Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T20:26:54.073Z Has data issue: false hasContentIssue false
  • English
  • Français

The elimination theorem when modality is present1

Published online by Cambridge University Press:  12 March 2014

Haskell B. Curry
Haskell B. Curry*
Affiliation:
The Pennsylvania State College
Get access Rights & Permissions [Opens in a new window]

Extract

The present paper contains an emendation to the theory of formal deducibility which was presented in lectures at the University of Notre Dame in the spring of 1948 [4]. The method of proof developed in these lectures for the elimination theorem, which corresponds to the “Hauptsatz” of Gentzen's thesis [6], failed when modal connectives were present. This deficiency was remedied soon thereafter; but it was not possible to include the new proof in those lectures. This new proof is presented below.

Before proceeding to this proof, the elimination theorem will be formulated, and its proof outlined, in an abstract form. Certain general conditions on the rules are formulated in §1; then, in §2, it is shown that the first two stages of the elimination theorem go through for any rules satisfying the conditions. The modifications necessary to extend this proof to cases where modal rules are present will then be taken care of in the later sections. A section on the extension of the elimination theorem to the singular form of LC is added at the end.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 17 , Issue 4 , December 1952 , pp. 249 - 265
Copyright
Copyright © Association for Symbolic Logic 1952

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

Presented at a joint meeting of the Association for Symbolic Logic and the American Mathematical Society, December 30, 1948 under the title The elimination theorem when necessity is present. The sections having to do with possibility were added in August 1950.

References

BIBLIOGRAPHY

[1]. Curry, H. B. On the definition of negation by a fixed proposition in inferential calculus, this Journal, vol. 17, pp. 98–104.Google Scholar
[2]. Curry, H. B. The system LD, this Journal, vol. 17, pp. 35–42.Google Scholar
[3]. Curry, H. B. A simplification of the theory of combinatora, Synthese, vol. 7 (1949), pp. 391–399.Google Scholar
[4]. Curry, H. B. A theory of formal deducibility. Notre Dame mathematical lectures, no. 6. Lithoprinted. University of Notre Dame, Notre Dame, Indiana, 1950, ix + 126 pp.Google Scholar
[5]. Curry, H. B. The permutability of rules in the classical inferential calculus, this Journal, vol. 17, pp. 245–248.Google Scholar
[6]. Gentzen, G. Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934), pp. 176–210, 405–431.CrossRefGoogle Scholar
[7]. Lewis, C. I. and Langford, C. H. Symbolic logic. New York (The Century Co.), 1932, xi + 506 pp.Google Scholar
[8]. McKinsey, J. C. C. A solution of the decision problem for the Leuns systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.Google Scholar