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 (P6) One, ‘P.6 Dialetheism and the Inclosure Schema’, summary

by

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

 

Summary of

Graham Priest

One:
Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness

Preface

P.6 Dialetheism and the Inclosure Schema

Brief Summary:

There are paradoxes of self reference. One example is the liar paradox (‘This sentence is false’. Is it true or false?), and another is Russell’s paradox (The set of all non-self-inclusive sets: is it included in itself or not?). Graham Priest uses his ‘inclosure schema’ to describe them, and he calls them inclosure paradoxes. This inclosure schema allows us to depict and understand how these paradoxes fit within a dialetheic logic. The paradoxical cases are not ‘enclosed’ within themselves or within an exterior set. They are ‘inclosed’, being on the boundary shared by both the interior and exterior of the sets, ‘in’ a special place where both sets overlap.

Summary

There is a kind of paradox called the inclosure paradox. The basic idea is that by modifying groups in a certain way, you will get a group that both belongs within itself and outside itself. Russell’s paradox and all other paradoxes of self-reference are types of inclosure paradoxes (Priest xx-xxi). Let us start with a famous and simple example: a barber who only shaves men who do not shave themselves. The question is, does he shave himself? If he does shave himself, then he is not the barber who only shaves non-self-shaving men. So he does not, meaning that he is not a self-shaving man. But by that criteria, he should then shave himself.

But Russell’s paradox is about groups. We will work our way gradually to this broader articulation of the inclosure paradox. So think of the barber as a group of one member. The question is, does the barber as a member of his grouping belong to that grouping or not? We think of him first as just a barber whose job it is to shave men. Then we apply a modification to him. We may think of it as enforcing a law on him that was not previously in place. He may now only shave men who do not shave themselves. So now we have both a category (‘barbers who only shave non-self-shaving men’) and a member (that barber himself), and the two will have to be identical; for, there is only one barber in this town. (If there were two barbers, then each could shave the other and never his own self). So this single member group, the barber himself, now both fits within the group of non-self-shaving men and does not fit within it.

This applies as well to larger groupings where self-reference can create paradoxes. Russell’s paradox is more generally the task of classifying the set of all non-self-inclusive sets: does that set belong within itself or outside itself? Like the barber example, if it does, then it does not, and if it does not, then it does.

Priest’s explanation of inclosure paradoxes is a bit technical, and for me personally it is tricky to grasp. Nonetheless, it is formulated with elegant simplicity. Let us take a look at it. I will take it apart piece by piece, and I invite corrections to my imperfect analysis.

We will work through it with the specific example of Russell’s paradox, then we will see how it can be generalized to other paradoxes of self reference.

The diagram and formulation is called the “inclosure schema”. There are enclosures which include items, and those outside the enclosure are excluded from the grouping. (‘Inclosure’ as we will see is being ‘in’ the boundary where the two groupings meet.) We will call the set in its entirety Ω (omega). In this Russell’s paradox example, it is the set of all sets.

Within it are items, other sets. These sets themselves contain members. These members of course are other sets, meaning that they themselves contain members. One such subset of Ω is x.

Now at this point we mention just one condition for inclusion in Ω and x, which is that they be self identical. We call this condition ψ (psi). So in our notation, ψ(Ω) means Ω is identical to itself (Ω = Ω), and ψ(x) means x is identical to itself (x = x).

[I suggest we here wonder about a possible formulation for this self-identity. We can think of x in terms of its totality, or we can think of it in terms of its plurality. It is both, of course. The totality is no more or less than the sum of the plurality, and the plurality of parts together define the totality. So both senses of x are identical. However, they can also be treated separately. Some sets can include themselves (the totality itself can be included among the plurality of its members), for example, “things”, “categories”, “nameables”, “ideas”, “abstractions”, “demonstrables (things you can point-out)”, “distinguishables (things you can distinguish from other things)”, and so on. The set of all things is itself a thing. If we call the set X, and if in it are such things as a, b, c, …, we would add X to that list. So,

a, b, c, … ∈ X,

but as well,

a, b, c, …, X ∈ X.

There is no paradox here. By the same token, we can consider a set which does not include itself, for example the set of all fruits (here the totality itself is not among the plurality of its members). That set is not itself a fruit. So for fruits a, b, c, …:

a, b, c, …, ∈ Y

but

Y ∉ Y

Here again there is no contradiction.] So we have described one condition or property, being self-identical, ψ. But we note another one, φ. In our Russell’s paradox example here, in order for something to belong to Ω, it must be a set that includes other sets, and we call this defining property of ‘includes other sets’ φ. (More generally speaking, φ is the defining property of Ω, which will be different for each application of the schema.)

This is important, because we will take a subset of x, which we will call y, but we want to be clear first that a certain operation works on the level of these subsets. (In the Russell’s paradox example) when that operation, called δ, is applied to a set, it converts that set in such a way that a subset of it is now a set of sets that do not include themselves. [I think the reason we complicate things now with a subset of x is because we need to distinguish the set of x’s members that includes itself and the set of x’s members that does not include itself. So y seems to be its members excluding x, which then allows us to ask whether or not x itself as a totality (and not the bare plurality of its members) belongs within or outside its own boundaries.]

δ(x) = {y ∈ x : y ∉ y}

So, the reason we speak of a subset of x is because as we will see, x itself is not placed within itself. [To understand why this might be, let us first conduct an exercise that we will then later conduct with Ω. Assume first on the one hand that x is included within itself. That means it is self-inclusive. But x only includes non-self-inclusive sets and thus cannot be included within itself, ultimately making it directed outside itself. If on the other hand it is not included in itself, then it lies outside itself, belonging to another set that includes certain non-self-inclusive sets. There is no contradiction in this case. It seems we conclude that it lies outside itself, because this situation is both a possibility and is also the least problematic option.]

[Let us reflect more on how we can interpret the inclosure schema diagram. We return to our examples above, X is the set of things, and is itself a thing. When we apply an operation making its members things which are not themselves things, X itself now does not lie within itself but with another set which can include non-self-including things. What seems to be important in the diagram above is that the arrow begins from x’s enclosing boundary itself, not from within it. This means that x can be seen as having a bounded limit which can be understood either as containing all its plurality of members including the whole totality itself or otherwise as excluding that totality itself. In this case, the arrow tells us that the boundary itself is directed outside itself, meaning that the set is not included in itself. The exterior encroaches entirely atop the boundary but not over it, and the interior comes up to but not atop of that boundary. If x would be included in itself, the interior would encroach atop but not over the boundary, while the exterior would come right up to it but not encroach atop it. The third possibility as we will see is that the exterior and the interior both encroach atop but not across and over the boundary, meaning that the set itself both belongs and does not belong within itself.]

[[But bare with me please once again. Before we continue, let us think of this diagram in terms of the boundary encroachment idea, as it might give a different visualization that can further help us grasp Priest’s insights. So we begin with Ω and its subset x.

We start first by making the set x be a set of non-self-inclusive sets (but not the one and only such set, just an instance of them, with there being other unnamed instances as well. So outside x can still exist non-self-inclusive sets, with x itself belonging to one of these exterior sets.)

Outside x is the rest of set Ω.

Let us think of these thick borders as representing the set itself in its entirety. The question is, is x included in itself? In that case, we would color the border the same as the set, to indicate that the set itself is enclosed within that set.

Or perhaps x is not included within itself. In that case, we would color the border blue, meaning that x as a totality that is distinguishable from the raw conglomerate of its parts lies outside itself and is included in the rest of Ω.

As we concluded above, it would lie outside itself, because that way it can itself be a set of non-self-inclusive sets without being included within itself.

Now we turn to Ω. It is the set of all sets. Outside it are no sets into which it may be included. We will designate that null space with yellow.

However, it can still be that the null outside encroaches over the border, meaning that Ω would not be enclosed in itself. Next we apply that operation δ to it, and we say that it is now the set of all non-self-inclusive sets. The question is, is Ω included within itself? In that case the border would be colored blue to show that the set itself is within that set.

Or is it not included within itself? In that case the border would be colored yellow to indicate that all within the set are included in the set, but the set itself is not.

This situation is similar to the barber paradox we mentioned above, and below we will look more at this paradoxical structure. But note for now how we saw that in a dialetheic logic, we could say that there is no logical problem here. Perhaps the set of all non-self-inclusive sets both does and does not include itself. We might depict that by shading the border  with green, the mixture of blue and yellow, to show that Ω itself both is found within itself, giving it blue tone, and not found within itself, giving it yellow tone.

Now let us return again to Priest’s diagram, with this possible visualization in mind.]

Not all of x remains in x after operation δ is applied, only subset y remains. Within x are y, which are non-self-including sets. But x itself cannot belong in itself, yet it does not have to, because x does not contain all possible sets, only some of them. So x is placed outside itself [its exterior encroaches atop its boundary but does not cross it]. What is important about y being φ (being a set that includes sets) is that we see a clear case where the operation δ works fine on items which have the defining traits of Ω. The question is, how does it work when we consider larger and larger sets x until they grow to exactly the size of Ω itself (the set of all sets)? [Since the operation works fine in all cases of subsets up to Ω, we would expect it to work on the limiting case of Ω itself, especially since all cases up to the limit share the same defining properties as the limit case itself.]

We apply the operation δ to Ω, and now all sets within Ω are non-self-inclusive sets. What do we say of set Ω itself? Can it be found within itself? If so, then it is a not a self-inclusive set, and thus its boundary is directed outward.

But if it cannot be found within itself, then it is a non-self-including set, qualifying it for inclusion within itself, and thus its boundary would be pointing inward.

[Unlike the case of x, there is no exterior set to which it may belong. So there is no unproblematic situation. It can only be that it both belongs to itself and does not belong to itself.] Priest depicts this double status of Ω by placing a cross on the border, meaning that it falls squarely on the boundary where the two contrary statuses meet.

[[As we noted in the previous section, this is a problem in classical logic, which does not allow for contradictions; but in dialetheic logic there are true contradictions. Russell’s paradox and other paradoxes of self-reference are good candidates for such dialetheias. Here we have situations where all the parts of the paradox make sense on their own, and their combination is made according to the normal rules of language, grammar, and logic. Some philosophers, dialetheists, think that instead of these paradoxes being problematic, they rather are evidence for the claim that there are true contradictions. For, how can the paradox be otherwise than true? Its meaning and formation are clear and proper. Just its logic is in question. Russell’s solution is to legislate that one level of a hierarchy refers to its lower level and not to itself. But language does not work by such rules. Language allows for many grammatically and semantically sensible combinations, and the trouble is found only on the level of logic. Maybe it is wiser to make our logic conform to the real way that language works, rather than insist on making language work in ways that it fundamentally does not, namely to say it cannot make logically self-contradictory combinations, when all its basic principles and mechanisms in fact allow for those combinations. Furthermore, philosophically speaking, these paradoxes, rather than being outlaws to the system, could lie at the very heart of its structure, so rather than exclude them, perhaps they should be treated as our best insights into the deeper structures of significance and meaning.]]

[Jc Beall looks at Graham Priest’s rendition of this enclosure schema as it is in Priest’s Beyond the Limits of Thought. He writes (citing Priest’s aforementioned text) (my comments are in double curly brackets):

The inclosure scheme involves a set Ω, two unary predicates φ and  ψ, and a function δ, where the following conditions are satisfied (see Priest, 2002, §9.4ff):
1. Ω = {y : φ(y)} and ψ(y)

{{The whole is made of subsets that have properties/predicates φ and ψ.}}

2. For any X ⊆ Ω such that ψ is true of X,

{{For any subset X of Ω where X has the predicate ψ, …}}

(a) Closure: δ(X) ∈ Ω

{{… the function applied to X leaves X belonging to Ω, …}}

(b) Transcendence: δ(X) ∉ X.

{{… but the function applied to X also makes it not included within itself}}
In the limiting case, where X = Ω, we have a contradiction: δ(Ω) ∈ Ω and δ(Ω) ∉ Ω.

{{In the largest set X of Ω, X is the same as Ω. In this case, the operation makes the set both included in itself and not included in itself.}}

Example: let Ω be the ‘collection’ of all truths (so that φ is truth). {{Ω is the set of all true matters. The defining property of that set, being true, is the predicate φ.}} Let ψ be the property of being well-defined {{I think this means in this case ‘unambiguous’, so before the function is applied, it is unambiguous, but in some cases that creates an ambiguity}}.  In turn, δ can be thought of as taking subsets of Ω to sentences, namely, the sentence ‘I am not in X’ {{Perhaps the “I” is like a variable into which the members are placed, and when applied to the set X itself, it just means that the set is not included in itself.}} By standard liar-like reasoning, we get a plausible argument for Closure and Transcendence: namely, that δ(Ω), which is the sentence ‘I am not true’ or | ‘I am not in the collection of truths’, is both in Ω and not in Ω. {{Perhaps Beall is saying that the liar sentence ‘I am not true’ can be understood with the “I” meaning not simply that very statement itself but the collection of all true statements. This equivalence between the singular statement itself and the collection of all true statements comes about through a limiting process. The difference would be between saying “I am not in a collection of truths” and “I am not in the one and only collection of all truths.” When we take the partial collections to the limit of the whole, its meaning is identical to the particular self-reference of “I am not true” or “I am not in the collection of truths”, because the “I” can refer to nothing other than that collection of truths.}} And precisely the same sort of situation exists in the case of the Russell-Zermelo-Cantor paradox (concerning all non-self-membered sets/collections, etc).
(Beall 11-12)

]

Priest continues by explaining how the inclosure paradox applies in other cases, namely, König’s paradox

Some of the paradoxes in question are paradoxes of definability. A paradigm of these is König’s paradox. Something is definable if there is a (non-indexical) noun phrase that refers to it. If a is a definable set of definable ordinals, then (since this is countable), there is a least ordinal greater than all the members of a. It is obviously not a member of a, but it is definable by the italicized phrase. Since the set of all definable ordinals is itself definable, we may apply this operator to it to obtain a set that cannot be referred to (defined), but which yet can. In this case, Ω is the set of all definable ordinals; ψ(x) is ‘x is definable’; and  φ(x) is the least ordinal greater than all the members of x.
(Priest xxi)

Priest, Graham. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University, 2014.

Beall, Jc. “End of Inclosure.” Mind (2014) 123 (491): 829-849.
http://mind.oxfordjournals.org/content/123/491/829.abstract
Quotation and page citation taken from the copy provided by Beall at:
http://homepages.uconn.edu/~jcb02005/papers/inclosure-Mind-wrphc.pdf