2 Jan 2015

Priest (4.7) In Contradiction, ‘Truth or Falsity: Truth Value Gaps’, summary


by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part II. Dialetheic Logical Theory

Ch.4. Truth or Falsity

4.7 Truth or Falsity: Truth Value Gaps

Brief Summary:

Priest gives his reasons for thinking that there are no truth value gaps, that is to say, that there are no sentences which are neither true nor false. Some hope that by designating the liar sentence as valueless that this will do away with liar-like inconsistencies. However, Priest in section 1.3 showed that even if we assumed there to be value-gaps, we still can have the problem of the liar paradox. Now in this section Priest goes further to argue that there can be no truth gaps anyway. He refutes the arguments for value-gaps one-by-one, with a special focus here on sentences which fail to refer, such as ‘the King of France is bald.’ Some value-gappers say that this sentence is neither true nor false, since there is no state of affairs, no Fact, which could confirm or deny it. Priest shows that while this might be true, that very lack of a Fact is itself a Fact to which its negation refers. So while ‘the King of France is bald’ may be valueless, we know that ‘it is not the case that the King of France is bald’ is true, since we know there is no Fact to confirm its affirmative form.


Previously Priest was discussing truth. Now he turns to falsity. He defines falsity in terms of truth and negation. [First recall our conventions for the Tarski (T) scheme. Tarski uses quotations around a sentence to mean the name for that sentence. For example:

“snow is white” is true if and only if snow is white

Now, we can abbreviate sentence formulations to such letters as α, and we can use underlining instead of quotation marks and F and T as the formulations “is true” and “is false”.]

We will say that a sentence, α, is false, Fα, just if its negation is true. We might write this thus:

(Priest, 64)

As we can see from the formulation, we are using negation to define falsity (a proposition is false if its negation is true). So we might now ask, what is negation? Priest admits he does not have a definition to offer. At best we have a circular definition: “Negation is that sentential function which turns a true sentence into a false one, and vice versa.” [This is circular of course because negation is defined as what changes a true sentence to a false one, and a false to a true; however, we already defined falsity as the value given to the negation of a true sentence. We understand negation as what generates falsity, but we understand falsity as what is generated by negation.] But this is a problem in classical logic as well. “Orthodox truth-tables define negation in terms of truth and falsity. But falsity can be defined only in terms of, or by using, negation.” (64)

So, we cannot define negation. However, we can still say certain intelligible things about it. For example, we can say things about the conditions “under which a negated sentence is true.” (64) [In the following we are perhaps assuming that because there are only two truth values, if something is not true, then it is false. Or put another way, if something is not true, then its negation is true.]

a sufficient condition for the truth of a negated sentence, ¬α, is the failure of the truth of α. In other words, if a sentence is not true, it is false:

(Priest, 64)

Priest then continues by explaining why there can only be two truth values, true and false. [Recall that Value-gappists argue that some sentences are neither true nor false. He will use an analogy.  His basic idea seems to be that when you assert something, you cannot be neither right nor wrong. In two person games, you can have a draw, when neither player achieves their goal, and thus neither one is winner; but, since neither player won some advantage over the other, neither player is loser as well. But asserting claims does not have this two person dynamic. In one person games, there is not another player whose success makes you fail and whose failure makes you win. So there is not a second player, and thus it is not the case that if she does not win and you do not win, neither one wins and neither fails. Rather, in one person games, you either achieve your aim or you do not, you either win or lose. This is similarly the case when asserting claims. There is just one ‘player’, the claim or claimer, and it either achieves its ‘aim’ of stating the truth, or it does not. You cannot have a ‘tie’ or ‘draw’ if there is only one ‘player’.]

This fact about falsity follows from the analysis of truth we have just had. To speak truly is to succeed in a certain activity. And in the context of asserting, anything less than success is failure. There is no question of falling into some limbo between the two. To use the game analogy again, a draw is possible in a two-player game, for neither player may achieve his end. In a one-player game either the player achieves his end or he does not: there is no third possibility. Asserting is a one-player game. The point, again, is Dummett’s. As he puts it,19 [the following is block quotation of Dummett] |

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.
[Footnote 19, quoting, “Dummett (1959a), p. 8 of reprint. Italics original.” From the bibliography:
Dummett, M. (1959a) ‘Truth’, Proceedings of the Aristotelian Society 59, 141–62. Reprinted in Dummett (1978).
Dummet, M. (1978) Truth and Other Enigmas, Duckworth.

Some people do, on the contrary, argue that there are truth value gaps, “that is, a limbo between truth and falsity” (65). Priest will now explain why their arguments are faulty. Priest first notes that he, in section 1.3, already showed how the arguments for the valuelessness of the liar sentences did not eliminate the problem. The next example he gives is “Aristotle’s argument in De Interpretatione, chapter 9, concerning future contingents” (65). He does not here address it, because its lack of cogency is well established, and he cites Susan Haack’s treatment in her Deviant Logic: Some Philosophical Issues. [We will just look at her formulation of it for now:

(1) If every future tense sentence is either true or false, then, of each pair consisting of a future tense sentence and its denial, one must be true, the other false.
(2) If, of each pair consisting of a future tense sentence and its denial, one must be true, the other false, then, everything that happens, happens 'of necessity'.
(3) But not everything that happens, happens of necessity; some events are contingent.
∴ (4) Not every future tense sentence is true or false.
Clearly, this argument is a valid one. But, equally clearly, Aristotle's arguements for the premisses, particularly (2), need examination.
(Haack p74)

Although Priest does not here examine the argument, he does however discuss some of it in his Logic: A Very Short Introduction. There he uses modal logic to show the problem we find in step 2. See pages 39-46.] And, the other arguments for truth gaps

appear to be a motley crew concerning non-denoting terms and other kinds of ‘‘presupposition failure’’; category mistakes and other ‘‘nonsense’’; sentences undecidable by the appropriate mathematical or empirical techniques; and so on. (65)

Yet despite this variety, they share a similar rationale, which Priest will now outline. [The basic idea here seems to be the following. Consider ‘the King of France is bald.’ Its meaning is clear. But it cannot be true or false, since it fails to refer to any real state of affairs which could confirm or deny it. Thus some sentences are undecidable for this reason.]

The correspondence theory of truth may not be correct, but it captures an important insight concerning truth: for something to be true, there must be something in the world which makes it so. This need not be a state of affairs as traditionally conceived of by correspondence theorists. It might, in the case of a mathematical truth for example, be our possession (in principle) of a proof. In the case of a statement of legal right, it might be certain activities of a legislature. But there must be something, some Fact, such that if (counterfactually) it did not hold, the sentence would not be true. The rationale can now be stated simply thus: for certain sentences, α, there is no Fact which makes ¬α true, neither is there a Fact which makes ¬α true. For example, in the case of reference failure, there is no state of affairs which is either the King of France’s being bald, or his not being bald. For the case of undecidable empirical sentences, there is no possible experiment which would verify either that a particle has a certain momentum, or that it does not have it. And so on.

There is a general reason why this argument fails. [So again, the problem these value-gappers seem to have is that there is no state of affairs which can prove a claim like the King of France is bald. However, what about the claim, it is not the case that the King of France is Bald? What would need to happen for this claim to be true? We are assuming that making something true requires some Fact which affirms it as such. If there is no reference, like there is no King of France, then there is no Fact to make it true or not true, and thus it is valueless.

The lack of a Fact means we cannot affirm it. So we might say there is this Fact: we cannot affirm ‘the King of France’ is bald, since no Fact exists to affirm it. That then is the fact which would affirm the sentence:

it is not the case that the King of France is bald.

This insight here seems to be that we can say non-referring sentence are valueless, but their negations are not, since their lack of evidence confirms that they are not true and thus that their negations are true. Please interpret the following for yourself for a better reading.]

there is a general reason why this argument fails. In a nutshell, if there is no Fact that makes α true, there is a Fact that makes ¬α true, viz. the Fact that there is no Fact that makes α true. Less cryptically, the point is this. Suppose that α is a sentence, and suppose that there is nothing in the world in virtue of which α is true—no fact, no proof, no | experimental test. Then this is the Fact in virtue of which ¬α is true. We may not know that this Fact obtains, but this is irrelevant. And we might be able to distinguish between different kinds of Fact which make ¬α true. For example, in the case of denotation failure, we might distinguish between the case where ‘John’s brother is a butcher’ is false because John has no brother, and that where it is false because he has a brother who is a French-polisher. But this is not a significant difference as far as truth and falsity simpliciter go.

[Next Priest addresses an intuitionist reply. I cannot explain this section, given my current unfamiliarity with the topic. Priest explains intuitionism in his book Introduction to Non-Classical Logic, chapters 6 and 20. Perhaps the main idea of this paragraph the following. Intuitionist logic does not have a strong principle of excluded middle. Priest, after examining the proof conditions for sentences in intuitionist logic, writes in this other book:

Note that these conditions fail to verify a number of standard logical principles – most notoriously, some instances of the law of excluded middle: A ∨ ⇁A.
(Priest, Introduction to Non-Classical Logic, 104)

However, as we saw in section 1.3, Priest uses the principle of excluded middle to show why the value-gap argument does not work. Please read the following to interpret it properly for yourself.]

There is one important reply here: the intuitionist one. It may be argued that the point that we cannot, in general, recognise when α fails is important. For Facts of this kind cannot play the required semantic role. This is, I think, incorrect. However, to discuss this issue here would take us too far away from the central theme of the book, and so I will not do so. In view of my rejection of the intuitionist claim and my consequent endorsement of the law of excluded middle and related principles, the position I am advocating might be called ‘‘classical dialetheism’’. It would be equally possible to have an ‘‘intuitionist dialetheism’’, which took a constructive stance on negation (so that a proof of the impossibility of a proof of α was required for the truth of ¬α) and the other logical constants. (We noted in section 1.3 that the proofs of many logical paradoxes do not require the law of excluded middle or other intuitionistically invalid principles.) The paradoxical features of intuitionist implication, such as ¬α⊃(α⊃(β), could not be incorporated. But these have always been dubious features of intuitionism anyway.

[Thus given the fact that classic dialetheism makes use of the principle of excluded middle:]

if α is any atomic sentence of a kind whose members have been proposed as truth valueless, ¬α is true. Thus, ‘Julius Caesar is not a prime number’, ‘The man next door does not have a television set’ (when there is no man next door), and so on are simply true.

While these sentences may seem strange, Priest describes some situations where they would be appropriate.

[I am uncertain of the details in the final paragraph, partly because he makes reference to section 4.3, which as of this time I have not summarized. At any rate, he discusses a sentence we saw already in 1.3, namely, This sentence is true. In section 1.3, we noted that it would be a good candidate for a value-gap. Here Priest is saying that the sentence is false and its negation is true. This seems to be because there is no fact which could make it true, and as we said already before in this chapter, that makes it false.]

As a final application of the position, let us return to the example given in section 4.3 of the sentence 0; in effect, ‘This sentence is true’. We saw there that the truth conditions of this sentence imply neither the truth of this sentence nor its falsity. There is therefore no question of an a priori proof (or refutation) of it. By its nature, this is the only kind of Fact which could make it true. No experiment is going to decide the issue. Hence, by the previous discussion, this sentence is simply false and its negation is true.

Citations from:

Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

Otherwise if indicated, from:

Priest, Graham. An Introduction to Non-Classical Logic: From If to Is. Cambridge: Cambridge University, 2001/2008.

Haack, Susan. Deviant Logic: Some Philosophical Issues. Cambridge: Cambridge University, 1974.


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