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The cylindric algebras of three-valued logic

Published online by Cambridge University Press:  12 March 2014

Norman Feldman
Norman Feldman*
Affiliation:
Department of Mathematics, Sonoma State University, Rohnert Park, California 94928, USA. E-mail: [email protected]
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Extract

In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.

What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 4 , December 1998 , pp. 1201 - 1217
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, 1974.Google Scholar
[2] Bolc, L. and Borowick, P., Many valued-logics, vol. 1, Springer-Verlag, Berlin, 1992.Google Scholar
[3] Cignoli, R. and de Gallego, M. S., The lattice structure of some Lukasiewicz algebras, Algebra Universalis, vol. 13 (1981), pp. 315–328.Google Scholar
[4] Henkin, L., Monk, D., and Tarski, A., Cylindric algebras, Part I, North-Holland, Amsterdam, 1971.Google Scholar
[5] Henkin, L., Cylindric algebras, Part II, North-Holland, Amsterdam, 1985.Google Scholar
[6] Kleene, S. C., Introduction to metamathematics, Van Nostrand, New York, 1952.Google Scholar
[7] Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, 1967.Google Scholar