p. 315. Self‐Reference: What is this Chapter About?
- Graham Priest
Sometimes things seem easy when we consider normal cases. This can be deceptive. If we consider more unusual cases, the simplicity can disappear. This is what happens with self-reference. ‘Self-Reference: What is this Chapter About?’ examines the concept of self-reference in a number of examples such as: ‘This very sentence that I am now uttering is false’. It is possible for a name to refer to something of which it, itself, is part. Some of the most ancient paradoxes fall into this area of logic, such as the liar paradox. They challenge the assumption that every sentence is either true or false, but not both. What does this tell us about validity? Sentences can be true, false, both, but never neither.
Often, things seem simple when one thinks about normal cases; but this can be deceptive. When one considers more unusual cases, the simplicity may well disappear. So it is with reference. We saw in the last chapter that things are not as straightforward as one might have supposed, once one takes into account the fact that some names may not refer to anything. Further complexities arise when we consider another kind of unusual case: self‐reference.
It is quite possible for a name to refer to something of which it, itself, is part. 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. Similar things happen in a set of regulations which contain the clause ‘These regulations may be revised by a majority decision of the Department of Philosophy’, or by a person who thinks ‘If I am thinking this thought, then I must be conscious’.
These are all relatively unproblematic cases of self‐reference. There are other cases which are quite different. For example, suppose someone says:
This very sentence that I am now uttering is false.
p. 32Call this sentence λ. Is λ 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. The sentence is like a Möbius strip, a topological configuration where, because of a twist, the inside is the outside, and the outside is the inside: truth is falsity, and falsity is truth.
Or 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.
These paradoxes are very ancient. The first of them appears to have been discovered by the ancient Greek philosopher Eubulides, and is often called the liar paradox. There are many more, and more recent, paradoxes of the same kind, some of which play a crucial role in central parts of mathematical reasoning. Here is another example. 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. 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. 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. And some sets are certainly not members of themselves: the setp. 33↵
p. 34of all people is not a person, and so not a member of the set of all people.
Now, consider the set of all those sets that are not members of themselves. Call this R. Is R a member of itself, or is it not? 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.
This paradox was discovered by Bertrand Russell, whom we met in the last chapter, and so is called Russell's paradox. 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.
What examples of this kind do, is challenge the assumption 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.
How can this idea be accommodated? Simply by taking these other 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. Recall from Chapter 2 that the truth conditions for negation, conjunction and disjunction are the following. 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 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.
p. 35a∨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 a and b have the value F.
Using this information, it is easy to work out the truth values of sentences under the new regime. For example:
Suppose that a is F but not T. Then, since a is F, ¬a is T (by the first clause for negation). And since a is not T, ¬a is not F (by the second clause for negation). Hence, ¬a is T but not F.
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). 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.
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). 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.
What does this tell us about validity? A valid argument is still one where there is no situation where the premisses are true, and the conclusion is not true. And a situation is still something that gives a truth value to each relevant sentence. Only now, the situation may give a sentence one truth value, two, or none. So consider the inference q/q∨p. In any situation where q has the value T, the conditions for ∨ assure us that q∨p also 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.
At this point, it is worth returning to the inference with which we started in Chapter 2: q, ¬q/p. As we saw in that chapter, given the assumptions made there, this inference is valid. But given the new assumptions, things are different. To see why, just take a situation where q has the values T and F, but p has just the value F. Since q is both p. 36↵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.
That's not the end of the matter, though. As we saw in Chapter 2, this inference follows from two other inferences. The first of these (q/q∨p) we have just seen to be valid on the present account. The other must therefore be invalid; and so it is. The other inference is:
Now consider a situation where q gets the values T and F, and p gets just the value F. It is easy enough to check 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.
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.
Which explanation of the situation is better, the one that we ended up with in Chapter 2, or the one we now have? That is a question which I will leave you to think about. Let us end, instead, by noting that, as always, one may challenge some of the ideas on which the new account rests. 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. p. 37↵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. 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.
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.