The liar paradox is the statement:





“This sentence is false.”



Let us refer to this sentence simply as L. The paradox comes when we try to assign a truth value to L. Is it true or is it false? If we declare it to be true, then what it says must actually be the case. But it says that it is false. We would thus have a sentence that is both true and false, which is not possible. On the other hand, if we declare it to be false then we are affirming precisely what the sentence says. That makes it true! So, once again, we would have a sentence that is both true and false. We get a contradiction either way.

Perhaps, though, we are making a category error. The concepts of truth and falsity apply to propositions, not just arbitrary sentences. Sentence L is not a proposition, we might argue. We should not be trying to assign it a truth value in the first place. It is neither true nor false.

Well, then. What are we to make of this:





“This sentence is either false or neither true nor false.”



Let us call this sentence M. Are you inclined to say that M is neither true nor false? Then you have just affirmed that what it asserts is actually the case, which makes it true. (Recall that an “or” statement is true if either of its parts is true.) Could the sentence be true? Certainly not, since in that case neither part of the or statement is correct. And could the sentence be false? Again, no, since in that case the first part of the or statement is true, and we once again have arrived at a contradiction.

But maybe you're still inclined to say that M just flat is not a proposition. Very well. Then try this one:





This sentence is not a true proposition.



Now what are we going to do? Let us refer to this as N. If you say that N is not a proposition at all, then it certainly is not a true proposition. So, by declaring N to be something other than a proposition, then you have just made it true. But if you say that it is true, then it must be a true proposition (since propositions are the only things with truth values), and you have just made the statement false. I'll let you work out for yourself the badness that ensues if you declare N to be false.

So how do we get out of this mess? How you answer that depends in part on whether you are a mathematician or a philosopher. Mathematicians just don't worry about it. For us, this sort of thing is comparable to the old joke about the man who approaches a doctor and says, “ My arm hurts when I move it like this. What should I do?” To which the doctor replies, “Don't move it like that.” Likewise, if you get into trouble applying concepts of truth and falsity to these weird, self-referential sentences, then don't apply concepts of truth and falsity to them. Problem solved!

Philosophers have different concerns. For them, these sorts of sentences are at least potentially telling us something important about the nature of truth. There's rather a large literature investigating this topic, much of it dense and technical. An especially audacious suggestion is simply to bite the bullet. If our attempt to assign a truth value to sentence L leads inevitably to the conclusion that it is simultaneously true and false, then maybe we should just accept that some sentences can be both true and false. Law of non-contradiction be damned! This view, which can also be expressed by saying that there are true contradictions, is known as dialetheism. It's one of those ideas that sounds so crazy that there must be something to it for clever people to be defending it. I've been reading about it lately, and, well, I still think it's kind of crazy. But interesting!

There's a parallel here with set theory. We have certain intuitive notions about collections of objects, which we call sets. But we run into trouble if we get too clever about defining our collections. Russell's paradox famously asks us to consider the set of all sets that are not elements themselves. If we call this set S, then there is no consistent way to answer the question, “Is S an element of itself?”

One way of dealing with this is to set-up elaborate rules regarding what is, and what is not, a legitimate set. This is the way of axiomatic set theory, and it is a long-standing and honored branch of mathematics. The fact is, however, that most mathematicians pay no attention to it most of the time. We prefer “naive set theory”. Set theory is just a useful formalism. If you get into trouble when you devise super-clever sets then stop being clever! Why should Russell spoil the fun for the rest of us?

So it is with logic. For most mathematicians, logic is just a tool. The principles of logic are the rules of the game. We accept those rules because we find it useful to do so. When it comes to various systems of logic, we don't think in terms of correct or incorrect. Instead, we think in terms of useful or not useful.

Philosophers are more interested in using formal languages to capture the way people use natural language. Systems of logic can be correct or incorrect, depending on how successfully they capture the way people reason and use language. Thus, classical logic, which is what most people think of when they think of logic, is just one more theory among many. For example, here's philosopher Graham Priest, an especially prominent defender of dialtheism, in his book Beyond the Limits of Thought:

That a contradiction might be true, or that dialetheism (the view that there are true contradictions) makes sense, may still be abhorrent, and even threatening, to many contemporary English-speaking philosophers. More likely than not, even the suggestion of it will be met with a look of blank incomprehension. How could a contradiction be true? After all, orthodox logic assures us that for every statement, S, either S or not S is true. The simple answer is that orthodox logic, however well entrenched, is just a theory of how logical particles, like negation, work; and there is no a priori guarantee that it is correct.

So that's it. I have no particular point to make, but this is the sort of thing I've been reading about, during all the time I've been spending not blogging!