Priest (7.7) An Introduction to Non-Classical Logic, ‘Truth-value Gluts: Paradoxes of Self-reference,’ summary

 

by Corry Shores

 

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

 

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part I:

Propositional Logic

 

7.

Many-Valued Logics

 

7.7

Truth-value Gluts: Paradoxes of Self-reference

 

 

 

 

Brief summary:

(7.7.1) We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts. (7.7.2) One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129). (7.7.3) Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “ (7.7.4) There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts. (7.7.5) We will now examine briefly a couple claims that these paradoxical arguments are not sound. (7.7.6) Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’. (7.7.7) Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither. (7.7.8) Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.

 

 

 

 

 

 

 

Contents

 

7.7.1

[Paradoxes of Self-Reference as Motivation for Gluts]

 

7.7.2

[The Liar’s Paradox]

 

7.7.3

[Russell’s Paradox]

 

7.7.4

[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]

 

7.7.5

[Turning to Claims that the Paradoxes are not Sound]

 

7.7.6

[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]

 

7.7.7

[Objection 2: The Liar Sentence Is Neither True nor False]

 

7.7.8

[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]

 

 

 

 

Summary

 

 

7.7.1

[Paradoxes of Self-Reference as Motivation for Gluts]

 

[We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts.]

 

[In the previous section 7.6, we examined a motivation for arguing for truth-value gluts, namely, inconsistent laws. We consider now another motivation: paradoxes of self-reference. There are both old and modern ones.]

A second argument for the existence of truth-value gluts concerns the paradoxes of self-reference. There are many of these; some very old; some very modern. Here are a couple of well-known ones.

(129)

[contents]

 

 

 

 

 

 

7.7.2

[The Liar’s Paradox]

 

[One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129).]

 

[(ditto)]

THE LIAR PARADOX: Consider the sentence ‘this sentence is false’. Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false.

(129)

[contents]

 

 

 

 

 

 

7.7.3

[Russell’s Paradox]

 

[Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “]

 

[(ditto) (See  section P.6 of One and ch.5 of Logic: A Very Short Introduction.)]

RUSSELL’S PARADOX: Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false.

(129)

 

[contents]

 

 

 

 

 

 

7.7.4

[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]

 

[There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts.]

 

[(ditto)]

These (and many others like them) are both prima facie sound arguments, and have conclusions of the form A∧¬A. If the arguments are sound, the conclusions are true, and hence there are truth-value gluts.

(129)

[contents]

 

 

 

 

 

 

7.7.5

[Turning to Claims that the Paradoxes are not Sound]

 

[We will now examine briefly a couple claims that these paradoxical arguments are not sound.]

 

[(ditto)]

Many people have claimed that the arguments are not, despite appearances, sound. The reasons given are many and complex; let us consider, briefly, just a couple.

(129)

[contents]

 

 

 

 

 

 

7.7.6

[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]

 

[Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’.]

 

[(ditto)]

Some have argued that any sentence which is self-referential, like the liar sentence, is meaningless. (Hence, such sentences can play no role in logical arguments at all.) This, however, is clearly false. Consider: ‘this sentence has five words’, ‘this sentence is written on page 129 of Part I of An Introduction to Non-Classical Logic’, ‘this sentence refers to itself’.

(129)

[contents]

 

 

 

 

 

 

7.7.7

[Objection 2: The Liar Sentence Is Neither True nor False]

 

[Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither.]

 

[The second objection is the most popular one. It says that the liar sentence is neither true nor false. This means that we cannot appeal to the law of excluded middle (because it is no longer the case that the sentence is either true nor false. How did we use it previously? I am not really sure. Maybe it goes like this, but I am guessing. We have the sentence, “this sentence is false.” Then we say, “Either it is true or it if false. If it were true, then it is false, and if it is false, then it is true. Either way, it is both true and false.” So maybe, the argument works by having the original proposal that it is either true or false. And maybe the idea now is that were it neither value, then the law of excluded middle does not hold, and so we cannot start off with the assumption “Either it is true or it is false; if true ...”. Or maybe we can start it that way, but it cannot end that way, because we have the third possibility to assess, that it is neither, meaning that we cannot further derive another value from it in addition to it being neither. I am not sure.) “Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too” (129).]

The most popular objection to the argument is that the liar sentence is neither true nor false. In this case, we can no longer appeal to the law of excluded middle, and so the arguments to contradiction are broken. (Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too.)

(129)

[contents]

 

 

 

 

 

 

7.7.8

[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]

 

[Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.]

 

[(ditto)]

This suggestion does not avoid contradiction, however, because of ‘extended paradoxes’.³ Consider the sentence ‘This sentence is either false or neither true nor false.’ If it is true, it is either false or neither. In both cases it is not true. If, on the other hand, it is either false or neither (and so not true), then that is exactly what it claims, and so it is true. In either case, therefore, it is both true and not true.

(130)

3. Moreover, and in any case, not all of the paradoxical arguments invoke the law of excluded middle. Berry’s paradox, for example, does not.

(130)

[contents]

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

 

 

 

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Priest, Ch5 of Logic: A Very Short Introduction, “Self Reference: What is this Chapter About?”, summary

by Corry Shores

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Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.5
Self Reference: What is this Chapter About?

Very Brief Summary:
Self-reference can present problems in logic, which leads us to conclude that there are actually four and not just the first two of the following possibilities. A sentence can be either 1) just true, 2) just false, 3) both true and false, or 4) neither true nor false. The liar sentence, “This sentence is false,” is a candidate for the third option, and its “cousin,” “This sentence is true,” for the fourth. There are certain inferences that intuitively seem valid, but under the first two “classic” assumptions they are deemed invalid. To their credit, the new assumptions make them valid. However, other inferences that the classic assumptions rightly deems valid are counter-intuitively deemed invalid under the new assumptions. There are further problems with the new assumptions. The valueless “cousin” sentence is assumed to be not true (and as well not false), but it in fact says of itself that it is true. And a stronger version of the liar sentence, “This sentence is not true,”  is not just true and false but in a more problematically contradictory way it is as well both true and not-true.

Brief Summary:
Paradoxical and otherwise problematic instances of self-reference lead us to suspect that we have more options than the following two: 1) a sentence can be just true, or 2) a sentence can be just false. Consider the “liar” sentence, ‘This sentence is false.’ If it is true, then it is false; but if it is false, then it is true. Either way, it’s truth-value will contradict what it says its truth-value is. So we have option 3) a sentence can be both true and false. Or consider the “liar cousin” sentence, ‘This sentence is true.’ Normally the terms in such a declarative sentence refer to things or situations by which we may determine the truth or falsity of the statement, that is to say, whether or not the indicated situation holds in reality or not. So if we say, “this chair is red,” we look to the indicated chair and its color, and we determine if the sentence is true or not. However, the terms in “this sentence is true” does not point us to such a determining situation, since we are only able to make two equally viable assumptions about its truth value, namely, that it is either true or that it is false; but, we have no way to make the determination one way or another, since it will always be consistent with what it says of itself under both assumptions. It would seem that we have no grounds that would allow us to determine whether it is true or false, and thus we have option 4) a sentence may be neither true nor false. The classical assumptions 1 and 2 lead us to conclude certain inferences are valid when our intuitions say otherwise. For example, “The Queen is rich,” “The Queen isn’t rich,” therefore, “Pigs can fly” (q, ¬q/p). Our intuitions tell us this seems invalid. But by just using assumptions 1 and 2, it is valid, since structurally speaking there is no situation where the premises are true and the conclusion is false. For, the premises can never all be true anyway. However, under the new assumptions, particularly that sentences can be both true and false, q, ¬q/p can be valid, if q is both true and false and p just false. For, q is at least true and ¬q is also at least true. However, our intuitions tell us that q∨p, ¬q/p is valid, but the new assumptions deem it invalid. Yet, perhaps it only seems intuitively valid if we forget that there are exceptional situations where sentences can be both true and false. There are other problems with the assumptions. When we assume that the liar cousin, “This sentence is true,” is neither true nor false, that means it cannot be true, but it says of itself that it is true. And while we might go along with saying that “This sentence is false” is both true and false, we might not feel the same way about “This sentence is not-true”. Here, we might conclude that it is both true and not-true (and not just true and false), which is a stronger contradiction that we may not want to accept.

Summary

 

This issue of reference is not a simple one, especially in cases of self-reference. Sometimes a name refers to something larger that it is a part of. “For example, consider the sentence ‘This sentence contains five words’. The name which is the subject of this sentence, ‘this sentence’, refers to the whole sentence, of which that name is a part” (31). We also have self-reference in the following other cases. There is the name “These regulations” in the sentence, “These regulations may be revised by a majority decision of the Department of Philosophy”. [Here the larger body of regulations of which this stipulation is a part is referred to by the name “These regulations”.] And there is the name “this thought” in the thinking of the person who says “If I am thinking this thought, then I must be conscious” (31).

All these instances above are not problematic cases of self-reference. But there are ones that are. Consider:

This very sentence that I am now uttering is false.
(31d)

We will call the sentence above λ. Now we ask, is it true or false?

Well, if it is true, then what it says is the case, so λ is false. But if it is false, then, since this is exactly what it claims, it is true. In either case, λ would seem to be both true and false.
(32a)

But matters are not much better with this statement:

This very sentence that I am now uttering is true.
(32)

Why? It would seem consistent. If it is true, then it is true, since it says it is true. And if it is false, then it is false, since it claims instead to be true. But, how would its truth or falsity be determined? [The only way it seems it can be determined is by assumptions, and even this does not determine it as one or the other, because both are equally valid. Since its truth or falsity cannot be determined anyway, it perhaps has neither value.]

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.
(32)

These are ancient paradoxes. The first one, “This sentence is false” is a form of the liar paradox, which was discovered by the ancient Greek philosopher Eubulides. Liar type paradoxes also have appeared in recent debates, “some of which play a crucial role in central parts of mathematical reasoning” (32). [The next paradox involves the mathematical and logical notion of a set. Sets are abstract in a way that allows for self-reference.] One such important paradox is found in set theory, and it involves the set of all non-self-including sets. To arrive at this paradox, we need to walk through some other concepts.

A set is a collection of objects. Thus, for example, one may have the set of all people, the set of all numbers, the set of all abstract ideas.
(32)

[But as we mentioned, there is an element of abstraction to sets that allows for them to be taken as members of other sets.]

Sets can be members of other sets. Thus, for example, the set of all the people in a room is a set, and hence is a member of the set of all sets.
(32)

[Furthermore, a set can even include its very own self. To understand this, we of course cannot imagine a set as being like a physical container like a jar, because then it could not physically fit within itself. Also, there is a strange doubling that seems to be happening which could not be understood with physical metaphors. We are dealing a self-inclusive set abstractly, which means there is one set, and thus it has just one name, but it is regarded doubly, namely, as being the set that is including itself and also the set that is included in itself.]

Some sets can even be members of themselves: the set of all the objects mentioned on this page is an object mentioned on this page (I have just mentioned it), and so a member of itself; the set of all sets is a set, and so a member of itself.
(32)

Many sets, however, cannot be self-inclusive.

And some sets are certainly not members of themselves: the set | of all people is not a person, and so not a member of the set of all people.
(32-34 [The text skips page 33, which is entirely an image])

[So we have the following progression of concepts, with a  new addition: 1) some simple set of things, 2) a set included in another set, as for example the sets included in the set of all sets, 3) a set that is included in itself, and thus the set to which this set belongs is not really other to it, and now, 4) a set that does not include itself, and furthermore 5) the set of all sets that do not include themselves.] So now we “consider the set of all those sets that are not members of themselves” (34). We will call this set of all non-self-inclusive sets R. We now ask, “Is R a member of itself, or is it not?” (34). [The problem will be that like the liar sentence, its status of self-inclusion is indeterminable. To write out the following more fully: If the set of all non-self-inclusive sets is included in itself, then it is not really a non-self-inclusive set, since it is self-inclusive. If the set of all non-self-inclusive sets is not included in itself, then it would belong within itself as as a member, because it is non-self-inclusive.]

If it is a member of itself, then it is one of the things that is not a member of itself, and so it is not a member of itself. If, on the other hand, it is not a member of itself, it is one of those sets that are not members of themselves, and so it is a member of itself. It would seem that R both is and is not a member of itself.
(34)

This paradox is called Russell’s paradox, named after its discoverer, Bertrand Russell. Just as we saw with liar paradox, it is also problematic to have the set of all self-inclusive sets.

Like the liar paradox, it has a cousin. What about the set of all sets that are members of themselves. Is this a member of itself, or is it not? Well, if it is, it is; and if it is not, it is not. Again, there would seem to be nothing to determine the matter either way.
(34)

[Recall what we said in Chapter 2. There we were looking at truth conditions and truth functions. On page 9 for example, Priest has us assume when making a truth table for negation “that every sentence is either true or false, but not both” (9).] These problematic examples of self-reference challenge our assumption that we made in Chapter 2 that “every sentence is either true or false, but not both. ‘This sentence is false’, and ‘R is not a member of itself’ seem to be both true and false; and their cousins seem to be neither true nor false” (34). [In section 1.3 of Priest’s In Contradiction, he explains these two situation using the terms gaps and gluts. To understand this distinction, we need to note a semantics issue. I think to follow through these ideas, we might consider a different class of statements altogether, namely, ones that refer to things other than themselves. So, “this chair is red” is true if the chair is red and false if it is not. Why is this different than “this statement is false”? We take note that the terms in “this chair is red” refer to a situation that may determine the truth value of the statement. How that truth value is determined is another matter. But the fact is, presumably, if certain basic conditions are met, that to which the terms refer can definitely determine the statement’s truth value. Now, what is it that terms in “This sentence is true” refer to? They refer back to the sentence itself. But the problem is that the sentence itself, unlike the chair’s color, cannot determine the truth value of the statement. As Priest writes in In Contradiction, “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 Contradiction 15). In this case, there is a truth-value “gap” since it can be determined neither as true nor as false. What about “This sentence is False”? Here we have the same structure of self-reference with the term “this sentence” and now we have the predication “is False”? For some reason that I do not quite grasp clearly, here the situation is different. I do not understand so well, because one could say that this second sentence is not doing anything different than the first. For the first case, the “truth-teller”, if we assume it is true then it is true and if we assume it is false it is false. In the second case, the “liar”, if we assume it is true it is false and if we assume it is false it is true. On those grounds, why do we not say that the liar also is neither true nor false, since we as well cannot on the basis of the terms determine one value or the other? This I do not understand. Perhaps the idea is the following. The truth-teller’s truth-value cannot be rightly found. It can only be endowed by means of assumption, which means that it intrinsically has no value on its own. The liar sentence always outputs the opposite of your assumptions, which contradicts what it says it should be. For the liar it does matter what your input is, because you get a self-consistent consistent output. But since the liar’s output is always inconsistent with its meaning, it does not matter what you input. For, its output can be inputted again to once more get the opposite output value. If it is true, then it is false, but if it is false, then it is true. So given this problematic circularity, it does not matter what assumption you begin with. But as you can see I am not certain what justifies us in distinguishing them fundamentally. At any rate, taking it for granted that the liar sentence is both true and false, we here have a “glut” since there is too much determination of its truth value, rather than a lack of it like in the truth-teller.]

Priest says we can accommodate this problematic situation by taking these other truth-status possibilities into account.

Assume that in any situation, every sentence is true but not false, false but not true, both true and false, or neither true nor false.
(34)

We recall the truth conditions for negation, conjunction, and disjunction from chapter 2.

In any situation:

¬a has the value T just if a has the value F.
¬a has the value F just if a has the value T.

a & b has the value T just if both of a and b have the value T.
a & b has the value F just if at least one of a and b has the value F.

|

a ∨ b has the value T just if at least one of a and b has the value T.
a ∨ b has the value F just if both of a and b have the value F.
(34-35)

[Instead of following the prior restriction that allowed only for true or false values,] we will “work out the truth values of sentences under the new regime” (35). [It seems here Priest is selecting as exercises three possible truth situations for negation, conjunction, and disjunction. 1) First we suppose a classical logic situation where a is false but not true. Here, negation simply flips the value. 2) Second we suppose a glut situation where a is both true and false, while b is just true, and the two are joined conjunctively. Here, the whole conjunct is both true and false. It seems the reasoning is this. Since a is at least true, that makes a & b true. But since it is also false, that makes a & b false as well. 3) Third we suppose a gap situation where a is just true, but b is neither true nor false, and the two are joined disjunctively. Here a ∨ b is merely true. The reasoning is as follows. What would make it false is only if both a and b are false. But since at least a is true, it does not matter that b is neither, and so the disjunction is always true.] [In the following, for the “clauses” (c1/c2), I insert them in curly brackets for convenience.]

• Suppose that a is F but not T. Then, since a is F, ¬a is T (by the first clause for negation).
{c1: ¬a has the value T just if a has the value F.} And since a is not T, ¬a is not F (by the second clause for negation). Hence, ¬a is T but not F.
{c2: ¬a has the value F just if a has the value T.}

• Suppose that a is T and F, and that b is just T. Then both a and b are T, so a & b is T (by the first clause for conjunction).
{c1: a & b has the value T just if both of a and b have the value T.}
But, because a is F, at least one of a and b is F, so a & b is F (by the second clause for conjunction). So a & b is both T and F.
{c2: a & b has the value F just if at least one of a and b has the value F.}

• Suppose that a is just T, and that b is neither T nor F. Then since a is T, at least one of a and b is T, and hence a∨b is T (by the first clause for disjunction).
{c1: a∨b has the value T just if at least one of a and b has the value T.}
But since a is not F, then it is not the case that a and b are both F. So a∨b is not F (by the second clause for disjunction). Hence, a∨b is just T.
{c2: a∨b has the value F just if both of a and b have the value F.}).
(35)

Now we wonder what this means for validity. Recall that “A valid argument is […] one where there is no situation where the premisses are true, and the conclusion is not true” (35). This is still the case, as is the fact that “a situation is […] something that gives a truth value to each relevant sentence” (35). The only difference is that situations now may also give either two truth values or none. We will now ask if the inference q/q∨p is valid. [The basic idea here seems to be that we cannot have a situation where the conclusion is not true while the premise is true. This is because we assume that q is true. That is enough to make the disjunct true, where q appears again. Thus it is valid. If q is false, then we cannot determine the validity anyway, so those cases do not matter. I wonder, what if q is both true and false? Perhaps that does not change the situation, since insofar as it is false, it has no bearing on the test for validity of the inference. I am not sure about this, but that might be what Priest is suggesting below in parentheses.]

So consider the inference q/q∨p. In any situation where q has the value T. (It may have the value F also, but no matter.) Thus, if the premiss has the value T, so does the conclusion. The inference is valid.
(35)

[Recall another inference from chapter 2: q, ¬q/p. We noted that we cannot have any situation where all the premises are true and the conclusion false. This is because we have both q and its negation, which means always at least one premise will be false. Because we do not even need to relate the premises to the conclusion, we called it vacuously valid. Here was the truth table:

] Under the old assumptions, q, ¬q/p is valid, but under the new ones, it is invalid. [The reasoning for this seems to be the following. q can be both true and false, and p just false. This means that ¬q is both true and false. Now since both ¬q and q are both true and false, they are both at least true, while the conclusion is false, thus making the inference invalid. Of course a concern could be that this reasoning does not work, since ¬q and q are also both false, and thus this case of glut values does not allow us to test the validity of the inference. Priest says that their additional falsity does not matter. I am not exactly sure why. It again could be the fact that even though they are no less false as true, that falsity is not relevant to the test of validity, and only their truth is relevant. I find these paraconsistency ideas absolutely fascinating philosophically. We do not take the joint truth and falsity as an unbreakable pair of values. They both stand independently on their own. Both values are absolutely affirmative in the sense that the one does not subtract from the other. I find this affirmative concept of “both” in application to truth and falsity to be quite interesting and powerful. It is addition without mixture or contamination, but the things being combined you would normally think would interfere with each other’s value.] Priest will explain why under the new assumptions q, ¬q/p is an invalid inference. [The fact that the new assumptions correspond more with our intuitions about the inference suggest that classical logic is inadequate and that we instead should consider a non-classical logic.]

just take a situation where q has the values T and F, but p has just the value F. Since q is both T and F, ¬q is also both | T and F. Hence, both premisses are T (and F as well, but that is not relevant), and the conclusion, p, is not T. This gives us another diagnosis of why we find the inference intuitively invalid. It is invalid.
(35-36)

Priest then says that “As we saw in Chapter 2, this inference follows from two other inferences,” namely, q/q∨p and

q∨p, ¬q
       p

[I recall the discussion of these other inferences, but at the time I did not realize that q, ¬q/p followed from them. I am still unsure how this is, but perhaps the idea is the following. The inference q, ¬q/p seems to throw in p at the end, and to all appearances it comes out of nowhere. So it would make sense if we introduce it in the premises, hence the need for q/q∨p, which seems to justify introducing other terms. But now that there are two terms, and we infer from them merely one of the two, we need a way to eliminate one of them. Hence the q∨p, ¬q/p. Most likely the above reasoning is not what Priest means by q, ¬q/p follows from these other two inferences. Perhaps he is just saying that if you begin with these other two inferences, you can combine them to get in essence q, ¬q/p.] Priest will now find a way to invalidate q∨p, ¬q/p by finding an instance (using our new assumptions) where the premises are true and the conclusion false. So we assume that p is just false, and p is of course the conclusion. But we assume that q is both true and false. This means “that both premisses get the value T (as well as F). But the conclusion does not get the value T. Hence the inference is invalid” (36).

[Previously these new assumptions allowed us to determine q, ¬q/p as invalid, which matched our intuitions about the inference. This was one advantage over the old (classical) assumptions. But now Priest acknowledges this case where the new assumptions make q∨p, ¬q/p invalid, which goes against our intuitions. Priest will still defend the new assumptions. His basic point seems to be that really it does in fact match out intuitions, but only when we are keeping in mind instances where q can be both true and false, as in the liar paradox. Then the inference intuitively seems valid.]

In Chapter 2, I said that this inference does seem intuitively valid. So, given the new account, our intuitions about this must be wrong. One can offer an explanation of this fact, however. The inference appears to be valid because, if ¬q is true, this seems to rule out the truth of q, leaving us with p. But on the present account, the truth of ¬q does not rule out that of q. It would do so only if something could not be both true and false. When we think the inference to be valid, we are perhaps forgetting such possibilities, which can arise in unusual cases, like those which are provided by self-reference.
(36)

Priest invites us to think about which explanation (the current one or the one from chapter 2) we find more compelling. Priest then notes other problems with the new assumptions. [I do not grasp the main ideas here clearly enough to restate them properly. The main idea is that even with our new ‘gap’ and ‘glut’ assumptions, we still have unresolved problems with the liar and its cousin. Regarding gaps, Priest discusses many problems with them in his In Contradiction. See section 1.3 and section 4.7. In our current treatment here, Priest shows that we still have a contradiction with the gap assumption applied to the cousin. Even though we begin by assuming that it has neither a true  nor a false value, we know from this that it is at least not true (for if it were true, then we are not using the gap assumption). However, it says of itself that it is true.]

Consider the liar paradox and its cousin. Take the latter first. The sentence ‘This sentence is true’ was supposed to be an example of something that is neither true nor false. Let us suppose that this is so. | Then, in particular, it is not true. But it, itself, says that it is true. So it must be false, contrary to our supposition that it is neither true nor false. We seem to have ended up in a contradiction.
(36-37)

Then he turns to the liar sentence, but now under a different formulation, “This sentence is not true,” which also presents a contradiction. [I think I do not adequately grasp the point here. We will conclude that the sentence results in a contradiction. I had thought that by saying it is both true and false we were already acknowledging there is a contradiction. Also, we are  making a distinction between not-true and false, which I do not know how to make. I will quote it below, because I cannot convey the meaning well in my own words and thinking. He does not present it this way, but I let me offer the following formulation. We begin with “This sentence is not true”. We say it is both true and false. Insofar as it is true, what it says of itself holds, and thus it is also not true. Insofar as it is false, what it says of itself does not hold. Thus it is not the case that it is not true, therefore it is true, but it says of itself that it is not true. So we have more than just the sentence being both true and false, as per our assumptions. It is as well both true and not true, in accordance with its stated self-determinations. So the idea here might be the following. Someone could think that it is one thing to say that a sentence is both true and false. But that is not as strong and as evident a contradiction as to say that it is both true and not true. So perhaps we might be willing to go along with saying that a sentence has both the values 1 and 0, or T and F. But we might not feel so sure if we take it another step to say that it is both 1 and not 1, or T and not T. I am not sure why someone would accept the first articulation but reject the second. And as I said, I also do not know how to distinguish not-T from F. If we only have two values, I would think that they would be equivalent. Perhaps the idea is that with the new assumptions they are not equivalent. The liar cousin under the gap assumption is not T but also not F. Thus not-T and F are not equivalent there. So his point might be that we need these extra values, like not-T vs. F, and thus we have extra complications.]

Or take the liar sentence, ‘This sentence is false’. This was supposed to be an example of a sentence that is both true and false. Let’s tweak it a bit. Consider, instead, the sentence ‘This sentence is not true’. What is the truth value of this? If it is true, then what it says is the case; so it is not true. But if it’s not true, then, since that is what it says, it is true. Either way, it would seem to be both true and not true. Again, we have a contradiction on our hands. It’s not just that a sentence may take the values T and F; rather, a sentence can both be T and not be T.
(37)

Priest concludes: “It is situations of this kind that have made the subject of self-reference a contentious one, ever since Eubulides. It is, indeed, a very tangled issue” (37).

 

[The following is quotation.]

 

Main Idea of the Chapter

● Sentences may be true, false, both, or neither.
(quoted from Priest, 37, boldface his)

 

 

From:

 

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.

Also mentioned:

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

 

 

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.

Summary

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:

Fα ↔ T¬α
(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:

¬Tα → Fα  
(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.
(64-65)
[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.
(305-306)]

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.
(65)

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.
(65-66)

[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.
(66)

[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.
(66)

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 Tα₀; 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.
(66)

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.

 

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.

Summary

[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.
(13)

[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.
(13)

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’.
(13)

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.
(13)

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.
(14)

[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.
(14-15)

[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.
(15-16)

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.
(16)

[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.
(16)

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.