Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T02:51:08.932Z Has data issue: false hasContentIssue false

Strong completeness of fragments of the propositional calculus

Published online by Cambridge University Press:  12 March 2014

Alan Rose*
Affiliation:
College of Technology, Manchester, England

Extract

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of the

Theorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1951

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 Henkin, Leon, Fragments of the propositional calculus, this Journal, vol. 14 (1949), pp. 4248Google Scholar.

2 Łukasiewicz, Jan and Tarski, Alfred, Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 3050, especially 42–43Google Scholar.

3 Op.cit.