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Simplest normal truth functions

Published online by Cambridge University Press:  12 March 2014

Raymond J. Nelson
Raymond J. Nelson*
Affiliation:
Owego, New York
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Extract

In [1] Quine has presented a method for finding the simplest disjunctive normal forms of truth functions. Like the tabular methods of [2] and [3], Quine's method requires expansion of a formula into developed normal form as a preliminary step. This aspect of his method to a certain extent defeats one of the purposes of a mechanical method, which is to secure simplest forms in complicated cases (perhaps by using a digital computer) [4]. In the present paper we develop a method for both disjunctive and conjunctive normal truth functions which is in some respects similar to Quine's but which does not involve prior expansion of a formula into developed normal form. Familiarity with [1] is presupposed.

We use the notations and conventions of [1] with the following exceptions and additions. ‘Φ’ names any formula, ‘Ψ’ any conjunction of literals, and ‘χ’ any disjunction of literals. Any disjunction of conjunctions of literals is a disjunctive normal formula and is designated by ‘ψ’; any conjunction of disjunctions of literals is a conjunctive normal formula and is designated by ‘X’. Note that we do not make use of Quine's notion of fundamental formulas. A formula Ψ occurring in a disjunctive normal formula ψ, provided it is a disjunct of ψ, is a clause; similarly for χ. We use ‘≠” for logical equivalence of formulas and ‘=’ for identity of formulas to within the order of literals in clauses and the order of clauses in normal formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 2 , June 1955 , pp. 105 - 108
Copyright
Copyright © Association for Symbolic Logic 1955

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References

REFERENCES

[1]Quine, W. V., The problem of simplifying truth functions, American mathematical monthly, vol. 59 (1952), pp. 521–531.CrossRefGoogle Scholar
[2]Staff of the Computation Laboratory, Synthesis of electronic computing and control circuits, the Annals of the Computation Laboratory of Harvard University, vol. 27, Cambridge, Mass., 1951.Google Scholar
[3]Veitch, E. W., A chart method for simplifying truth functions, Proceedings of the Association for Computing Machinery, Richard Rimbach Associates, Pittsburgh, 1952, pp. 127–133.Google Scholar
[4]Nelson, Raymond J., Review of [1], this Journal, vol. 18 (1953), pp. 280–282.Google Scholar
[5]Quine, W. V., Methods of logic, New York, 1950.Google Scholar
[6]Burkhart, W. H., Theorem minimization, Proceedings of the Association for Computing Machinery, Richard Rimbach Associates, Pittsburgh, 1952, pp. 259–263.Google Scholar