Saturday, March 17, 2018

The Liar Proof

This assertion is not true.
Let that assertion – if it is an assertion – be called ‘L’.
If L is an assertion – the assertion that L is not true – then L is an assertion that it is not true that L is not true, and so L is also 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 is fairly clearly 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, so there is that consistency. Note that L is not a simple conjunction of those two assertions; it is wholly the assertion that L is not true (if it is an assertion), and it thereby asserts that L is true. And note that no assertions are perfectly straightforward; all are to some extent vague, for example.
Nevertheless, logic seems 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.) 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 (which seems implausible) 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.

Tristan Haze said...

Hi Martin,

Interesting post. You may be interested in a paper by N.J.J. Smith which also contains the interesting idea of viewing the Liar reasoning as a reductio proof.

Semantic Regularity and the Liar Paradox
Monist (Special Issue on Truth)
89 (2006) 178-202

Published version: http://www-personal.usyd.edu.au/~njjsmith/papers/smith_liar.pdf

Longer version: http://www-personal.usyd.edu.au/~njjsmith/papers/smith_liar_melbourne.pdf

Smith's conclusion, however, is different, and to me very implausible: it is that the expressions involved "misfire" and end up having different semantic values, and that we cannot know which those are. (I have never understood why Smith felt driven to this conclusion instead of the conclusion that the expressions involved simply fail to get any semantic values. I once wrote to him about this paper but he didn't really address the worry in his reply.)

Martin Cooke said...

Thank you Tristan,

Of course, I cannot really view the Liar as a reductio because I am concluding that it takes us beyond the black-and-white logic in which reductios operate. But I think that it must be something similar.

Formal work is all very well, but as I say in the post, axioms are abstracted from pre-formal or informal-but-rigorous work, which also needs to be done, preferably by logicians. I think that too many analytic philosophers are blinding themselves with the science ...