2 Jan 2015

Priest (1.3) In Contradiction, ‘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 I. The Logical Paradoxes

Ch.1 Semantic Paradoxes

1.3 Truth Value Gaps

Brief Summary:

Priest is arguing that inconsistencies like the ones produced by liar-like paradoxes are inevitable in a natural language. He addresses arguments against this position. The strongest among them is the argument that such paradoxical sentences are neither true nor false, since they are valueless. Priest demonstrates how even if we take this assumption, we still do not avoid the liar-like paradoxes.


[Priest’s basic aim in this chapter is to show that dialetheias, true contradictions, are generated necessarily by both formal and natural languages.] Previously Priest showed how the Tarski conditions lead to a liar-like paradox, that is, to a dialetheia. Those three conditions, basically, are that 1) the language in question is able to give names to all its sentences, that 2) names for sentences can be equated with the sentence they name (such that self-reference is also possible), and 3) we can draw the inference that if a formula holds if and only if it does not hold, then you have both that formula and its negation. Now in this section Priest notes that in order to deny that English satisfies the Tarski conditions and thus does not fall prey to the inconsistencies it leads too, then one  must deny that at least one of these conditions holds for English. (12) It would seem very difficult to deny conditions 1 and instead claim that not every sentence in English can have a name in English; for, all we need to do is put quotation marks around any phrase in English and we get its name. So we must look to see if we can deny conditions 2 and 3, and Priest will begin first with 3. (12d)

So recall condition 3.

(3) The rule of inference {α ↔ ¬α} ⊢ α ∧ ¬α is valid in the logic underlying the theory.
(Priest 11)

[Priest will claim that the best and maybe the only reason to reject this is if you think there are truth value gaps, perhaps meaning that some sentences are neither True nor False. This would perhaps have to be the case, since if α is True, then ¬α is False, and vice versa. Thus if you have both, then you have a sentence (their conjunction) which is both True and False. If you accept the validity of dialetheias, this is fine. But if you do not, then you cannot say that α is True nor can you say that it is False. Thus you must think it has a third value which is neither True or False or that it has no value. What Priest says about intuitionism I will have to return to later when I understand it better. Priest’s book Introduction to Non-Classical Logic looks promising for grasping better this topic.]

There is one (and perhaps only one), plausible reason for rejecting the reductio principle of 3, and this is the existence of truth value gaps, sentences that are neither true not false. Not that an intuitionist will think that these cause the principle to fail: the principle is valid intuitionistically.

[In the next part, Priest will use the term “gap-in/gap-out conditional”, which seems to mean that say you have p → q. And suppose also that p for example does not have a truth value. The question is, does that mean the whole conditional does not have a truth value? If the whole would not have a value as a consequence of only one part not having a value, then it is a “gap-in/gap-out” conditional (called such perhaps because we put one gap into the formulation and we get a gap output for the whole). But if one part can be valueless but the whole have a value, then it is not a “gap-in/gap-out” conditional. I am guessing. I am also not certain about the rest, so this is another guess to his meaning. But let us return to his formulation for condition 3:

{α ↔ ¬α} ⊢ α ∧ ¬α

The final point he might be making is that if we allow for truth gaps, then the left side of the formula can be true but the right side not true. I am not sure how to do this. But maybe he is saying that with regard to this idea of truth gaps, we might say that α is true but ¬α has no value. Nonetheless, a conjunction requires both terms to be true. So α ∧ ¬α would not then be true. Now, recall the only way that a conditional can be untrue. That would happen if the antecedent is true while the consequent is false. Assume again that α is true but ¬α has no value. That means

α → ¬α

is: true implies no value (so not false), and thus the conditional would still be true. And,

¬α → α

would be: no value implies true, and thus it would be true. Then put them together into the biconditional (or just conjunction of the two conditionals) and you still would have true. Thus if you thought that there could be truth gaps (valueless terms), then you can reject the third condition, which would then allow you to say that the dialetheias produced by the three conditions together are not necessary for a language that can have truth gaps. Priest is defending the idea that dialetheias are necessary. He later argues that truth gaps cannot exist. But for now, we will suppose they do, and still he will show that dialetheias will result from the three conditions]

But suppose we are thinking in more classical terms and that we have a sentence, α, such that both α and ¬α fail to be true. Then α ∧ ¬α will fail to be true (assuming a normal conjunction, as I will do throughout). But, given a conditional that is not simply a gap-in/gap-out conditional (where the valuelessness of a part spreads to that of the whole), α → ¬α and its converse may hold, and their conjunction may be true. In this case the inference fails. In fact, under very weak conditions the reductio scheme is equivalent to the law of excluded middle, whose failure can very naturally be taken to express the existence of truth value gaps. Hence if we may take paradoxical sentences to be neither true nor false, this particular argument to dialetheism may be blocked. I shall argue in section 4.7 that there are no truth value gaps. However, for the present let us suppose, at least for the sake of argument, that there are. I will argue that dialetheism is not to be avoided in this way.

Priest then notes that there are two main sorts of theses holding that sentences may be truth-valueless.

1) Some sentences are neither true nor false, even though sentences are the sort of thing that can be either true or false.

2) It is not sentences themselves that can be true or false, but rather only that which they express can have truth value. And some uses of sentences fail to express something that can have truth value. This theses also has two subvarieties, depending on whether the holder thinks it is statements or propositions that are being expressed. Priest will address both parties simultaneously in the following way:

I intend my discussion to apply to all versions and subversions of the thesis. To this end, I will now write ‘true’, ‘false’, and their cognates with initial capitals. Those who think sentences are true/false can read ‘True/False sentence’ in the obvious way. Those who think that it is statements or propositions that are true/false can read it as ‘sentence (the use of) which makes a true/false statement/proposition’, depending on their preferred theory. The thesis that there are truth valueless sentences can now be expressed as: there are some (indicative) sentences that are neither True nor False. Let us call such sentences ‘Valueless’.

Priest has some main points. The first is that:

even granted that Valueless sentences vitiate the reductio scheme, this does not, per se, solve the paradoxes.

Priest now looks at some arguments for why the paradoxical sentences are valueless, but “none of them is very satisfactory.” (14) [Understanding the following seems to require a gasp of the referenced texts and ideas. I do not have this familiarity, but let us make sense of what Priest is doing as best we can for the time being. Priest cites Ryle as one philosopher who thinks that one of the paradoxical sentences are valueless. The counter example Priest gives is this. My father says that all one-legged men in town lie (do not tell the truth), and the one and only one-legged man in town says that my father always tells the truth. Here we have a liar paradox. If either claim is true then it is also false. Priest then says that these sentences stand all the tests for making a statement: “I understood what he said; I can draw inferences from it; I can act on the information contained in it, and so on” (14). So Ryle seems to be saying that the expressions can be statements, meaning that their content is obvious enough that it can be affirmed or denied, and yet there somehow is no truth value to them. Before going further, let us look at a little of the Ryle text that Priest cites. What Ryle seems to be saying is that in the sentence, ‘This sentence is false’, the part ‘this sentence’ fails to refer to something, because of an infinite recursion of substitutions. So Ryle has us consider the statement:

That statement is false.

He says that such sentences come with a ‘namely-rider’. So which other sentence is this sentence referring to? How about, ‘Today is Tuesday.” So we might substitute:

That statement [namely that today is Tuesday] is false.

So far, no problem. But what happens with sentences like:

This statement is false.

? Let us try to insert the namely-rider:

This statement [namely that this statement is false] is false.

But here again we have another reference, ‘this statement’. So we need another namely-rider to know what it is referring to. So we get:

This statement [namely that this statement {namely that this statement is false}] is false.

But still again in curly brackets we have another instance of ‘this statement’ that requires a namely-rider. This will have no end, and thus we never get to the predication ‘is false.’ Since it does not succeed in making a reference that would establish the subject of the sentence, we cannot give it a truth value (it seems more like a sentential function like x is false, where it would only have a truth value when we substitute a value for x.) Here are some passages from the Ryle text:

The same inattention to grammar is the source of such paradoxes as ‘the Liar’, ‘the Class of Classes . . .’ and ‘Impredicability’. When we ordinarily say ‘That statement is false’, what we say promises a namely-rider, e.g. ‘ . . . namely that to-day is Tuesday’. When we say ‘The current statement is false’ we are pretending either that no namely-rider is to be asked for or that the namely-rider is ‘ . . . namely that the present | statement is false’. If no namely-rider is to be asked for, then ‘The current statement’ does not refer to any statement. It is like saying ‘He is asthmatic’ while disallowing the question ‘Who?’ If, alternatively, it is pretended that there is indeed the namely-rider, ‘ . . . namely, that the current statement is false’, the promise is met by an echo of that promise. If unpacked, our pretended assertion would run ‘The current statement {namely, that the current statement [namely that the current statement (namely that the current statement . . .’. The brackets are never closed; no verb is ever reached; no statement of which we can even ask whether it is true or false is ever adduced.
(Ryle 67-68)

Many of the Paradoxes have to do with such things as statements about statements and epithets of epithets. So quotation-marks have to be employed. But the mishandling which generates the apparent antinomies consists not in mishandling quotation-marks but in treating referring expressions as fillings of their own namely-riders.
(Ryle 69)

Priest’s point seems to be that in the case of the one-legged man, we do not have this problem of infinite self-nestings of namely riders. We would have something like, the father says “All one-legged men lie,” and the one-legged man says “Everything (namely that all one legged men lie) that the father says is true.” Here when we think of truth values, we do not necessarily have self-reference problems. If we say the father’s claim is true, that means the one-legged man’s claim is false, which means the father’s claim is false. Same if we assume that the one-legged man’s claim is true: if that is true, then what the father says is false, meaning that what the one-legged man said is also false. Perhaps Priest is saying that because the statement passes all tests for being a statement, we cannot say that it fails to refer. Priest then mentions another philosopher, Kripke, who takes a similar strategy. I am not familiar with this, but Priest seems to be saying that for Kripke, you can have a sentence which at some ‘fixed point’ has  no truth value. But if the sentence has  no truth value, then it cannot be true, so it is untrue. However, even though it would have this negative truth value, we do not assign it any truth value at all. So in sum, those who try to say that the sentences in the liar paradox do not have value still encounter inconsistency. Thus we do not have these as strong justifications to deny that natural language necessarily leads to inconsistency. See the first full paragraph on p.14 for Priest’s discussion, as you should interpret it better for yourself. I provide some of it here in the following:]

Suppose that my father asserts the mendacity of all one-legged men in town; suppose also that there is only one one-legged man in town who, unbeknown to us, has asserted the veracity of my father. If Ryle is right, then either my father or this one-legged man failed to make a statement. Without loss of generality, let us suppose it to be my father. Yet, by all the standard tests for making a statement, he did. I understood what he said; I can draw inferences from it; I can act on the information contained in it, and so on. Alternatively, take Kripke’s position. Let α be any sentence that obtains no truth value at a fixed point. Then, obviously, ‘α is not true’ should be true at the fixed point (at least if truth at the fixed point models the behaviour of truth in English!), though in the construction it receives no truth value. Hence it seems that none of the motivations will do what is required.

[Priest will now give more reason to think that even if we have truth value gaps, we would still obtain paradoxical sentences. I do not grasp this paragraph sufficiently, but let us work through the ideas. He will give two sentences:

(1) This sentence is true

(2) This sentence is false

Of these, perhaps only the second one is paradoxical or  inconsistent, since if it is true it is false and if it is false it is true. Priest’s point however has to do with the semantic rules that govern the meanings of the sentence parts and that determine its truth value. I am not sure exactly how to understand this, but let us try the (T) scheme first for a non-problematic sentence:

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

So if y is snow, then “y is white” is true, but if y is coal, then it is false. Perhaps the important thing here is that we can determine the second half of the formulation, since we can know whether or not the substitution for y is white or not. But what happens if our formulation is

“y is true” is true, if and only if y is true.

and we substitute “This sentence” in for y?

“This sentence is true” is true if and only if this sentence is true.

Recall in the case of

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

we could know whether or not the substitution in the underlined part held or not, depending on whether or not the substitution was something white. However in

“This sentence is true” is true if and only if this sentence is true.

We are not talking about something outside this formulation, like snow is to the variable y, but rather something inside the formulation and whose meaning and truth value are conditioned by that sentence. So we do not have enough information to know if the underlined part is a true substitution. I think that might be Priest’s point, but I am unsure. Priest explains it another way in his book Logic: A Very Short Introduction

suppose someone says: This very sentence that I am now uttering is true. Is that true or false? Well, if it is true, it is true, since that is what it says. And if it is false, then it is false, since it says that it is true. Hence, both the assumption that it is true and the assumption that it is false appear to be consistent. Moreover, there would seem to be no other fact that settles the matter of what truth value it has. It’s not just that it has some value which we don’t, or even can’t, know. Rather, there would seem to be nothing that determines it as either true or false at all. It would seem to be neither true nor false.
(Priest, Logic: A Very Short Introduction, p.32)

Now consider the second sentence again:

(2) This sentence is false

Which in the (T) scheme would be:

“This sentence is false” is true if and only if this sentence is false.

Recall that for sentence 1, if it is true, then it is true, and if it is false, then it is false. It gives out a consistent value, but it is hard to know which one. But in the liar paradox, if it is true then it is false and if it is false then it is true. So regardless of what you assume, you get two truth values rather than one like sentence 1 had. So the second sentence is not a truth value gap but rather it is a glut, because there is too much value output. Priest’s argument here seems to be that those who would want to say there are truth value gaps would not apply it in the cases which produce the liar paradox. This is because the problem is not that we are lacking a way to determine their truth output – we know the output is both true and false – but rather the problem is that there is too much output. Hence again arguing for truth value gaps does not do away with the liar paradox, since it is not a case where truth value is lacking or is underdeterminable. Please interpret the paragraph for yourself to be sure of what it means.]

Any doubts that we might have that, even if there are Valueless sentences, paradoxical sentences are not among them are magnified when we consider the pair

This sentence is True          (1)

This sentence is False         (2)

| There is something odd about both these sentences, but, prima facie at least, it is not the same in both cases. In the case of (1) the semantic rules governing the use of the demonstrative ‘this sentence’ and those governing the predicate ‘is True’ appear not to be sufficient to determine the Truth value of the sentence. In other words, the semantic rules involved underdetermine its Truth value. Such a sentence is an obvious candidate for a Truth value gap. By contrast, in the case of (2) the semantic conditions of the words involved seem to overdetermine its Truth value. (2) would therefore seem a much more plausible candidate for a Truth value ‘‘glut’’ than a truth value gap, which is exactly, of course, what it is.

[Now Priest will now make another point in his argument against those who claim the liars paradoxes can be resolved by saying some sentences are valueless. This section is beyond my ability to summarize. But let us work through it gradually to at least follow the points in his argument. First we suppose there can be valueless sentences. So

‘This sentence is False’ is True iff it is False.

We can say that this sentence is valueless. So it is neither true nor false. That means we do not encounter the contradiction that results when we assume it is true then find it is false and vice versa. Now Priest will show why this argument will not work. So consider a sentence like

This sentence is not true

which we will call α and consider as valueless. Now, what  happens if we say:

‘α is not True’ is not True

? That means α is True. But that contradicts our assumption that α has no value. So we cannot say that. Or what if we say:

‘α is not True’ is Valueless

? We are already saying that α is Valueless. That means it is neither True nor False that α is not True. But that cannot be right, because we know that since α is Valueless, it cannot be True or False. By definition, it is not True, so we cannot be indifferent about whether or not it is not True. We rather have to admit that:

‘α is not True’ is True

This is fine, because α is not True, as well, it is also not False. But does this then imply that:

‘α is True’ is False

? No, because there is a third value, Valuelessness. (I am confused here. It seems that we can rightly assert that

‘α is True’ is False

since α cannot have the value of Truth. Perhaps Priest is saying that

‘α is not True’ is True

does not directly imply

‘α is True’ is False

because there is the third option of valueless. But I am not sure.) Priest says that it is beyond question that

if α is not True, ‘α is not True’ is True    (3)

(We will see in a bit why he establishes this.)

Then Priest has us consider the “extended” or “strengthened” liar paradox:

(4) is not True.       (4)

This can take one of three values, true, false, and valueless. If it is true, then it is not True. If it is not true (because it is either false or valueless), then it is True (since it says of itself that it is not True). But someone might say that if we suppose 4 is valueless, that does not imply it is true. However, recall again 3, which was undeniable:

if α is not True, ‘α is not True’ is True    (3)

4 is an instance of α, that is, of a valueless statement, hence:

if (4) is not True, then ‘(4) is not True’ is True

Since ‘(4) is not True’ is identical to 4 itself, we get:

i.e. if (4) is not True, then (4) is true.

In sum, Priest seems to be saying that even if you claim that the liar sentence is valueless, you still are committed to saying that it is not True, and in the end this will create the inconsistency of saying that the sentence which is not true is true. Please read this paragraph yourself to find a better interpretation.]

The second main point against Value gap solutions to the semantic paradoxes concerns extended paradoxes. Let us suppose that there are Valueless sentences, and that the claim that paradoxical sentences are Valueless can be substantiated. This allows us, in effect, to maintain that, although a paradoxical sentence such as ‘This sentence is False’ is True iff it is False, since it is neither, the derivation of a contradiction is blocked. There is, however, a standard argument to show that this ploy will not work. Some sentences are neither True nor False. Obviously we are capable of expressing this idea in English: we have just done so. (Moreover anyone who maintains that paradoxical sentences are Valueless must accept this on pain of obvious self refutation.) In particular, for any sentence a that is neither True nor False, ‘a is not True’ must be True. (Again, anyone who maintains a Value gap solution to the paradoxes must accept this, or face a devastating ad hominem argument.) This does not necessarily mean that ‘a is True’ is False, since it is possible to maintain that ‘a is True’ is Valueless, and that negation transforms a Valueless sentence into a True one. It is beyond question, though, that

if a is not True, ‘a is not True’ is True.       (3)

Now consider the ‘‘extended’’ (or ‘‘strengthened’’) liar paradox:

(4) is not True.                                (4)

This sentence is either True, False or Valueless. If it is True, then (by the T-scheme, which is not here at issue) it is not True. Similarly, if it is not True (i.e. False or Valueless), then it is True. Hence, whatever it is, we have a contradiction. One might object to the inference from (4)’s being Valueless to its being True. If, for example, we suppose that (4) makes no statement, then it should not follow that it makes a true one (see Goddard and Goldstein 1980). Yet we have agreed (and the Valuegappist is committed) to (3), an instance of which is

if (4) is not True, then ‘(4) is not True’ is True

i.e. if (4) is not True, then (4) is true. Hence there is no way out here.
(Priest 15)

[Now notice that when we deny something being true, we say it must then either be False or Valueless. This means we do not allow for an additional option, meaning that we still use the law of excluded middle. It is not clear to me why this might constitute an objection, or how Priest defends against that objection. Perhaps the basic idea is that a value gappist has no other options, but I am unsure. Please consult the following:]

It may be objected that the above argument still uses the law of excluded middle in the form of the assumption: (4) is True or it is not True (False or Valueless). However, this is just an instance of the law of excluded middle, and | one, moreover, that is unimpeachable for classical logic augmented with truth value gaps. (For the intuitionist the situation might be different, but we have already dealt with him.) Indeed, given that the Valuegappist is committed to the view that (4) is Valueless, and hence that it is not True, he can hardly deny that it is either True or not True.

There is another objection the Valuegappist might make, which is that “the notions necessary for the formation of the paradox (and in particular the notion of Valuelessness) are not expressible in the language in question” (16). [This might have something to do with the meta-language object language distinction. Priest replies by writing:]

But if this is right, it is an admission that the language for which the semantics has been given is not English, since these notions obviously are expressible in English. Thus the problem, which was to show how the English concepts are consistent, has not been solved.

[Again recall that Priest’s strengthened liar formulation and argument used the principle of excluded middle, since not-True implied either False or Valueless.] Still someone might deny the law of excluded middle [and thus affirm the invalidity of the strengthened liar paradox argument above]. Priest says this will not eliminate the problem, since there are “proofs of contradictions which do not use it” (16). Priest offers as an example Barry’s paradox:

Take Berry’s paradox, for example: English has a finite vocabulary. Hence there is a finite number of noun phrases with less than 100 letters. Consequently there can be only a finite number of natural numbers which are denoted by a noun phrase of this kind. Since there is an infinite number of natural numbers, there must be numbers which are not so denoted. Hence there must be a least. Consider the least number not denoted by a noun phrase with fewer than 100 letters. By definition, this cannot be denoted by a noun phrase with fewer than 100 letters, but we have just so denoted it. Contradiction. This argument appeals nowhere to the law of excluded middle. Both horns of the dilemma are given a direct proof. Reductio, or its equivalent, the law of the excluded middle, is not appealed to at all.

Citations from:

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

Priest, Graham. Logic: A Very Short Introduction. Oxford / New York: Oxford, 2000.

Ryle, G. (1950) ‘Heterologicality’, Analysis 11, 61–9.


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