Was the Liar solved?

I vaguely recall that somewhere Karl Popper expresses his suspicion that some philosophers are unwilling to accept a solution to a challenging problem because they have become so fond of the problem. I wonder if this applies to the Liar Paradox and a “simple solution” that Eugene Mills supports. As Mills sees it (and I’m effectively convinced) the Liar is not paradoxical, but plain false, and though it appears to truthfully say it is false, it does not truthfully say so. So it’s just false.

After a very brief introduction to the Liar and some attempted solutions that fail, I’ll summarize Mills’s solution. For more depth, please see the SEP and IEP entries.

Consider a typical expression of the Liar sentence.

L: This sentence is false.

The case for paradox often runs like this.

If L is true, then what it says is the case, and so it’s false. On the other hand, if L is false, then since L says that it is false, L is true.

We have an alleged paradox. I think the issue of whether we have a real paradox is important for reasons both the SEP and IEP entries mention.

Some thoughts come to the minds of people who like to avoid paradox. One attempted solution is to propose that the Liar has some third truth value which is neither True nor False; perhaps it is Meaningless. But while this avoids the alleged paradox for the Liar, it does not avoid it for another sentence:

U: This sentence is not true.

We call this the “revenge of the Liar”. (Homework: See for yourself how U “avenges” L, that is, how it engenders apparent paradox as L did.)

Another attempted solution is to reject self-reference by sentences. There are two problems with this proposal. First, it comes at the cost of rejecting sentences that seem clearly true:

This sentences has five words.

This sentence contains no semicolons.

This sentence is in English.

But second, and more importantly, this proposal does not solve the alleged paradox which arises from Liar Pairs, neither of which are (at least explicitly) self-referential.

A: B is true.

B: A is false.

(Again, homework for the reader.)

The Liar is old, and it has weathered numerous attempted solutions over the centuries. However, as mentioned, I think Mills has solved it. Here I will only summarized the argument in his paper, ‘A Simple Solution to the Liar’ (1998), which is here and here.

Mills begins by acknowledging that every proposition entails a proposition expressing its own truth. For instance, the proposition that grass is green entails the proposition that it is true that grass is green. And the proposition that polar bears are made of snow entails the proposition that it is true that polar bears are made of snow. In short, each proposition entails its own truth; thus, the proposition expressed by L entails its own truth. (This of course is not to say that the entailed propositions are therefore true. The proposition that it is true that polar bears are made of snow is not true, but false.)

However, a self-referential use of L also entails its own falsity simply because L in that case expresses the proposition that L is false. We have (not a paradox, but) a contradiction: L entails that it is both true and false.

But any proposition that entails a contradiction is just false, so L is false.

What remains to be shown is that though L is false, we cannot derive that it is true. After all, if we can derive L’s truth, then we are back to where we started, with an apparent paradox.

To show that we cannot derive that L is true, Mills relies on the notion that every proposition “attributes truth to itself.” (I think here is where critics may attack; however, I also suspect that Mills does not need such a strong claim.) L then does not express only the proposition that L is false, but the proposition that L is false and L is true. Now, of course, there is a sense in which L does “say” that L is false, and in fact according to Mills’s solution L really is false, but the key point is that L does not consistently say that L is false, because L expresses a contradiction. L no more truthfully asserts that it is false than this sentence expresses that Koko is a primate:

K: Koko both is and is not a primate.

Koko is a primate, and there is a sense in which K says so, but this much is insufficient for K’s truth. This difference between K and L is that K’s inconsistency is obvious whereas L’s is not. But both, I allege in agreement with Mills, are inconsistent and thus simply false. To paraphrase it all, I take Mill to have shown the following.

Because the Liar entails both that it is true and that it is false, it entails a contradiction and is thus false. But even though the Liar is false and appears to say just that, it does not consistently say just that, and therefore it does not truthfully say it. The Liar is thus false, and not true.

While Mills’s argument certainly relies on assumptions which some philosophers will challenge, I find that these assumptions are (my opinion, of course) fairly innocuous when compared to the more popular solutions: rejecting bivalence or the Law of Excluded Middle, holding that some contradictions are true, constructing “fixed point” systems, forbidding sentences about everything—to name a few. Moreover, it is encouraging that Mills’s “simple solution” survives both Revenge Cases and, as I argue in this old unpublished piece, Liar Pairs.



Mills, E. “A Simple Solution to the Liar” Philosophical Studies (1998) 89: 197. https://doi.org/10.1023/A:1004232928938