The liar’s paradox

There are a number of logical paradoxes that philosophers don’t quite know what to do with. The most famous is the liar’s paradox:

This sentence is false.

If it’s true, then it is false. If it’s false then it is true.

So which is it?

The typical answer is to claim that the sentence is not a genuine statement; that is, that it is neither true nor false. Indeed, we can prove that it is not true or false by simply considering each the two cases above.

There are other sentences that do not have truth values: “Shut the door.” and “How are you today?”, for instance. Perhaps the liar’s paradox is just an odd example of something similar.

One problem with this is that the liar’s sentence certainly seems to have a truth value. If sentences that appear to be statements are not, then what prevents us from discovering one day that some other thing we thought was a statement isn’t one after all? It’d be nice to be sure that what we have been claiming over the years is at least coherent!

Perhaps the reason the liar’s sentence is not really a meaningful statement has something to do something with its self-reference. However,

This sentence has five words.

and

This sentence is in English.

seem unproblematic. What justifies our accepting them as meaningful statements and rejecting the liar’s sentence?

To put the issue more pointedly, consider the related sentence known as the liar’s revenge:

This sentence is not a true statement.

If this is a true statement, then it is not, but if it is not a true statement, whether that is because it is false or because it is not a statement at all, then it certainly speaks the truth about itself, and so is clearly a true statement after all.

I have lots of thoughts about the liar’s paradox, but one of them is that most analytic philosophers should take it a lot more seriously than they do. They seem to regard it as an oddity which can be safely ignored. It’s “just playing with words”, and has no real import.

I think the man or woman on the street is justified in thinking this way, but not analytic philosophers, because “just playing with words” is what analytic philosophy is all about. The analytic philosophy game is to translate subtle ideas into precise statements and then manipulate those statements logically to see what else we get as a result. The liar’s paradox calls the usefulness of that procedure into question.

Personally, I think the liar’s paradox runs deep. I think that any time we translate ideas of any kind into precise statements, we introduce a gap in translation between the idea we are trying to articulate and the actual thing we ended up saying. We never directly prove anything about the ideas we started with, we just prove something about the statements we used to articulate those ideas. The liar’s paradox, in my opinion, is a way to exploit that gap, to show that what we say and what we mean by what we say are never completely the same. (That includes, of course, this post.)

What we say and what we mean by what we say are never completely the same. Continental philosophers seem to realize this. Analytic philosophers, as far as I can tell, conveniently ignore it, but they shouldn’t.

One thought on “The liar’s paradox

  1. “This statement is false” is true if and only if it is false.

    But there can exist no well-defined statement that is true if and only if it is false. The proof is simple, and very similar to the reasoning behind the resolution of Russell’s Paradox:

    Suppose to the contrary that x is a well-defined statement, and that x is true if and only if x is false. Now, the last part is nothing more than a simple self-contradiction, so the original premise must be false. Generalizing on x, we conclude as required.

    In short, “This statement is false” cannot be a well-defined statement — it is pure nonsense!

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