Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-qn7h5 Total loading time: 0.2 Render date: 2022-10-06T01:51:30.002Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": true, "useSa": true } hasContentIssue true

Simplest normal truth functions

Published online by Cambridge University Press:  12 March 2014

Raymond J. Nelson*
Affiliation:
Owego, New York

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
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. 521531.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. 127133.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. 259263.Google Scholar
66
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Simplest normal truth functions
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Simplest normal truth functions
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Simplest normal truth functions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *