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True Contradictions

Published online by Cambridge University Press:  01 January 2020

Terence Parsons
Terence Parsons*
Affiliation:
University of California/Irvine, Irvine, CA92717, U.S.A.
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Extract

In In Contradiction, Graham Priest shows, as clearly as anything like this can be shown, that it is coherent to maintain that some sentences can be both true and false at the same time. As a consequence, some contradictions are true, and an appreciation of this possibility advances our understanding of the nature of logic and language.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 20 , Issue 3 , September 1990 , pp. 335 - 353
Copyright
Copyright © The Authors 1990

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References

1 Graham Priest, In Contradiction (Dordrecht: Martinus Nijhoff 1987); hereafter, [IC]. This paper originated in a talk given to the Pacific division A.P.A. in Spring 1989. I am indebted to Peter Woodruff and Graham Priest for discussion of earlier drafts; neither is responsible for the final form (even according to Priest's paraconsistent logic).

2 The notation Γ | S is read: from Γ you may validly infer S. This includes cases in which Γ contains sentences that are both true and false. That is, the condition is to be read: If every sentence in Γ is either simply true or both true and false, then S is either simply true or both true and false.'

3 Gottlob Frege, The Thought: a Logical Inquiry, trans. A.M. Quinton and M. Quinton, Mind 65 (1956) 289-311. The earlier tradition, e.g. in the Part Royal Logic, held that assertion and denial were never the same. But that tradition analyzed assertion and denial as always having two objects - two ideas - not as having a single content, and so the views may not be comparable.

4 Since falsehood is defined in terms of comfortable negation, and comfortable negation is defined in terms of a truth-table using falsehood, there is a circularity here, as Priest notes ([IC], 81). It is not difficult to show that there are constraints on such circularity. In particular, if a notion of negation is going to be used to define falsehood (via the definition: F'A = dt T'⇁A) then the following are the only possible entries in the truth-table for that negation (assuming that the falsehood used in its truth-table is the same as the one it defines):

Selecting the classical entries for the classical cases leaves these options:

Comfortable negation is the version of this table that selects the non-classical valuations for the non-classical inputs.

5 The discussion of truth is in [IC] section 4.5. The Dummett quotation is from Michael Dummett, Frege (London: Duckworth 1973), 320.

6 This business of truth-value gaps and gluts can be fairly subtle. Let me illustrate this in connection with the argument that Priest gives against the gappers, and ask whether this argument is really any good even if we grant Priest the principle that if a isn't true then it is false. Does this refute the existence of truth-value gaps? The answer is, it does not do this from Priest's point of view, though it does do this from the point of view of the pure gappers. It is a classic case of a good ad hominum argument.

Here is how the argument would have to go. The point is to refute the claim that a is neither true nor false, using the inference from nontruth to falsehood. The argument is:

Suppose a is neither true nor false. Then, since a isn't true, it is false, by the inference in question:

So it is false and it isn't, a conclusion that is unacceptable for the pure gapper.

(Actually, what the argument establishes for the gapper is that the original claim is not true. It does not show it to be false, it only shows it either to be false or lacking in truth-value. But that is enough to undercut the gapper's asserting it.)

For the gapper, this appears to be a good criticism; the claim that a is without truth-value leads to the conclusion that it is false. The gapper must then refrain from endorsing any claim of truth-valuelessness.

For Priest, on the other hand, this argument does not refute the premise, i.e., the argument does not prove that the premise isn't true; you can't prove that by a reductio argument within a dialetheist framework. The argument does prove that the premise is false, but it doesn't rule out the possibility that the premise is both true and false. Small comfort to the pure gappers, of course.

We need not pursue this line further, since Priest has not established the key inference anyway.

7 Priest's second argument, which he sees to be equivalent to the first, he quotes from Michael Dummett, Truth and Other Enigmas (London: Duckworth 1978):

A statement, so long as it is not ambiguous or vague, divides all states of affairs into just two classes. For a given state of affairs, either the statement is used in such a way that a man who asserted it but envisaged that state of affairs as a possibility would be held to have spoken misleadingly, or the assertion of the statement would not be taken as expressing the speaker's exclusion of that possibility. If a state of affairs of the first kind obtains, the statement is false; if all actual states of affairs are of the second kind, it is true. (8)

There are at least four difficulties with this argument. (1) The very first sentence of Dummett's remarks would be disputed by those gappers who think that truthvalueless sentences make statements which have no truth-value. Others might agree with Dummett's sentence under the proviso that truth-valueless statements are vague in Dummett's sense; for them, the passage is irrelevant to the issue of truth-value gaps. Still others would grant that statements do not lack truthvalue, but maintain that some sentences or propositions lack truth-value just because they express no statements; for these people the passage is irrelevant to the issue of truth-value gaps. (2) The last sentence does not follow from anything said previously. This is because the notion of falsity does not occur previously in the passage. This criticism remains if we replace falsity by its definition, rewording the last phrase as the negation of a is true'; the conclusion still does not follow. (3) There is no connection made between Dummett's notion of falsehood and Priest's. Indeed, given the clause following the semicolon, in the light of the preceding clause, it looks as if Dummett is using a is false where Priest would use a is not true. If so, the issue of gaps has not even been raised by Dummett's passage. (4) Suppose that we accept the last sentence of Dummett's quotation, and suppose that we even interpret the if then .. . in the quotation as entailment, not as the material conditional. (For otherwise, the remaining argument will fail in Priest's view.) We still have not reached the conclusion. For we will need the further assumption that either no actual states are excluded by a or some are, an assumption that seems to presuppose the law of excluded middle. But excluded middle is part of what many gappers are willing to give up. (Others of them find a way to save excluded middle, by means of supervaluations, for example; for them, the argument also fails, for more complicated reasons involving the details of how supervaluations work.)

Priest's third argument is a hypothetical one, since it rests on this account of truth, which he himself does not accept, but which he attributes to some gappers:

a is true iff some Fact makes a true.

He argues that if there is no Fact that makes a true, there is a Fact that makes , a true, viz. The Fact that there is no Fact that makes a true ((IC] 83). Here there are at least two difficulties. (1) Whether the quoted claim is true depends on what Facts there are and how they are related to language; this is a complex business, and many different theories about them are possible. (2) If the quoted claim were true, Priest would have given an RAA of the truth of the view that a is neither true nor false. But this only shows that that claim is either false or lacking in truthvalue. The gapper already believes the latter, and so the argument only establishes what is already known about the gapper view. (See below for difficulties that the gapper has regarding whether s/he may assert his/her own view.)

8 See n.9.

9 The reason is this. Suppose that the liar sentence is ⇁ Ta, where a refers to that very sentence. Now suppose that you assert that the liar sentence is not true. Your assertion has the form:

(1) ⇁Tb

where b is the liar sentence. But a also names the liar sentence, so this is true:

(2) a=b.

Now (1) and (2) together immediately let us infer (3):

(3) ⇁Ta.

But then we have validly inferred the liar sentence itself from two true claims. The standard gapper solution is to consider (1) as on a par with the liar sentence itself, and to deny that either of these have truth-value.

10 Priest could indicate genuine disagreement with you if he could assert ⇁ and also say that is not a dialetheism. However, the usual way to say this is to say is not both true and false, i.e. ⇁ ( is true & is false). This, however, gets us nowhere, since if is the liar sentence, it is a dialetheism, yet this statement about it is true. (If is the liar sentence, then it is both true and not true. Since it is not true, is true is false, and so is the conjunction is true & is false. Thus the negation of this conjunction is true. [It is also false, but that is not relevant to the point at issue.]) Priest has no means within his symbolism of adequately expressing not being a dialetheism'; see [IC] page 139 for discussion of this point.

11 Terence Parsons, Assertion, Denial, and the Liar Paradox, Journal of Philosophical Logic 13 (1984), 137-52

12 In more complex cases the gapper must carefully choose the mode of expression. For example, the gapper should not assert some sentences lack truth-value, for this is itself without truth-value on most gapper accounts; instead, the gapper should deny every sentence has truth-value. For any claim in predicate calculus notation, if the gapper rejects its truth, the right way to express this is to deny its classical negation.

13 The point here is not a practical one. The problem is not that Priest cannot express disagreement; he does so successfully throughout his book, and communicates this disagreement to his readers. The point is that if he is doing nothing but asserting, then according to his own theory he is not ruling out the views that he clearly intends to rule out by his writings. The answer is that he must sometimes be interpreted as denying things, rather than asserting their negations. This puts his position on a par with that of the gapper, who must do the same thing.

14 In introducing this connective I do not mean to revise the definition of falsehood proposed earlier in terms of comfortable negation. That is, falsehood is still defined in terms of comfortable negation, and super exclusion negation is simply a new and prima facie unrelated notion.

15 This is why I call the original truth-tables truth-tables for comfortable connectives': both parties are comfortable with these connectives, because they have solutions to the paradoxes when the paradoxes are formulated in terms of these connectives. For other connectives these solutions do not seem to work.

16 I choose my words carefully here. Each party must deny that there is such a thing as super exclusion negation. Neither party should assert that there is no such thing as super exclusion negation, because that will misrepresent their views.

In Parsons (1984] the point is expressed differently. Instead of questioning whether super exclusion negation is different from comfortable negation, I assumed that it was different and questioned whether it exists. The formulations are equivalent, but the earlier way of putting the point is unnecessarily mysterious.

17 This does not provide a new solution to any of the problems raised above. One still needs to distinguish assertion from denial, and acceptance from rejection, and one still needs to evade the issue raised by super-exclusion negation. (Such issues will arise in familiar ways upon replacing the N under the ⇁A by T.)

More needs to be said about the synthesis view. For one thing, if we retain our previous definition of valid inference, it appears that the Ns are just gaps in disguise. We can instead define another notion of valid inference:

┌ // ⸫ S = If no sentence in + is false,

then neither is S.

This makes the system resemble one in which the Ns are gluts in disguise. Fortunately, the choice between these definitions of valid inference is not a substantive question. The notions of inference are used in constructing proofs, and once we have two methods of constructing proofs we then have two kinds of proofs: one establishes that its conclusions are true when its premises are, and the other establishes that its conclusions are non-false when its premises are. (If the liar proof is construed as the former kind, then it has a fallacy at line 4, as in the gapper solution; if it is construed as the latter kind, then its last line is an aberrancy, but the significance of the proof is only that its last line is not false.)

The more substantive issue turns out to be the question of acceptance and assertability. What do you say, for example, about the acceptability of asserting a sentence with aberrant status? If this is uniformly unacceptable, then the Ns look like gaps; if it is uniformly acceptable, they look like gluts.