Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T04:28:32.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of the Lewis system S51

Published online by Cambridge University Press:  12 March 2014

Schiller Joe Scroggs
Schiller Joe Scroggs*
Affiliation:
Oklahoma Agricultural and Mechanical College
Get access Rights & Permissions [Opens in a new window]

Extract

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 16 , Issue 2 , June 1951 , pp. 112 - 120
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.)

Footnotes

1

This paper was prepared under the direction of Dr. J. C. C. McKinsey as a Master's thesis at Oklahoma Agricultural and Mechanical College.

References

BIBLIOGRAPHY

[1].Bergraann, G., The finite representations of S5, Methodos, vol. 1 (1949).Google Scholar
[2].Dugundji, J., Note on a property of matrices for Lewis and Longford's calculi of propositions, this Journal, vol. 5 (1940), pp. 150–151.Google Scholar
[3].Huntington, E. V., Postulates for assertion, conjunction, negation, and equality, Proceedings of the American Academy of Arts and Sciences, vol. 72 (1937), pp. 1–44.CrossRefGoogle Scholar
[4].Lewis, C. I., and Langford, C. H., Symbolic logic, New York 1932.Google Scholar
[5].McKinsey, J. C. C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.Google Scholar
[6].McKinsey, J. C. C., On the number of complete extensions of the Lewis systems of sentential calculus, this Journal, vol. 9 (1944), pp. 42–45.Google Scholar
[7].McKinsey, J. C. C., and Tarski, Alfred, Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), pp. 1–15.Google Scholar
[8).Parry, William Tuthill, Modalities in the survey system of strict implication, this Journal, vol. 4 (1939), pp. 137–154.Google Scholar
[9].Tarski, Alfred, Der Aussagenkalkül und die Topologie, Fundamenta mathematicae, vol. 31 (1938), pp. 103–134.CrossRefGoogle Scholar