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On Słupecki T-functions

Published online by Cambridge University Press:  12 March 2014

Trevor Evans  and
P. B. Schwartz
Trevor Evans
Affiliation:
Emory University
P. B. Schwartz
Affiliation:
Emory University
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Extract

In [2] E. L. Post defined a many-valued propositional logic to be functionally complete if, for every function on the set of truth-values, there exists a formula of the logic having that function as its associated truth-value function. He proved that the logic with truth-values 1, 2, …, m and (i) a unary connective ∼ such that ∼p has truth-value i+1 (mod m) when p has truth-value i, (ii) a binary connective ∨ such that p ∨ q has truth-value min(i, j) when p, q have truth-values i, j respectively, is functionally complete.

The many-valued logics described by Łukasiewicz and Tarski [1] are not functionally complete. These logics have truth-values 1, 2, …, m and (i) a unary connective ~ such that ~p has truth-value m−i+1 when p has truth-value i, (ii) a binary connective → such that if p, q have truth-values i, j respectively, then p → q has truth-value 1 for i ≧ j, and truth-value 1 for i ≧ j. The functional incompleteness of these logics is immediate, since there exists no formula in p having truth-value i (≠ 1 or m) when p has truth-value 1.

In [4] Słupecki showed that if a new unary connective T, such that T(p) has truth-value 2 for all truth-values assigned to p, is added to the 3-valued Łukasiewicz-Tarski logic, then the resulting logic is functionally complete. In [3] Rosser and Turquette proved this result for the m-valued (m ≧ 3) logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 3 , September 1958 , pp. 267 - 270
Copyright
Copyright © Association for Symbolic Logic 1958

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References

REFERENCES

[1]Łukasiewicz, J. and Tarski, A., 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
[2]Post, E. L., Introduction to a general theory of elementary propositions, American journal of mathematics, vol. 43 (1921), pp. 163–185.CrossRefGoogle Scholar
[3]Rosser, J. B. and Turquette, A. R., Many-valued logics, Studies in logic and the foundations of mathematics, Amsterdam (North Holland Pub. Co.), 1952.Google Scholar
[4]Słupecki, J., 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