoś, Jerzy, O matrycach logicznych (On logical matrices), Prace Wroclawskiego Towarzystwa Naukowego, series B, no. 19, Wrocław 1949, 42 pp.Google Scholar
[4]Martin, Norman M., Some analogues of the Sheffer stroke function in n-valued logic, Indagationes mathematicae, vol. 12 (1950), pp. 373–400.Google Scholar
[5]Martin, Norman M., A note on Sheffer functions in n-valued logic, Methodos, vol. 3 (1951), pp. 240–242.Google Scholar
[6]Martin, Norman M., Sheffer functions and axiom sets in m-valued propositional logic, Ph. D. thesis (University of California at Los Angeles, 1952).Google Scholar
[7]Myhill, John R., A complete theory of natural, rational, and real numbers, this Journal, vol. 15 (1950), pp. 185–196. See Errata, ibid., p. iv.Google Scholar
[8]Piccard, Sophie, Sur les fonctions définies dans les ensembles finis quelconque, Fundamenta mathematica, vol. 24 (1935), pp. 183–185.CrossRefGoogle Scholar
[9]Post, Emil L., Introduction to a general theory of elementary propositions, American journal of mathematics, vol. 43 (1921), pp. 163–185.CrossRefGoogle Scholar
[10]Rose, Alan, Review of [4], this Journal, vol. 16 (1951), pp. 275–276.Google Scholar
[11]Słupecki, Jerzy, Kryterium pełności wielowartościowych systemów logiki zdań (A criterion of fullness of many-valued systems of propositional logic), Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 32 (1939), pp. 102–109.Google Scholar
[12]Swift, J. Dean, Algebraic properties of n-valued propositional calculi, American mathematical monthly, vol. 59 (1952), pp. 612–621.CrossRefGoogle Scholar
[13]Webb, Donald L., Generation of any n-valued logic by one binary operator, Proceedings of the National Academy of Sciences, vol. 21 (1935), pp. 252–254.CrossRefGoogle Scholar
[14]Webb, Donald L., Definition of Post's generalized negative and maximum in terms of one binary operator, Bulletin of the American Mathematical Society, vol. 58 (1936), pp. 193–194.Google Scholar