The Liar paradox refutes by

assertions having to be either true or else not true :

*reductio ad absurdum*assertions having to be either true or else not true :

**This assertion, which you are currently considering, is not true.**Let that assertion – if it is an assertion – be called ‘L’.

If L is an assertion that L is not true, then L is an assertion that it is not true that L is not true, and so it is also an assertion that L is true. Of course, L is not simply an assertion that L is true. Nor is it the conjunction of those two assertions, because it is wholly the assertion that L is not true, if it is an assertion, and only thereby an assertion that L is true. That is unusual, to say the least; but it is clear enough what is being asserted – how else could we know that it had that unusual property? – and so L clearly is an (unusual) assertion. And if L is as true as not (see below), then it is as true to say that L is true, as it is to say that it is not, and 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 arguments, arguments so rigorous that we regard them as proofs. Were such a proof to include a step that did not correspond to any axiom, we should have a reason to revise our formal logic; we should have no reason to reject the proof. So, 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, of course; 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. The negation of that assumption is not
logically impossible – see below – and so it is that assumption that logic is
taking to a contradiction. That assumption is certainly very plausible, of
course. To want the truth of a matter is to want things to be made clear. It is
to want the vagueness to be eliminated. 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). More precisely, @ 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. It is much better to say ‘it is as much an apple as not’ instead of ‘it is an apple’ when the former is true, the latter only as true as not.
But we cannot express L better, 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 if L is an assertion then L is true insofar as L is not true, and hence that L is as
true as not. There is no contradiction, and so either L is not even an assertion, or else 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 would be inappropriate for an assertion as unnatural as R.