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Gaps and Gluts: Reply to Parsons

Published online by Cambridge University Press:  01 January 2020

Graham Priest*
The University of Queensland, Brisbane, Queensland, Australia4072


1 Introduction

Numerous solutions have been proposed to the semantic paradoxes. Two that are frequently singled out and compared are the following. The first is that according to which paradoxical sentences are neither true nor false — as it is sometimes put, they are semantic gaps. The second is that according to which paradoxical sentences are both true and false — as it is sometimes put, they are semantic gluts (dialetheias). Calling the first of these a solution is, in fact, somewhat misleading: it is rather like calling an opening gambit a game of chess. For the solution runs into severe problems almost immediately, and so can be only the first of a series of (often ad hoc) moves made to defend the original weak opening. Nonetheless, the symmetry involved in the gap and glut solutions is obvious enough to make the comparison a natural one.

Research Article
Copyright © The Authors 1995

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1 Thus, variations on the gambit, agreeing on nothing but the first move, abound in the literature. To cite but a few: Barwise, J. and Etchemendy, J. The Liar (Oxford: Oxford University Press 1987)Google Scholar; Goldstein, L.“This Statement is not True” is not True’, Analysis 52 (1992) 15Google Scholar; Kripke, S.Outline of a Theory of Truth,Journal of Philosophical Logic 72 (1975) 690716Google Scholar; Sainsbury, M. Paradoxes (Cambridge: Cambridge University Press 1988), ch. 5Google Scholar; and Smiley, T.True Contradictions, I,Proceedings of the Aristotelian Society, Supplementary Volume 68 (1993) 1733CrossRefGoogle Scholar. Those familiar with the literature will be able to cite numerous others.

2 ‘True Contradictions', Canadian Journal of Philosophy 20 (1990) 335-53. Unless otherwise stated, page references are to this paper. Parsons's essay is based on a talk given at the Pacific DivisionAPA in Spring 1989. This paper is an extended version of the reply given there. I am very grateful to him for written comments on an earlier version of it.

3 Parsons, T.Assertion, Denial, and the Liar Paradox,’ Journal of Philosophical Logic 13 (1984) 137–52CrossRefGoogle Scholar

4 Priest, G. In Contradiction (The Hague: Kluwer Academic Publishers 1987). I will refer to this as IC.CrossRefGoogle Scholar

5 Arguments from the Hegel/Marx tradition are not employed, though occasionally connections are hinted at. Parsons comments that he thinks these allusions misleading (336). I think the matter is a complex one, but certainly some of the true contradictions of this tradition are dialetheias. See my ‘Dialectic and Dialethic,’ Science and Society 53 (1989) 388-415; and ‘Was Marx a Dialetheist?’ Science and Society 54 (1990) 468-75.

6 IC, ch. 4. Both the account and the observation that it appears to rule out truth value gaps are Dummett's.

7 Parsons objects to two other arguments he claims to find in the text (343, n. 7). The first of these is a quotation by Dummett. I take this merely to express the argument that we have just considered, but I do not want to enter into questions of exegesis here. The second is actually an objection to an argument for the existence of truth value gaps, rather than an argument against them.

8 I even indicate how gaps can be incorporated into dialetheic semantics (95, n. 3).

9 There are, of course, many other suggested solutions. One is to suppose that different tokens of the same type can have different truth values. In the form required here, this has always stuck me as a rather desperate move, which throws spanners into numerous works. It is dealt a damning blow in Hazen, A.Contra Buridanum,Canadian Journal of Philosophy 17 (1987) 875–80CrossRefGoogle Scholar. Another tack is taken in Smiley's ‘True Contradictions, I,’ and criticized in my ‘True Contradictions, II,' Proceedings of the Aristotelian Society, Supplementary Volume 68 (1993) 17-54. A full discussion of all the moves suggested is too long to be undertaken here.

10 Here, I am asserting that something is irrational. One should, presumably, deny the claim that it is rational, too, though this is weaker. It is the stronger position that is warranted since the weaker one is compatible with the person asserting <β>being neither rational nor irrational.

11 This argument is due to G. Littman, ‘What Problems does Dialetheism Pose for Rationality?’ (Honours Dissertation, University of Queensland 1991 ). Note that it is not a ‘performative paradox,’ such as an utterance of the sentence ‘I cannot utter this sentence.’ The paradox does not depend on anyone actually uttering <β>.

12 ‘Against Global Paraconsistency,’ Studies in Soviet Thought 39 (1990) 209-29. See esp. §§ 4 and 5. Parts of the discussion (at least as directed against me) are incorrect, since Batens assumes that falsity entails non-truth. However, the central point survives the correction of this.

13 See IC, 8.5.

14 The possibility of this is hinted at in IC. 123, n. 10. It is elaborated in ‘True Contradictions, II.’

15 On the intensional paradoxes, see my ‘Intensional Paradoxes,’ Notre Dame Journal of Formal Logic 32 193-211.

16 See, further, IC 13.1, 13.2, and 13.5. The main notion of obligation discussed there is legal obligation. However, as I point out, there is no reason to suppose that there is anything special about legal obligation in this regard. Rational obligation, it appears, may well be the same.

17 Parsons writes ‘-’ as a pair of stacked tildes. I have changed this for typographical reasons. Something seems to have gone wrong with Parsons's definition of ‘-’ as a sentential operator since two distinct definitions are given on separate lines. Presumably, a negation sign has been missed out in the second line of the definition. But sorting through the issues this raises merely cloaks the main point of the present discussion, so I have formulated what follows slightly differently from Parsons, but in a way that I feel sure that he would find acceptable.

18 See, e.g., Geach, P. Logic Matters (Oxford: Blackwell 1972), 8.1 and 8.2.Google Scholar

19 See my ‘Boolean Negation and All That,’ Journal of Philosophical Logic 19 (1990) 201-15, esp. § 5. The discussion there is in terms of Boolean negation, which is slightly different from ‘-'; but in the relevant ways, it is the same.

20 Some discussion can be found in my ‘Logic of Paradox Revisited,’ Journal of Philosophical Logic 13 (1984) 153-79, § 2.

21 Thus, pace Parsons (ibid.), the question of designated values is a substantive issue.