Resolving the Liar’s Paradox

The following is an excerpt from my book Square One: The Foundations of Knowledge.

The most popular paradox used when trying to demonstrate a “true contradiction” is the liar’s paradox. Philosophers have been writing about it for thousands of years. It’s supposed to be a proposition that is true and false at the same time, thereby demonstrating that the laws of logic are neither universal nor inescapable. If this is correct, then logic would not be a certain foundation for our worldview, and no such foundation could exist. The liar’s paradox can be formulated many ways. I will focus on one formulation, which I think ultimately resolves all the others. The paradox is:

(1) This sentence is false.

Think about it. Is that sentence true or false? If it’s true that “this sentence is false,” then it must actually be false. But if it’s false that “this sentence is false,” then it must actually be true. This presents us with a problem. If it’s true, it’s false, which means it’s true, which means it’s false, and so on. Thus, many philosophers have concluded that the liar’s paradox is both true and false at the same time – a true contradiction. Of course, given what we mean by “true” and “false,” we can know that the liar’s paradox must have a resolution. It’s not an empirical question. Nothing can be both true and false at the same time – such an idea doesn’t even make sense.

So here’s my preferred resolution to the paradox. The liar’s paradox is a linguistic error. It is not a meaningful proposition, though it appears to be at first glance. The problem is with the first two words: “this sentence.” “This sentence” is either impossible to define, or it’s impossible to evaluate as true or false. To illustrate, let’s begin by re-stating the paradox.

(1) This sentence is false.

One question gets at the heart of the issue: what exactly is false? In order to claim that something is false, we must know what we’re evaluating. We need to know the precise function of the words “this sentence.” There are two possible scenarios. “This sentence” could either be a reference to something, or it could be what’s being evaluated as true or false. Both options turn out to be linguistic errors. Let’s look at the latter first. If “this sentence” is what’s being evaluated as true or false, then we can quickly see the problem. “This sentence” simply isn’t a truth claim. There’s nothing to evaluate as true or false. It’s just two words put next to each other.

Imagine somebody came up to you and said only the words, “This sentence!” You wouldn’t respond, “That’s true!” That wouldn’t make sense. So in order to be sensible, “this sentence” must be referencing something. We again have two options. “This sentence” can either be referencing the entire liar’s paradox or only part of it. In other words, the claim is either:

(1) “This sentence is false” is false. Or,

(2) “This sentence” is false.

Both options fall apart under scrutiny. The second option runs into the error mentioned previously. If “this sentence” is only referencing the words “this sentence,” then it cannot be evaluated as true or false. “This sentence” is not a truth claim. Thus, the only attempt at creating a true contradiction must be formulating the liar’s paradox as such:

(1) “This sentence is false” is false.

But this is merely one step removed from the original problem. What is the sentence being evaluated as true or false? If we’re trying to evaluate the words in the quotation marks – “this sentence is false” – then we must ask the question, “What do the words ‘this sentence’ reference?” If “this sentence” references only “this sentence,” then as we established earlier, it’s not a truth claim. But if “this sentence” references “this sentence is false,” then the liar’s paradox is really claiming:

(1) “‘This sentence is false’ is false” is false.

And we’ve gotten no further. This can continue ad infinitum. Every time you ask, “What exactly does ‘this sentence’ reference?” you’re stuck with the impenetrable response “this sentence is false” or the non-evaluable “this sentence.” It’s like peeling back all the layers of an onion. Once you get to the true subject of the argument – something that isn’t referencing something else – you’re left only with the words “this sentence,” which isn’t anything meaningful to evaluate. Parentheses help illustrate more clearly. The liar’s paradox is saying:

(1) Proposition X is false.

What is proposition X? It’s “Proposition X is false.” Therefore, the liar’s paradox really formulated as:

(1) (Proposition X is false) is false.

Alright, it looks like we’re a step closer. Now we have to evaluate the proposition within the parenthesis. Within the parenthesis, it’s claiming “proposition X” is false. So what exactly is proposition X? It’s “Proposition X is false.” Therefore, the liar’s paradox is actually saying:

(1) [(Proposition X is false) is false] is false.

And again, we’re no closer to finding a concrete proposition to evaluate. When “proposition X” references “proposition X is false,” we’re stuck generating an infinite regress. It continues:

(1) {[(Proposition X is false) is false] is false} is false…

And so on. This is not a paradox. It isn’t a true contradiction. It’s simply a linguistic error. Some people will object to this resolution by claiming that we need to reformulate the paradox another way. Instead of saying, “This sentence is false,” they will try:

(1) The following sentence is true.

(2) The previous sentence is false.

If you understand the laws of logic, you can know that this formulation must also fail. In this case, both sentence 1 and 2 fall into the same error as the original formulation. The same question illustrates it: what sentence exactly is true or false? Examine the phrases “the following sentence” and “the previous sentence.” Those phrases are either references to something, or they are what’s being evaluated as true or false. If they are what’s being evaluated, then it’s clear they can’t be true or false – “the following sentence” is as meaningful a proposition as “this sentence.” It’s just three words put together. It cannot be true or false. But, if “the following sentence” is a reference to something else, we run into the same infinite regress problem. Parentheses help illustrate:

(1) (The following sentence) is true.

(2) (The previous sentence) is false.

If “the following sentence” is a reference to something, then we could rephrase proposition 1 into:

(1) [(The previous sentence) is false] is true.

Now we must analyze the proposition within the parenthesis. What is “the previous sentence”? It’s “The following sentence is true.” So, the claim is actually:

(1) {[(The following sentence) is true] is false} is true…

And so on. Again, we’ve gotten no closer to a proposition to evaluate as true or false. If “the following sentence” and “the previous sentence” are references, then there will never be a truth claim being made. They are simply two phrases pointing to each other. There are other ways to formulate the liar’s paradox, and they all follow the same pattern. They appear to be sensible at first glance, but once you unpack the meaning of the terms, they are revealed to be linguistic errors.