Hostname: page-component-6d856f89d9-xkcpr Total loading time: 0 Render date: 2024-07-16T08:18:42.980Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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