Recently, I had a conversation with my friends Matt Dillahunty and Ozy about philosophy. At about the 1:25:00 mark (the link above should be timestamped), we started talking about how there may be considerations which lead philosophers to rationally question the basic ‘laws of logic’, such as the law of non-contradiction (for all p: ~(p & ~p)) and the law of excluded middle (for all p: (p ∨ ~p)). I brought up the liar paradox, as an example of this sort of thing. Matt objected that it is actually an instance of a ‘gappy’ sentence, which is neither true nor false. At the time, I knew there was a phenomena called ‘revenge’ which poses big problems for this strategy, but annoyingly I couldn’t bring the details to the bit of my brain that makes my mouth work. Here I want to right that wrong.
- The Liar
The liar sentence is of the following form:
a) This sentence is false.
The issue with a) is that it leads to a contradiction.
We only have to assume what seems like a very natural assumption about how the word ‘true’ works to get there. This is that if p is true, then p. We can think of this principle like this; if I say that it is daytime, and if what I say is true, then it is daytime. Alternatively; if I say a declarative sentence, and if it is true, then what it says correctly describes the thing that the sentence is about. This seems to be at the very core of idea of truth.
A corresponding idea is there for ‘false’ as well; if p is false, then it is not the case that p. If I say that it is daytime, and if what I say is false, then it is not daytime. If I make a declarative sentence, and it is false, then it incorrectly describes the thing that the sentence is about.
So let’s apply these principles to a):
If a) is true, then a) correctly describes what it is about. But a) is about itself, and it says about itself that it is false. So if it is true, then it correctly describes itself as false. So if it is true, it is false. And that is a contradiction.
So maybe a) is false. And if a) is false, then it incorrectly describes itself; yet what it says about itself is that it is false. If its self description is incorrect, then it isn’t false; and the only other option is that it is true. So if it is false, then it is true. Contradiction again.
So if it is true, it’s false; but if it is false, it’s true. Either way you go, you run into a contradiction. This is the paradox.
Yet, maybe there is a solution here. Matt certainly proposed a solution here. His idea was that a) is neither true nor false. So, let’s run through the options and see how it works.
a) says about itself that it is false. And we are now saying that it has no truth-value at all. Well, it certainly doesn’t correctly describe itself, because it says that it is false, and it is ex hypothesi neither true nor false. If something is neither true nor false, then it is not false. So it’s own self-description fails. This seems to leave no reason to consider it true. It says about itself that it is false, but we cannot derive that it is true. So far, no contradiction.
But, it says about itself that it is false, and this is incorrect (because being neither true nor false, it is not false). And it’s hard to see why this wouldn’t count as a case of a falsity. After all, it says that it is false, yet (ex hypothesi) it isn’t (because it’s gappy). It is certainly not true that it is false; it’s own self-description fails. But does this mean that it is false? Well, only if ‘not true’ means false. And, on this assumption, where we have some sentences which are ‘gappy’ (i.e. neither true nor false), there is a difference between being not-true and being false. If we listed all the not-true sentences, it would include all the ones which had no truth-value, and all the ones which were false. Thus, being not-true does not entail being false. Thus, we seem to have got out of the trap.
It is neither true nor false, and when it says about itself that it is false we can consider it’s incorrect self-description to be a case of being not-true, rather than false.
Strictly speaking, this does work as a consistent (i.e. contradiction-free) way to think about a).
So far, so good. However, things are not over. There is a second round.
Consider the ‘strengthened liar’ sentence:
b) This sentence is not true.
We have, on our assumption of ‘gappyness’, three options. Either b) is true, or it is false, or it is neither true nor false. Let’s take them one at a time:
If b) is true, then it correctly describes itself. Yet it says about itself that it is not true. So if it is true, it is not true. This is a contradiction.
If b) is false, then what it says about itself is incorrect. Yet, if it is false, then it does come under the category of not-true, which is what it says about itself. So if it is false, then what it says about itself is correct, and so it is true. And we have another contradiction.
The only other option is the one we used for a), which is that it is neither true nor false. Yet, if b) is neither true nor false, then it is in the not-true category as well (because anything which is neither true nor false is not true). But, as it says about itself that it is not true, it would seem like it has correctly described itself. If it has correctly described itself, then it is not in the not-true category, but in the true category. So if b) is neither true nor false, then it is true! This is, again, a contradiction.
So, while the gappy proposal got rid of one liar sentence (i.e. a)), it fails to help us with another one (i.e. b)). As a strategy, gappiness won a battle, but it loses the war.
The problem that the liar paradox presents is subtle, and still an open question in philosophy and logic. It may be that a solution to the generalised problem exists which involves adopting a logic which has truth-value gaps. That may be the case for all I know. But it seems clear that simply adopting truth-value gaps does not solve the underlying phenomenon. It merely pushes the problem to somewhere else. Even if a) can be got around by postulating truth-value gaps, b) cannot be. The liar paradox has had its revenge.
As the philosopher Tyler Burge put it:
“Any approach that suppresses the liar-like reasoning in one guise or terminology only to have it emerge in another must be seen as not casting its net wide enough to capture the protean phenomenon of semantical paradox.” (Tyler Burge, Semantical Paradoxes, p. 173, (1979))