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Modalities in Ackermann's “rigorous implication”

  • Alan Ross Anderson (a1) and Nuel D. Belnap (a1)

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.

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[1]Ackermann, Wilhelm, Begründung einer strengen Implikation, this Journal, vol. 21 (1956), pp. 113128.
[2]Anderson, Alan Ross, review of [l], this Journal, vol. 22 (1957), pp. 327328.
[3]Johannson, I., Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio mathematica, vol. 4 (1936), pp. 119136.
[4]Parry, William Tuthill, Modalities in the “Survey” system of strict implication, this Journal, vol. 4 (1939), pp. 137154.
[5]Robinson, Abraham, review of [1], Mathematical reviews, vol. 18 (1957), p. 271.
[6]von Wright, Georg H., An essay in modal logic, Amsterdam, 1951.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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