by Tim Sommers
There’s something wrong with the sentence, “This sentence is false.” Is it true or false? Well, if it’s true, then it’s false. But then if it’s false, it’s true. And so on. This is the simplest, most straightforward version of the “Liar’s Paradox”. It’s at least two thousand five hundred years old and well-known enough that you can buy the t-shirt on Amazon.com.
I’ve been thinking about the “Liar’s Paradox” lately, because I’m teaching an “Introduction to Philosophy” class on paradoxes (and writing a book) called “Life’s a Puzzle: Philosophy’s Greatest Paradoxes, Thought-Experiments, Counter-Intuitive Arguments, and Counter-Examples from AI to Zeno”. It starts with the “Liar’s Paradox” because it’s one of the oldest and most well-known, but also simplest and most daunting, of philosophical paradoxes. Some people think that while “puzzle” cases in philosophy are fun and showy, they are not where the real action is. I think every real philosophical puzzle is a window onto a mystery. And proposed solutions to that mystery are samples of the variety and possibilities of philosophy.
So, let’s start with this. Why is it called the “Liar’s Paradox”? Let’s go to the Christian Bible for that one, specifically, “St. Paul’s Letter to Titus” (Ch. 1, verses 12-14)
“They must be silenced, because they are disrupting whole households by teaching things they ought not to teach – and that for the sake of filthy lucre.12 One of Crete’s own prophets has said it: ‘Cretans are always liars, evil brutes, lazy gluttons.’13 This statement is true.14”
Verse 12 has philosophers dead to rights. We are disrupting whole households, teaching things we ought not to teach and – speaking for myself at least – it’s all about the filthy lucre (hence, the book). But verse 13 is what we want here. It has “Cretan’s own prophet” saying “Cretans are always liars.” Now, if that just means that all Cretans lie a lot, but not all the time, there’s no problem. But if it means that Cretans are always lying whenever they speak, given that this is asserted by a Cretan (read: liar), we have a paradox. This then is the primordial, liar’s version of the “Liar’s Paradox”. If that’s unclear you can simplify the liar’s version down to: “I am lying right now.”
By the way, “Crete’s own prophet” was “Epimenides of Crete”. Crete is the largest of the Greek islands, only 99 miles from the mainland, but (at that time) culturally distinct. It was the home of the Minoans. You might think you’ve never heard of the Minoans, but you have. They built a very famous labyrinth. Anyway, Epimenides was a semi-mythical figure. For example, he supposedly fell asleep while tending his father’s sheep and slept for 57 years, then awoke with the gift of prophecy. Imagine sleeping for 57 years. What I wonder is why no one tried to wake him up. The first year or two, okay, he’s really tired, but after that? I had a roommate that didn’t wake me up one time, even though he knew I was late for work, and I was mad at him for like two years. So…
What should we say about, “This sentence is false”? What if we said not every sentence has to be true or false and leave it at that? But “This sentence is not true” works just as well. Or rather works just as paradoxically. It’s true, if it’s not true. If it’s not true, it’s true. Even if we say not every sentence has to be true or false, we can’t have sentences that are both true and not true. That’s a straight-up contradiction.
What’s wrong with a contradiction? Well, for one thing, there’s “the principle of explosion”. Sounds bad, right? It is. We can’t have explosions in the classroom. If you accept a contradiction as true, you can prove that everything is true. Everything. That view is called “trivialism” and it’s bound to lead to chaos. If you are really curious about how that works, I’ll show you at the very end exactly how a contradiction allows you to prove anything. It’s not that complicated.
There’s also something called the Prior solution to the “Liar’s Paradox”, not because it’s prior to anything, but just because that’s the guy’s name, Arthur Prior. Prior says every sentence already implicitly implies its own truth. So, “This is fun”, really says “This sentence is true and this is fun.” Apply that to “This sentence is false” and you get “This sentence is true and this sentence is false”, which asserts a contradiction and so is just false now and not paradoxical. Voila! Problem solved.
Except, did you ever see the episode of “Rick and Morty” where Morty said, “What Rick just said is false”, and Rick said, “What Morty just said is true”? That’s the two-sentence version of the “Liar’s Paradox”. To be clear: (1) The next sentence is false. (2) The previous sentence is true. (There’s a t-shirt for that, too.) Prior’s solution can’t help us here. (1) This sentence is true and the next sentence is false. (2) This sentence is true and the previous sentence is true. The paradox does not go away.
Maybe, what’s wrong with “This sentence is false” is that it’s self-referential. But, in general, it’s not a problem if a sentence is self-referential. Consider, “This sentence is six words long”. That’s just true. No problem. Or “This sentence is seven words long”. That’s just false. No problem. But “This sentence is false” is not just self-referential, it self-referentially assigns itself a truth value. So, let’s take a step back.
You have to admit, our ordinary, natural language is a mess. It’s imprecise and ambiguous and, most importantly, as any computer programmer will tell you, natural languages are over-flexible. They allow sentences to do things like self-referentially assign their own truth value. Computer languages can be thought of as artificial languages that replace natural languages in some contexts. Some universities now have departments of “logic and computation” that are no longer part of the philosophy department. This kind of thing happens all the time historically, by the way – some part of philosophy morphs into a science. Four hundred years ago physics was “natural philosophy” and now they expect their own offices.
Anyway, in such an artificial language you might think there should be a kind of hierarchy of types of sentences. So, in computing, or in Tarski’s logic, sentences can only assign truth values to sentences that are lower in the hierarchy. They are forbidden to assign truth-value to themselves or any of the sentences that outrank them.
“This sentence is false” is just nonsense, then, because it violates this rule. On this way of looking at it, the “Liar’s’ Paradox” is evidence that you need to enforce such a rule and/or develop an artificial language – or you get insoluble paradoxes.
This may seem like a weird solution, but a tremendous amount of philosophy in the twentieth century revolved around fixing up language in one way or another (Frege, Russel, early Wittgenstein, Tarski). The thought behind the “ideal language” version of “analytic” philosophy was that all, or at least most, philosophical problems could be solved by fixing up language. Wittgenstein, for example, in the “Preface” to his “Tractatus Logico-Philosophicus”, says “the problems of philosophy” are “misunderstanding[s] of the logic of our language”. (He also says, with a hubris that might have made Plato or Kant blush, that “the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems [of philosophy] have in essentials been finally solved.”)
Of course, we don’t really need to construct a whole new, artificial and perfect language to take action here. We could just stipulate that “This sentence is false” is false. To paraphrase Humpty Dumpty, we don’t work for our words, they work for us.
But isn’t this a dodge? We’re just stipulating the problem away. But if this is a dodge, isn’t saying you are going to stipulate the problem away as part of building a shiny, new logical language, just a more elaborate dodge?
Another way to go is to embrace contradictions. There are philosophers who defend “dialetheism” – the view that there are true contradictions. Maybe, “This sentence is false” is one. Dialetheists try to construct “paraconsistent” logics, embracing contradictions while trying to avoid the principle of explosion. That seems like a long way to go to avoid dealing with four-word sentence. Maybe this approach, like the ideal language approach, is just a way of changing the subject. I mean, sure, if contradictions are not a problem, then this contradiction is not a problem.
But wait, is “This sentence is false” a contradiction? Unless you Prioritize it (“This sentence is true and this sentence is false”), it’s not a contradiction. In fact, we could evade the Prior solution with the two-sentence version, because it’s not a contradiction. It’s a paradox.
So, how can we resolve the “Liar’s Paradox”. Beats me. I am hoping you will leave the solution in the comments section so I can go tell my class.
In the meantime, here’s why assuming a contradiction implies that everything is true.
Assume that both X and not-X are true. There’s your contradiction.
Since X is true, we can put it into a true disjunction with any sentence whatever.
So, X or Y. [“Or” just means “at least one of these is true”.]
But we also assumed not-X,
If X or Y and not-X, then Y follows.
But Y, remember, is any sentence at all. In the whole world. Ever. So, one at a time, we can go on to prove that all sentences are true.
Finally, a bonus. Here’s a riddle for you.
You are trying to get to a town called “Truth”. It’s called Truth because everybody who lives there always tells the Truth. You come to a fork in the road.One fork leads to Truth, the other fork leads to Lie – a town where everyone always lies. You have no idea which is which and no idea where to go.
Fortunately, there’s person sitting smack dab in the middle-of-the-fork. You decide to ask them for directions.
“Are you from around here?” you ask.
“Yes,” she answers, “I am from the Two Towns.” This what they call Truth and Lie around here since they are such close neighbors. Then she says, “I’ll answer one question and no more, then you’re off. Get it?”
“Sure,” you say. Suddenly, it hits you. She’s either from Truth or Lie, but you don’t know which one. If you ask her for directions, you won’t know if she is being truthful or not. Oh, no!
What’s the one question you can ask that will surely lead you down the right path and, so, to Truth?