Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T17:46:25.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Simplest normal truth functions

Published online by Cambridge University Press:  12 March 2014

Raymond J. Nelson
Raymond J. Nelson*
Affiliation:
Owego, New York
Get access Rights & Permissions [Opens in a new window]

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

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]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