The Philosophical Puzzle Of The Liar Paradox

The Puzzle of the Liar Paradox (Copyright © 2009 by Jesse Butler)

Let's start with three basic assumptions: (1) Meaningful sentences of English have "truth conditions" – a set of conditions or circumstances under which sentences are true. For example, the sentence labeled 'A', A: 'Florida is north of the equator.' is true because its truth condition is met: Florida is north of the equator. On the other hand, there are sentences we can understand, but which are false. For example sentence labeled 'B', B: 'Abraham Lincoln was the president of the US in 2005.' is meaningful, but false, because the sentence's truth conditions are not met: since Abraham Lincoln died in 1865, he couldn’t have been the president in 2005. (2) Every meaningful sentence of the English language is true if, and only if, it's truth conditions are met, and false if, and only if, its truth conditions are not met. (3) Every meaningful sentence is either true or false. A sentence labeled by 'S' is false if, and only if, the sentence, 'It is not the case that S.' is true. These seemingly innocent assumptions get us into trouble because they lead us to an inconsistent set of beliefs. A set of beliefs is inconsistent if two beliefs contradict each other. We’ll explain exactly what this means in a moment, but first we need some terminology. Let P and Q stand for propositions – roughly that which is expressed by a sentence – and not-P and not-Q stand for the negations of those propositions. If we let P stand for the proposition expressed by the sentence, ‘Christopher Columbus discovered America’, then not-P stands for the proposition expressed by the sentence, ‘It is not the case that Christopher Columbus discovered America.’ So to return to our discussion of belief and consistency, we say that a set of beliefs is inconsistent if it contains both P and not-P. For example, if I believe both that a body must have a soul to have a mind AND that a soulless robot can have a mind, then I have an inconsistent set of beliefs. It’s the aim of Western philosophy to examine one's beliefs, check for inconsistencies and if any are found to try and remove them – by revising beliefs or perhaps by removing some. So how do assumptions (1)-(3) lead to inconsistency? Consider sentence labeled 'L'. L: 'Sentence L is false.'

Certainly L is meaningful because we can understand it, so by assumption (1) it has truth conditions, by (3) it is either true or false, and so by (2) it is true if and only if its truth conditions are met and false if, and only if, its truth conditions are not met. So L must be true or false. Which one is it? Well, if it’s true, then its truth conditions are met, but its truth conditions include that L is false, so if L is true it must be false. The conclusion is unacceptable, because to believe that one sentence, L, is true AND false is to hold an inconsistent set of beliefs: in that case, we would believe both L and the negation of L. What about the case in which L is false? Since the truth conditions of L are met just in case L is false, then, since we've assumed that L is false, L's truth conditions are met and we can reason that L must be true after all. This conclusion is also unacceptable because, as we noted before, to believe that one sentence is true AND false is to hold an inconsistent set of beliefs. To achieve a consistent set of beliefs, we must give up one of (1)-(3), or modify one of them. What should we do?