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Modalities in Ackermann's “rigorous implication”

Published online by Cambridge University Press:  12 March 2014

Alan Ross Anderson  and
Nuel D. Belnap Jr.
Alan Ross Anderson
Affiliation:
Yale University
Nuel D. Belnap Jr.
Affiliation:
Yale University
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Extract

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 2 , June 1959 , pp. 107 - 111
Copyright
Copyright © Association for Symbolic Logic 1959

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References

[1]Ackermann, Wilhelm, Begründung einer strengen Implikation, this Journal, vol. 21 (1956), pp. 113–128.Google Scholar
[2]Anderson, Alan Ross, review of [l], this Journal, vol. 22 (1957), pp. 327–328.Google Scholar
[3]Johannson, I., Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio mathematica, vol. 4 (1936), pp. 119–136.Google Scholar
[4]Parry, William Tuthill, Modalities in the “Survey” system of strict implication, this Journal, vol. 4 (1939), pp. 137–154.Google Scholar
[5]Robinson, Abraham, review of [1], Mathematical reviews, vol. 18 (1957), p. 271.Google Scholar
[6]von Wright, Georg H., An essay in modal logic, Amsterdam, 1951.Google Scholar