Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-12T21:20:51.505Z Has data issue: false hasContentIssue false
  • English
  • Français

A minimal canonically complete m-valued proper logic for each m

Published online by Cambridge University Press:  12 March 2014

Richard L. Call
Richard L. Call*
Affiliation:
Lafayette College
Get access Rights & Permissions [Opens in a new window]

Extract

In [1], William H. Jobe has shown that a certain three-valued proper logic is canonically complete, i.e., each first order J is expressible by a formula that is in M-normal form. The first order operator Jk(p)1 represents the truth table of a possible formula which has the truth value k in the ith row and the truth value 1 in all other rows. “The M-normal form of a given formula is another formula which has the same truth table as the given formula but which contains no operation symbols other than those of conjunction, disjunction, and undefined unary operations, with no nonunary operation, i.e., conjunction or alternation, included in the scope of an operation which is unary, and no alternation included in the scope of a conjunction.” Here it is shown that it is possible to construct a canonically complete proper logic for each choice of M which contains, in addition to alternation and conjunction, exactly two unary operations. Moreover, it is further shown that at least two unary operations are necessary in order for such a logic to be canonically complete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 1 , 26 April 1968 , pp. 108 - 110
Copyright
Copyright © Association for Symbolic Logic 1968

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

[1]Jobe, William H., Functional completeness and canonical forms in many-valued logics, this Journal, vol. 27 (1962), pp. 409–422.Google Scholar
[2]Post, Emil L., Introduction to a general theory of elementary propositions, American Journal of mathematics, vol. 43 (1921), pp. 163–185.CrossRefGoogle Scholar