By JD Snodgrass

This sentence is false.

Is the above sentence true or false? If it’s true, then what it says of itself — that it’s false — is true. In other words, it’s false. So it can’t be true. But can it be false? If it’s false, then what it says of itself — that it’s false — is false. In other words, it’s true. So it can’t be false. AHHHHHH!!!!!!!

This is the infamous Liar Paradox. The paradox has a storied history and continues to fascinate philosophers, logicians, and mathematicians alike. It originated in a list of logic puzzles written by the Greek philosopher Eubulides of Miletus in the fourth century BCE. Eubulides wrote, “A man says that he is lying. Is what he says true or false?” This eventually evolved into the famous Liar Paradox we all know and love, “This sentence is a lie,” or more to the point, “This sentence is false.” The problem with the Liar should be obvious. If it’s true, then it’s false. But if it’s false, then it’s true. So what do we do? How do we solve the paradox? Can it be solved? And why the hell should we care?

Regarding the last question, we need to know which statements are true and which ones aren’t. Without a formal theory of truth, we’re at a loss to say, for example, whether a given mathematical theory is true or false, as we don’t know what “true” and “false” really mean. The Liar Paradox opens holes in our theory of truth that must be closed in order to know that the statements we make about mathematics — or, in fact, anything — hold.

At this point, you might be tempted to say that the Liar is neither true nor false. I’m not going to address that solution here. Instead, I’ll refer you to the Stanford Encyclopedia of Philosophy entry on the Liar Paradox for an explanation of why that solution is ultimately problematic. In short, you actually end up creating another paradox.

So if we can’t say it’s neither true nor false, then what do we do? Numerous solutions have been devised over the years. They all fall into two camps: those that retain classical logic and those that reject it. I’m only going to address one solution in this post, a very recent one that works within the constraints of classical logic.

A few years ago, philosopher Kevin Scharp developed a new theory of truth that solves not just the Liar Paradox but all so-called alethic, or truth-based, paradoxes. In short, Scharp claims that truth is an inconsistent concept. Most of the time, we don’t notice the inconsistency. It’s not until we start examining certain kinds of self-referential sentences like the Liar that we run into problems. His solution is to replace our concept of truth with a pair of concepts he calls “ascending truth” and “descending truth.” According to Scharp, replacing truth with these new concepts dissolves the paradox.

Background

Before jumping into Scharp’s theory, we need to talk about an important concept in theories of truth. In the 1930s, Polish mathematician and philosopher Alfred Tarski developed a theory of truth for formal languages (think mathematics and logic). In it, he claims that for a theory of truth to be adequate, it must entail all possible T-sentences, or sentences of the form

“P” is true if and only if P.

Using logical notation, we can restate this as

T(P) ↔ P.

This double-arrow signifies a biconditional. It combines the following two statements into one:

If P then T(P), or using logical notation, P → T(P), and

If T(P) then P, or using logical notation, T(P) → P.

In the first statement, you can infer from a sentence P that it’s true that P, or T(P). Put another way, you can “capture” P with the truth predicate T. In the second statement, you can infer the opposite, and thereby “release” P from the truth predicate T. For Tarski, a theory of truth must entail both capture and release.

In more familiar terms, an adequate theory of truth in English should entail the sentence

“Snow is white” is true if and only if snow is white.

This should make intuitive sense: a sentence is true if and only if the proposition it expresses is true. Likewise, you can turn it around and say that snow is white if and only if the sentence “Snow is white” is true. Most philosophers, logicians, and mathematicians agree with Tarski that an adequate theory of truth must entail all possible T-sentences.

We can rewrite the Liar according to the T-schema. When we do, we get the following problematic T-sentence:

“This sentence is false” is true if and only if this sentence (the one in quotation marks) is false.

The issue is that we don’t want our theory of truth to entail this T-sentence because that would mean our theory is incoherent. But how do we ensure our theory doesn’t entail paradoxical T-sentences? Scharp thinks he has an answer.

Replacing Truth

In his book Replacing Truth, Scharp claims that truth is an inconsistent concept and that this inconsistency is what causes problems like the Liar Paradox. Scharp claims that the T-schema T(P) ↔ P is really two separate concepts that we’ve improperly confused as one. He calls these two concepts “ascending truth” and “descending truth,” or AT and DT. They correspond to the capture and release concepts mentioned above: AT captures but doesn’t release, and DT releases but doesn’t capture. Put into logical form,

Ascending truth: P → A(P)

Descending truth: D(P) → P.

Scharp’s key insight was that these concepts cannot properly be combined into a biconditional. They are separate. He is denying the T-schema.

So how does this help with the Liar Paradox? Well, because truth is two concepts, the Liar sentence needs to be slightly modified. Let’s take a look at the Descending Liar, or D-Liar, a sentence P that says of itself that it’s not descending true: “This sentence is not D-true.” Is this sentence D-true or not D-true? If it’s D-true, then what it says of itself — that it’s not D-true — is D-true, and therefore it’s not D-true. Put another way, we’re just restating the release relationship D(P) → P; we’re moving from an assertion that a sentence is D-true to the sentence itself.

So far, the D-Liar is acting just like the traditional Liar. But, unlike the traditional Liar, our analysis ends here. You might be tempted to continue the analysis and say, “but if it’s not D-true, then what it says of itself — that it’s not D-true — is not D-true, and therefore it’s D-true.” Resist the temptation! Because of how DT works, you can’t make that move. Remember, DT only releases; it doesn’t capture. The move from P to D(P) doesn’t exist in our new scheme. That move is only possible for AT. So rather than being a paradox, the sentence is simply not D-true (it also works out as not D-true if you begin the entire analysis with the claim that it’s not D-true, as you can’t infer the sentence P from not-D(P)). It is, however, A-true, since the sentence P implies A(P). Separating truth into AT and DT prevents the paradox from arising in the first place. In the end, the Liar sentence is A-true but not D-true. Although not paradoxical, Scharp calls such sentences “unsafe.” Clever, right?

Scharp’s solution is incredibly powerful and convincing. However, it’s not without its problems. As philosopher David Ripley points out in his review of Replacing Truth, Scharp’s theory has some rather unpalatable limitations. First, we want a theory that allows us to use truth as a device of endorsement. We want to be able to assert that true things are true; we want to be able to assert both P and T(P). But Scharp’s theory makes this difficult. The strongest assertion we can make is that something is D-true, as the statement D(P) implies P itself. But, as Ripley points out by way of example, it’s impossible to endorse the contents of Scharp’s book without repeating all the sentences in it. If we say the book is D-true, then we’re actually disagreeing with it because it contains unsafe sentences like the Liar. But AT doesn’t work as a device of endorsement either. Claiming that P is A-true doesn’t give us P itself. The most we can say with AT is that if P is not A-true, then not-P. In other words, AT cannot serve to endorse. It can only serve to reject.

Second, Ripley claims that all this focus on truth is unwarranted. Scharp’s theory only helps us with alethic paradoxes and doesn’t apply to any other kind of paradox. Ripley points out that there’s one concept present in all paradoxes, and it’s not truth. Rather, it’s validity. Ripley thinks the concept of validity is the one worth pursuing as inconsistent.

Regardless of its limitations, Scharp’s theory is striking. And it upends a concept most of us never would have thought about as particularly problematic. It’s a fascinating addition to the literature on truth and a real sweet piece of brain candy.

JD Snodgrass is Philosopher-in-Chief of The Will to Power Hour. He is currently making great use of his philosophy degree as a professional contract negotiator. Who knows, maybe the whole philosophy thing will pan out and he can live out his days contributing to the literature on truth. But first, he’ll need to purchase a tweed jacket with leather elbow patches. He’ll also need to take up pipe smoking, although he could probably get away with one of those cool little plastic pipes that blows bubbles.