I show that the Liar paradox is essentially a proof by reductio ad absurdum that it is not the case that assertions must be either true or else not true. For definiteness, let L be an assertion that L is not true, such as the assertion expressed by the following sentence: This assertion, which you are currently considering, is not true.
Since L is an assertion that L is not true, L is an assertion that it is not true that L is not true, and so it is an assertion that L is true, as well as not true. L is not simply an assertion that L is true, of course; nor is it a conjunction of those two assertions: L is wholly the assertion that L is not true, it is just that an aspect of that is that L is thereby an assertion that L is true. That is unusual, to say the least. One might wonder if L is indeed an assertion. But it is clear enough what is being asserted, clear enough for us to know what we are considering. And if L is as true as not – as shown below – then it is as true to say that L is true, as it is to say that it is not. So there is that consistency.
But, logic does seem to take L to a contradiction. By ‘logic’ I mean that which formal logics model mathematically. Formal axioms are abstracted from informal but rigorous proofs. So, were some such proof to seem good to us, then had it used something other than an axiom, we might have a reason to formulate a new axiom, or to reformulate our axioms, but we would have no reason to reject the original proof. Let me briefly revise how logic seems to take L to a contradiction. If L is true – if it is true that L is not true (and that L is true) – then L is not true (and true). But L cannot be true and not true – the ‘not true’ rules out its being true – and so if L must be either true or else not true, then it follows that L is not true. But if L is not true – if it is not true that L is not true (and that L is true) – then L is true (and not true). And L cannot be true and not true.
So, logic takes L to a contradiction if – and, as shown below, only if – we assume that assertions must be either true or else not true. And as I argue next, the negation of that assumption is not logically impossible. It follows that it is that assumption that logic is taking to a contradiction. Now, that assumption is certainly very plausible. To want the truth of a matter is to want things to be made clear. It is to want vagueness to be eliminated. Truth stands opposed to vagueness. Nevertheless, there are a variety of abnormal situations where it would be highly implausible for the assumption in question to be true. And L is not a normal assertion.
Suppose, for example, that @ is originally an apple, but that it has its molecules replaced, one by one, with molecules of beetroot. The question ‘what is @?’ is asked after each replacement, and the reply ‘it is an apple’ is always given. Originally that answer is correct: originally it is true that @ is an apple. But eventually it is incorrect. And so if the proposition that @ is an apple must be either true or else not, then an apple could (in theory) be turned into a non-apple – some mixture of apple and beetroot – by replacing just one of its original molecules with a molecule of beetroot. And that, of course, is highly implausible. What is surely possible, since far more plausible, is that @ is, at such a stage, no less an apple than apple/beetroot mix, that it is as much an apple as not, so that the assertion that @ is an apple is as true as not. That assertion could not be true without @ being an apple, nor not true without @ not being an apple (and we can rule out neither true nor not true, because that is just not true and true). Of course, @ is likely to move from being an apple to being as much an apple as not in some obscure way that is to some extent a matter of opinion. In between true and not true we may therefore expect to find states best described as ‘about as true as not, but a bit on the true side’, ‘about as true as not’ (a description that would naturally overlap with the other descriptions) and ‘about as true as not, but a bit on the untrue side’. For such abnormal situations, formalistic precision would be quite inappropriate, because the truth predicate is indeed suited to the elimination of vagueness. We might say, for example, ‘it is as much an apple as not’ instead of ‘it is an apple’ precisely because the former is true, the latter only as true as not.
But, we cannot express L differently, we have to understand it as it is. Fortunately, if we do not assume that assertions must be either true or else not true, then from the definition of L it follows only that L is true insofar as L is not true, which means that L is as true as not. There is no contradiction, and so the Liar paradox is a disguised proof by reductio ad absurdum that it is not the case that assertions must be either true or else not true.
Note that there is no ‘revenge’ problem with this resolution. E.g. consider the strengthened assertion R, that R is not even as true as not (which is thereby also an assertion that R is at least as true as not). If R is true then R is false (and true), if R is as true as not then R is false (and true) and if R is false then R is true (and false); but, if R is about as true as not, a bit on the untrue side, then it would be about as false as not to say that R was not even as true as not (and about as true as not to say that R was at least as true as not). Greater precision than that really would be inappropriate for an assertion as unnatural as R.