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Are Some Propositions Neither True Nor False?

Published online by Cambridge University Press:  14 March 2022

Charles A. Baylis
Charles A. Baylis*
Affiliation:
Department of Philosophy, Brown University, Providence, R. I.
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Extract

Though some doubts about the principle that every proposition is either true or false were entertained even by Aristotle, both the number and the vigor of criticisms of this principle have been increasing in recent years. This paper attempts a restatement and a re-examination of the issues involved in this dispute, and in particular an evaluation of the effects on the argument of such recent discoveries as that of the “many-valued logics.”

Type
Research Article
Information
Philosophy of Science , Volume 3 , Issue 2 , April 1936 , pp. 156 - 166
Copyright
Copyright © Philosophy of Science Association 1936

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References

1 The Search for Truth, Baltimore, The Williams & Wilkins Co., 1934; pp. 254–255.

2 Symbolic Logic, by C. I. Lewis and C. H. Langford, New York, the Century Co., 1932; p. 222.

3 Even so careful a writer as H. Margenau, in a discussion (“On the Application of Many-Valued Systems of Logic to Physics,” Philosophy of Science, Vol. 1, No. 1, Jan. 1934, p. 119) in which he rightly opposes the much more radical suggestion of F. Zwicky that the application of the many-valued logics to physical reasoning gives rise to a “principle of flexibility of scientific truth,” feels constrained to admit: “Many-valued logic admits of a series of truth values including the equivalents of true and false, and perhaps including doubtful. This means that every proposition is either true, or false, or doubtful.”

4 E. T. Bell, op. cit., p. 273.

5 “Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls” in Comptes Rendus des Stances de la Société des Sciences et des Lettres de Varsovie (Sprawozdania 2 pozesiedeń Towarzystwa Naukowego Warszawskiego), Classe III, Vol. XXIII, 1930, Fascicule 1–3, pp. 51–77.

6 Op. cit., pp. 64, 75.

7 “Ich kann ohne Widerspruch annehmen, dass meine Anwesenheit in Warschau in einem bestimmten Zeitmoment des nächsten Jahres, z. B. mittags den 21 Dezember, heutzutage weder im positiven noch im negativen Sinne entschieden ist. Es ist somit möglich, aber nicht notwendig, dass ich zur angegebenen Zeit in Warschau anwesend sein werde. Unter dieser Voraussetzung kann die Aussage: „ich werde mittags den 21 Dezember nächsten Jahres in Warschau anwesend sein,” heutzutage weder wahr noch falsch sein. Denn wäre sie heutzutage wahr, so müsste meine zukünftige Anwesenheit in Warschau notwendig sein, was der Voraussetzung widerspricht; und wäre sie heutzutage falsch, so müsste meine zukünftige Anwesenheit in Warschau unmöglich sein, was ebenfalls der Voraussetzung widerspricht. Der betrachtete Satz ist daher heutzutage weder wahr noch falsch und muss einen dritten, von „0“ oder dem Falschen und von „1” oder dem Wahren verschiedenen Wert haben. Diesen Wert können wir mit, ,½;“ bezeichnen; es ist eben das „Mögliche,” das als dritter Wert neben „das Falsche“ und „das Wahre” an die Seite tritt.

“Diesem Gedankengang verdankt das dreiwertige System des Aussagenkalküls seine Entstehung.” Op. cit., p. 64; translation mine.

8 Łukasiewicz speaks of his future presence in Warsaw as not now “decided” (entschieden). He might conceivably mean by his sentence either: (a) “I can assume without contradiction that my presence in Warsaw … is today determined in neither a positive nor a negative sense,” or (b) “I can assume without contradiction that today it neither is nor is not the case that I shall be in Warsaw …” If he meant (b) he would be begging the whole point at issue and his argument which follows would be entirely superfluous. It is more charitable to assume that he meant (a). That he did mean (a) is indicated also by his statement two paragraphs earlier that “The dispute about the principle of two-valuedness has a metaphysical background: the advocates of this principle are resolute determinists while the opponents of the principle incline to an indeterministic Weltanschauung.”

9 “P” is used here as an arbitrary abbreviation for “I shall be in Warsaw at noon on the 21st of December of next year.”

10 Alice Ambrose, “Finitism in Mathematics,” Mind, Vol. XLIV, No. 174, April 1935, pp. 186–203; and No. 175, July 1935, pp. 317–340.