Sentences That Eat Themselves

This sentence is false. If it is true, then what it says holds, so it is false. If it is false, then what it says fails, so it is true. There is no third option available and no assignment that works.

It is tempting to treat this as a word game. It is a proof. Given a sentence that ascribes falsity to itself, plus the assumption that every sentence is either true or false and not both, you derive a contradiction by valid steps. Since the steps are valid, at least one of the assumptions has to go. Every candidate for removal is expensive, which is why the liar has resisted a settled solution for two and a half thousand years rather than for want of clever people looking at it.

Alfred Tarski's response was to deny that a consistent language can contain its own truth predicate. On his picture there is an object language that talks about the world, a metalanguage that talks about truth in the object language, a meta-metalanguage above that, and so upwards. The liar cannot be formulated because the sentence would have to use a predicate it is not permitted to contain. The technical achievement is enormous and the philosophical cost is immediate: English plainly does talk about the truth of its own sentences, constantly and without catching fire, so the solution works by declaring natural language inconsistent rather than by explaining how it survives.

The sorites is the other great paradox of ordinary language and it is in some ways worse, because it attacks words you cannot avoid. One grain of sand is not a heap. Adding one grain to something that is not a heap never makes it a heap. Therefore a million grains is not a heap, which is false. The argument is valid and each premise looks obvious. Every escape is unattractive. Either there is a precise grain at which a heap comes into existence and no one can ever know which: a real position, called epistemicism, and much less popular than it deserves. Or the predicate takes intermediate values, which means giving up on true and false as the only options. Or classical logic does not apply to vague language, which is a very large bill.

Since almost every useful word is vague, this is not an edge case. Bald, tall, rich, reasonable, safe, dangerous, adult: all of them behave exactly like heap. Law deals with it by imposing sharp lines and accepting that they are arbitrary. Fewer than fifty employees makes you a small business, and the firm at forty-nine changes legal character by hiring one person although nothing about it has changed. That is not a failure of drafting. It is a considered judgement that vagueness in a rule is worse than a line in the wrong place, made by people who cannot postpone the decision until philosophy is finished.

The strongest evidence that none of this is decorative comes from computation. Suppose a program exists that takes any program and decides whether it halts. Build a second program which asks the decider about itself and then does the opposite of whatever it is told: if the decider says halt, loop forever; if it says loop, halt immediately. That program halts exactly when it does not. The contradiction proves no such decider can exist. It is the liar written in machine code, and Turing's result and Godel's incompleteness theorems both come out of self-reference used in precisely this way.

The general lesson is what makes paradoxes worth the hours. A genuine paradox is not a puzzle with a trick answer waiting to be spotted. It is a proof that a set of individually plausible commitments cannot all be kept, and the only question is which one you are willing to give up. That choice has consequences elsewhere. Abandon bivalence to handle vagueness and you have changed what every argument in this library is entitled to assume. There is no free repair, which is exactly why the arguments continue.