Usually, when we say how things are, we’re talking about non-linguistic bits of the world. “Gemini is a dog” is a sentence about a particular dog; so is “Gemini wants cheese.” Sometimes, though, we do talk about pieces of language. “Why did you say that?” asks for the reasons (psychological or otherwise) for someone else’s assertion. A teacher might write it in the margins of an essay in red ink.
Consider the Liar sentence (which we’ll call “L”):
L is not true.
L is about a piece of language — in fact, it’s about L itself — and it says of itself that it’s not true. We normally don’t say sentences that are about themselves, but it’s not ungrammatical. Here’s another sentence we’ll call “E”:
E is a sentence in English.
No problem. E is a well-formed sentence in English, and because E accurately describes itself, it is true.
L is an odd duck because it says that it’s not true. If L accurately describes itself, then L has to be true. But then, if L is true, then what L says of itself (that it isn’t true) must hold true. So if L is true, then L is not true. But if L is not true, then it is. L is true if and only if it isn’t.
The unhappy implication is that L must be true and not true, and this goes against the presumption against contradiction in classical logic. For that reason, a lot of philosophers have thought that L must be defective somehow. Even though it breaks no obvious logical rules, it must be violating some logical principle somewhere. Otherwise, we have to live with L, countenance contradictions, and give up some principle or other in classical logic. In any event, without some rational explanation to the Liar, we might end up with trivial languages that say, of any sentence you like, that it’s true, since in classical logic, contradictions entail any sentence you like.
Another possibility, though, is that paradoxes like the Liar are semantic or logical illusions. Perceptual illusions are anomalies of perception. In the Muller-Lyer illusion, for example, we see two lines of equal length as unequal.
The standard explanation of this is that we interpret the angles at the ends of the lines as corners, which cue our capacities for making guesses about distance and depth. Some angles make their conjoining line segment look farther away.
For simplicity, let’s suppose that, when we look at a line, our brains get to work on processing the retinal image by sending it to two teams of neurons. One team only works on answering the question, “How long is the line?” The other team devotes itself entirely to “How far away is it?” The teams submit their respective results, and what we’re aware of in conscious experience is the interpretation of both teams’ work. The answer is that the two lines are unequal, and it’s hard (for most us anyway) to undo this result, even after confirming with a ruler or common measure that the lines, minus the angles at the end, are actually the same length.
In this simplified model, neither of our unconscious teams have to make a mistake. They’re both using time-tested methods for completing their work. But when their work is combined to work out this particular problem (“Are these two lines equal?”), they get the wrong answer. The result is an illusion.
Arguably, some illusions are purely cognitive, in the sense that they produce illusory effects without sensation or perception. The conjunction fallacy (a.k.a. the Linda problem) could fit this category. Two processes get to work on the same problem and throw out different answers, without either process making a mistake.
Illusions are really interesting. They seem to show that we don’t come to grips with the world as it really is, but in a way that’s good enough, though not good enough to prevent anomalous results. Despite their mind-bending effects, we tend not to despair too much. Things aren’t always what they seem, and that’s okay.
Philosophers have worried about the Liar and similar semantic paradoxes for centuries, but what if the Liar is an illusion produced by the way we process sentences? Considering sentence L, one team in our brains asks, “What does this sentence say?” Another asks, “What formally follows from this sentence?” Combining the results of each team’s work yields a paradoxical conclusion: L is true and not true, a real contradiction. But instead of patching up logic, we might just set the result aside. Semantic and logical illusions are fascinating objects of study, but they need not be existential problems for language, in the same way that perceptual illusions do not show that everything is a hallucination. What, after all, is a language? Is it a system that gives every possible sentence one and only one truth value? Or is it a jumble of processes that are good enough for their typical application?