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Some theorems about the sentential calculi of Lewis and Heyting

Published online by Cambridge University Press:  12 March 2014

J. C. C. McKinsey  and
Alfred Tarski
J. C. C. McKinsey
Affiliation:
Oklahoma A. and II. College, and, The University of California
Alfred Tarski
Affiliation:
Oklahoma A. and II. College, and, The University of California
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Extract

In this paper we shall prove theorems about some systems of sentential calculus, by making use of results we have established elsewhere regarding closure algebras and Brouwerian albegras. We shall be concerned mostly with the Lewis system and the Heyting system. Some of the results here are new (in particular, Theorems 2.4, 3.1, 3.9, 3.10, 4.5, and 4.6); others have been stated without proof in the literature (in particular, Theorems 2.1, 2.2, 4.4, 5.2, and 5.3).

The proofs to be given here will be found to be mostly very simple; generally speaking, they amount to drawing conclusions from the theorems established in McKinsey and Tarski [10] and [11]. We have thought it might be worth while, however, to publish these rather elementary consequences of our previous work—so as to make them readily available to those whose main interest lies in sentential calculus rather than in topology or algebra.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 13 , Issue 1 , March 1948 , pp. 1 - 15
Copyright
Copyright © Association for Symbolic Logic 1948

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References

[1]Dugundji, James, Note on a property of matrices for Lewis and Langford's calculi of propositions, this Journal, vol. 5(1940), pp. 150–151.Google Scholar
[2]Gödel, Kurt, Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisseeines mathematischen Kolloquiums, Heft 4(1933), pp. 39–40.Google Scholar
[3]Gödel, Kurt, Zum intuitionistischen Aussagenkalkül, Ergebnisseeines mathematischen Kolloquiums, Heft 4(1933), p. 40.Google Scholar
[4]Heyting, A., Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1930, pp. 42–56.Google Scholar
[5]Lewis, C. I., and Langford, C. H., Symbolic logic, New York 1932.Google Scholar
[6]McKinsey, J. C. C., A reduction in number of the postulates for C. I. Lewis' system of strict implication, Bulletin of the American Mathematical Society, vol. 40(1934), pp. 425–427.10.1090/S0002-9904-1934-05881-6CrossRefGoogle Scholar
[7]McKinsey, J. C. C., Proof of the independence of the primitive symbols of Heyting's calculus of propositions, this Journal, vol 4(1939), pp. 155–158.Google Scholar
[8]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
[9]McKinsey, J. C. C., On the syntactical construction of systems of modal logic, this Journal, vol. 10(1945), pp. 83–94.Google Scholar
[10]McKineey, J. C. C., and Tarski, Alfred, The algebra of topology, Annals of mathematics, vol. 45(1944), pp. 141–191.10.2307/1969080CrossRefGoogle Scholar
[11]McKinsey, J. C. C., and Tarski, Alfred, On closed elements in closure algebras, Annals of mathematics, vol. 47(1946), pp. 122–162.10.2307/1969038CrossRefGoogle Scholar
[12]Parry, William Tuthill, Modalities in the Survey system of strict implication, this Journal, vol. 4(1939), pp. 137–154.Google Scholar
[13]Tarski, Alfred, Der Aussagenkalkül und die Topologie, Fundamenta mathematicae, vol. 31(1938), pp. 103–134.10.4064/fm-31-1-103-134CrossRefGoogle Scholar
[14]Wajsberg, M., Untersuchungen über den Aussagenkalkül von A. Heyting, Wiadomości matematyczne, vol. 46(1938), pp. 45–101.Google Scholar