Resolving the Liar’s Paradox

Perhaps the most famous and persistent paradox of all time is the “Liar’s Paradox”. Whenever somebody tries to prove paradoxes are real, it’s almost inevitable that the following sentence will appear:

“This sentence is false.”

Sounds simple, but think about. Is this proposition true or false?

If “This sentence is false” is true, then the sentence must be false, because the sentence is claiming it’s false.

If “This sentence is false” is false, then it must be true, because the proposition is claiming “this sentence is false” is false.  But, then again, if it’s actually true, then it must be false… which would mean it’s actually true.

You get the point.

This has perplexed many philosophers through the years, many of whom have concluded that paradoxes must exist in reality, because that sentence is both true and not-true at the same time. I am not sympathetic to this view.

The paradox can actually be resolved rather simply, without needing to throw out classical logic. I will elaborate on this point in the future, but suffice to say: it shouldn’t be surprising that the Liar’s Paradox can be resolved because all paradoxes, by their nature, can be resolved. They don’t exist because they can’t exist.

So, without further ado, here’s the resolution: “This sentence is false” is not a meaningful proposition. The terms don’t actually reference anything, though they appear to at first glance.

To evaluate a proposition, the words we use need to have definite meaning. In this case, the error is with the first two words: “this sentence”. As I’ll show in a minute, “this sentence” is impossible to define.

Which Sentence, Exactly?

Let’s begin with a restatement of the proposition:

“This sentence is false.”

The next step is to ask, “Exactly what sentence is false?” Is the reference to the entire sentence, or just part of it? In other words, is the claim:

“This sentence is false” is false?

Or only,

“This sentence is false.”

Let’s examine the latter. If the full proposition is “This sentence is false”, then we can quickly see the proposition is actually nonsense. Concretely speaking, it’s evaluating the words “this sentence”, and concluding that it’s false. But that doesn’t make any sense! The sentence “this sentence” is not a meaningful proposition, so it literally can’t be true or false. It’s like saying the words “this cat” or “this finger” is true or false.  It’s a non-sequitur.

“This sentence” is just two words placed next to each other, not a proposition to evaluate as true or false.

So, that leaves us with the first option. Perhaps what’s actually being claimed is:

“This sentence is false” is false.

But we immediately run into the same problem. Is this proposition referencing the entire sentence as false, or just part of it? If it’s referencing the entire sentence, then the full proposition is actually:

”’This sentence is false’ is false” is false.

And we’ve gotten no further. This can continue ad infinitum, without ever reaching a concrete proposition to evaluate as true or false. So, if we’re going to make any meaningful analysis, we must interpret the full Liar’s Paradox as:

“This sentence is false” is false.

But this merely one step removed from our first problem: “This sentence is false” cannot be evaluated as true or false, because it’s not a meaningful proposition! It’s like trying to evaluate whether or not “This haircut is false” is true or false.

It’s like peeling back all the layers of an onion. When you get to the very “core” of the argument – what is actually being claimed – you’re only left with the words “this sentence”, which isn’t anything meaningful to evaluate.

Parentheses Show The Error

 We can use parentheses to highlight the problem more clearly. Let’s reformulate the paradox as:

“Proposition X is false.”

And we define:

Proposition X = (Proposition X is false).

Then it’s a simple matter of substituting Proposition X for itself.

(Proposition X is false) means that:

[(Proposition X is false) is false].

And:

{[(Proposition X is false) is false] is false}.

And:

|{[(Proposition X is false) is false] is false} is false|.

Thus, it becomes obvious: there is no concrete definition for “Proposition X”, and there can literally never be. The only way for “Proposition X” to have a concrete meaning is if it already has a concrete meaning. Its definition depends on the definition of itself, which is never defined – which means it will never be defined.

In other words, the Liar’s Paradox is a syntax error – albeit a tricky one that isn’t immediately self-evident. This is the nature of paradoxes in general. They give the illusion of sensibility, while containing subtle problems that have to be sorted out.

Imagine a mathematician shows you an extremely complex formula proving that 2 = 3. He thinks he’s proven a paradox – that logic isn’t absolutely true in all cases. Then, he goes and develops his entire worldview based on the premise that paradoxes are possible. But upon inspection, you examine his formula and discover he divided by zero. Everything else about his argument was sound, but buried in a little equation was an accidental syntax error – he incorporated a little bit of nonsense into his proof.

This is what happens with philosophic paradoxes and with the Liar’s Paradox in particular. If you don’t examine it carefully, you might end up thinking that a “real paradox” is possible. Well, I can assure you that whenever you encounter a logical contradiction, somebody has accidentally divided by zero. You just have to sit down and carefully sort it out.

There are other resolutions to the Liar’s Paradox, a few of which are quite clever, but I’ll cover those in the future. I don’t find them nearly as compelling as what is outlined above.

As is true with every other paradox, the appropriate response to someone claiming “This sentence is false” is quite simple:

“Nonsense.”