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True, False, Etc.*

Published online by Cambridge University Press:  01 January 2020

Hans G. Herzberger
Hans G. Herzberger*
Affiliation:
University of Toronto
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How many truth-values are there? Although this appears to be a very simple question, in my opinion it defies any very simple answer. Some of us have trouble making up our minds. Frege, who invented the term “truth-value”, declared that apart from Truth and Falsity “there are no further truth-values”; and yet Frege, who introduced many of us to semantic presuppositions, acknowledged truth-value gaps. Now how many values would that be? On one way of counting, True, False and Gap make three. To be sure it's not Frege's way of counting; but it's defensible. And we are left with the historical puzzle that Frege, who founded his semantics on the insistence that functions be everywhere defined—without any gaps—has come to be known as the author of the doctrine of semantic presuppositions and truth-value gaps.

Type
Research Article
Information
Canadian Journal of Philosophy Supplementary Volume , Volume 6: New Essays in Philosophy of Language , 1980 , pp. 1 - 14
Copyright
Copyright © The Authors 1980

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Footnotes

*

Delivered to the University of Toronto Linguistics Club and to the University of Calgary Ninth Annual Colloquium on Directions in Linguistics, March 1978. Many other authors have arrived at conclusions similar to or relevant to my own; apart from Emil Post, Dana Scott and Merrie Bergmann, whose work has been cited in the text or footnotes, a few others who might be mentioned in this connection are Michael Dummett, Timothy Smiley, Karel Lambert, Ronald Scales, S.-Y. Kuroda, and “Mr. Turquer” of “Turquer and Rossette”. The principal novelty of the present approach lies in the general conception of relativized semantics as an instrument for the analysis of phenomena previously held to require many-valued or nonbivalent treatments. In a sequel, “Supervaluations Without Truth-Value Gaps”, other techniques will be applied to the same end.

References

1 “On Sense and Reference”, p. 63 in Geach, P. and Black, M., editors, Translations from the Philosophical Writings of Gottlob Frege (Blackwells, Oxford 1966).Google Scholar

2 “On Sense and Reference”, pp. 62, 69, and the second footnote top. 71.

3 See his “requirement of sharp delimitation”, p. 33, “Function and Concept”, in Geach and Black.

4 I owe these two “modes of presentation” to Dr. Tom Matthien.

5 In his Introduction to Logical Theory (Methuen, London 1952). p. 175, P. F. Strawson wrote: “if S’ is a necessary condition of the truth or falsity of S … let us say, as above, that S presupposes S’ “;and this characterization has been widely adopted as definitional for “semantic presuppositions”.

6 Łukasiewicz's three-valued matrix Ł 3 fails to validate the classical tautology ∼(P & ∼ P), Kleene's three-valued matrix K3 validates no classical tautologies, and Post's three-valued matrix P3 validates the classical contradiction ∼∼∼(Pv∼Pv ∼∼P). See Nicholas Rescher, Many-valued Logic (McGraw-Hill, New York, 1969), Ch. 2 for a description of these matrices.

7 Kempson, Ruth, Presupposition and the Delimitation of Semantics (Cambridge U. P., Cambridge, 1975), p. 53.Google Scholar

8 Op.cit. p. 3.

9 Łukasiewicz, Jan, “A System of Modal logic”, The Journal of Computing Systems, 1 (1953), 111·149.Google Scholar Reprinted in Borkowski, L., editor, Jan Łukasiewicz: Selected Works (North Holland, Amsterdam, 1970).Google Scholar

10 Aristotle,Categories Ch.10, 14a12ff.

11 The reader should not be misled by similarities between Rustler's opening remarks and a certain passage in Bertrand Russell's “Mr. Strawson on Referring”, Mind, (1957). let me stress once more that Rustler and Strawman are fictitious characters engaged in a fictitious debate.

12 This question of “partial semantic closure” is examined at length in my paper “Presuppositional Policies”, in A. Kasher, editor, Language in Focus (D. Reidel, Dordrecht, 1976).

13 See for example Böer, Steven and Lycan, William, “The Myth of Semantic Presupposition”, Ohio State University Working Papers in Linquistics, 21 (1976).Google Scholar

14 For related developments in connection with sortal correctness, see Bergmann, Merrie, A Presuppositional Theory of Semantic Categories, Ph. D. Dissertation, University of Toronto, 1976;Google Scholar and her “Logic and Sortal Incorrectness”, The Review of Metaphysics, 31 (1977), pp. 61-79.

15 The term “presuppositional plug” is from Karttunen, L., “Presupositions of Compound Sentences”, Linquistic Inquiry, 4 (1973), 169-93.Google Scholar For the treatment of secondary positions in terms of the scope of connectives - rather than in terms of the scope of definite descriptions, as in Russell - see Scales, Ronald, Attribution and Existence, Ph. D. Dissertation, University of California, Irvine, 1969.Google Scholar

16 In the works cited in footnote 14, Merrie Bergmann obtains much the same effect with the syntactic device of “correctness operators”.

17 Emil Post, “Introduction to a General Theory of Elementary Propositions”, American Journal of Mathematics, 43 (1921), 163-185; reprinted in Heijenoort, Jean van, editor, From Frege to Gödel (Harvard U.P., Cambridge, 1967).Google Scholar

18 Scott, Dana, “Background to Formalization” in LeBlanc, H., editor, Truth, Syntax and Modality (North Holland, Amsterdam, 1972).Google Scholar

19 The non-Postian character of her framework results from what she calls “separating truth-value and sortal status“; see p. 77 of “Logic and Sortal Correctness”, and the “Separation Principle” of Ch. Ill. 2 in A Presuppositional Theory of Semantic Categories.

20 Some preliminary studies of this question have been undertaken in my unpublished “Notes on Product and Semi-Product Logics” 1975-6, and in Ch. IV.5 of Merrie Bergmann's dissertation.

21 This has been undertaken in the works cited in footnote 14.