Thursday, June 02, 2011

The Liar Paradox

Over two and a half thousand years ago, the Cretan Epimenides called all Cretans liars. But while there is an air of paradox about that, liars do not always lie. So, to see the paradox more clearly, suppose that Tiberius says ‘this that I am now saying is a lie’ (without having said anything else to which he might be referring).
......If what he said was true, he would have been – as he said he was – deliberately saying something false. So unless what he said was true and false, it was not true. So he did not deliberately say something false. Did he mistakenly believe that he was telling the truth? But how could he have believed that what he said was true – that he was lying – without thereby believing himself to be saying something false?
......Perhaps he did not know what he was saying, but if that is the only coherent possibility, then he could not possibly have known what he was saying. And yet what he said was not nonsense. Had it been, the above would have been impossible to follow. What he said was therefore paradoxical. And to see the paradox even more clearly, consider the simpler assertion ‘this is not true’, where that ‘this’ refers to that very assertion.
......If an assertion is true, then what it asserts is the case, so if ‘this is not true’ is true, then since it is self-referential, it is not true. Does that mean that it is not true? That would follow from it being either true or not (since even if it is true, it is not). The paradox is that, were it not true, its description of itself as not true would be correct. In general, if what is asserted is the case, then the assertion is true.
......Assertions are true when, and only when, what they assert is the case. E.g. the description ‘snow is white’ is true if, and only if, snow is white, which clearly generalises to any description. And the self-description ‘this is not true’ is true if, and only if, it is not true. Since ‘not true’ applies when, and only when, ‘true’ does not, hence our self-description cannot be true and not true. So it cannot be true – since if it is, it is not – but what is the alternative? If it is not true, then it is true. And we cannot even conclude that it is neither true, nor not true, because that is just to say that it is not true, and true.
......Perhaps the sentence ‘this is not true’ cannot coherently be interpreted as describing itself. There would be no paradox if, from that sentence failing to express a truth, it did not follow that it was a true self-description, but rather that the attempt at self-reference had failed. And we did leave Tiberius unable to know what he was saying, as though there was nothing for him to know. However, self-reference is not usually a problem, e.g. ‘this is not French’ seems true enough. And a paradox without self-reference, but otherwise very like the Liar, was introduced by Stephen Yablo in 1993.
......In our version of Yablo’s paradox, Tiberius has been around forever, and until today the only claims he ever made were a rather repetitive ‘no claim made earlier by me was true’, which he said once a year throughout his infinite past. Had none of those claims been true, each would thereby have been true. But if any one of them had been true, then none of those made earlier would have been true. And in particular, the one made a year earlier would not have been true, even though none of the earlier claims would have been true.
......Even so, it remains possible that the sentence ‘this is not true’ cannot coherently be interpreted as describing itself. E.g. Alfred Tarski suggested in 1935 that ‘true’ was equivocal – if not inconsistent – in such paradoxical contexts. So maybe each claim made by Tiberius gave ‘true’ a slightly different sense. Nevertheless, we intuitively take ‘true’ to be unequivocal, at least in descriptive contexts. And while there are lots of other possibilities – e.g. Graham Priest suggested in 1987 that ‘not’ allows descriptions to be true and false – common sense must make us wonder whether we are forced, by the Liar paradox, to entertain such counter-intuitive possibilities.
......I shall be arguing that we are not, because there is a common-sense resolution. Our words do not describe a black-and-white world, and so truth is not an all-or-nothing affair. So self-descriptions like ‘this is not true’ are neither simply true, nor simply not true, but are rather vaguely true. My explication of that will be as simple as possible, in order to show how it is little more than common sense. And to begin with, note that colours do not divide into those that are blue and those that are not blue.
......Some analytic philosophers would disagree, but it is only common sense that there is no dividing line between the blue and the other colours. Given some spectrum, the colours on either side of any such line would be indistinguishable, but colours that appear identical will both be blue enough to count as blue if one is. So there is no such dividing line. Rather, there are colours that are about as blue as not. We might call such colours ‘vaguely blue’. And if you said ‘that is blue’ of something vaguely blue, would what you said not be vaguely true (about as true as not)? Let me explain why I think that it would be.
......Descriptions are true when they describe how things are, rather than how they are not, but descriptive accuracy is in general a matter of degree. E.g. ‘blue’ describes royal blue more accurately than it describes a faintly greenish turquoise. So we should say that descriptions are true when, and insofar as, they are accurate. E.g. ‘snow is white’ is true insofar as snow is white. Of course, ‘snow is white’ is usually true enough to count as simply true. But snow can also be a bit bluish, or discoloured by dirt, or sparkle with all the colours of the rainbow because it is, on closer inspection, transparent. (Whether or not snow is white therefore depends on the context of ‘snow is white’.)
......Descriptions that are not true enough to count (in the given context) as true can usually be replaced with more accurate descriptions, e.g. ‘that is vaguely blue’. But we are considering particular descriptions – ‘that is blue’ and ‘this is not true’ – and wondering just how true they are. Since descriptions are true insofar as they are accurate, hence ‘that is blue’, said of something vaguely blue, is as true as not. Similarly, ‘this is not true’ is true insofar as it is not true, so it is as true as not. In other words, such descriptions are vaguely, but only vaguely, true. To see more clearly how that is only common sense, we should go more slowly through another example.
......Suppose that, out of the blue, Tabitha says ‘this that I am now saying is not true’. She has said, in effect, that what she said was not a good enough description of itself for it to count – in the context of her utterance – as simply true. And what she said was nothing if not self-contradictory, so it was not describing itself very well. But therefore it seems to have been describing itself quite well.
......What that shows is that self-contradictions are not always false. Usually they are, e.g. ‘this is not an assertion’ is simply false. But what Tabitha said was almost true enough to count as fairly true, falling short of that rather vague standard in order to avoid paradox. That is a coherent possibility because if what she said was vaguely true – if it is vaguely true that what she was saying was not true – then it need only follow that what she said was vaguely untrue (about as untrue as not), which clearly coheres with it being only vaguely true (about as true as not). And since all the other possibilities appear to be incoherent (or at least implausible), then that was what it was (or probably was).
......What Tiberius said would also have been vaguely true, if he had known what he was saying (had he not, what he said would have been false). And for yet another variation, suppose that Tabitha knew that what she was saying was only vaguely true, so that she said ‘this that I am now saying is not true’ with the intention to say something true. Since she did not actually say ‘is vaguely untrue’, what she said would still have been only vaguely true.
......What she said did seem true when we thought of it as not true, and then untrue when we thought of it as true. But that was when those two inaccurate descriptions were each creating a misleading context for the other. In fact, what she said had only the one context, that of its utterance. A nice analogy is someone wondering whether the colour of some blue-green object is really a sort of green (a bluish green). As she thinks of it as possibly green – and hence sees it, in her mind’s eye, against the various shades of green that it might be – it would probably look bluer, because the contrast would tend to enhance its bluishness. She might even wonder if it was really a sort of blue (a greenish blue). But similarly, it might thereby seem not to be, especially if it was really as blue as not.
......For yet another kind of Liar paradox (with indirect self-reference), consider the following pair of sentences (read in the obvious way): The next description is true. The previous description was not true. They are paradoxical because if the first description is true then, via the second, it is not, and vice versa. But if the first is vaguely true then it follows that the second is vaguely true, and hence that the first is vaguely untrue, which coheres with it being vaguely true. Indeed, if that is the only coherent possibility – within the bounds of common sense – then those descriptions are both vaguely true.
......We can hardly check all variants of the Liar paradox one by one, to see that they can all be resolved like that. But we can – and should – examine the most difficult to resolve. Suppose that ‘this is not even vaguely true’ was said, self-referentially. This new self-description seems, in effect, to have asserted its own untruth, much like the others. Yet how could it be vaguely true? Were it vaguely true, what was said – that what was said was not vaguely true – would seem false, not just vaguely untrue. Indeed, it would then seem true, since false. So this new self-description may well be hard to resolve.
......We have been using ‘vaguely true’ to mean about as true as not, though. And a description that is not even vaguely true in that sense need not be completely untrue, so long as it is significantly less true than untrue. So I was equivocating when I took the new self-description to be asserting its own untruth. I was taking ‘not even vaguely true’ to mean the same as ‘not true’. For clarity, we should stick with the former sense.
......It is still true that if the new self-description is vaguely true, then the assertion that it is not even vaguely true will be false. But we also know, from the previous paragraph, that if this self-description is a lot less true than untrue, then the assertion that it is not even vaguely true will be true. And if we look in between those two extremes, we find that this self-description can be a bit more vaguely true – a bit more untrue than true – while the assertion that it is not even vaguely true is also, coherently, a bit more untrue than true.
......Our difficult self-description is therefore more vaguely true (less vaguely untrue). And similarly, the self-description ‘this is only vaguely true’ would seem true if vaguely true, and vaguely true if true, and is therefore less vaguely true (more vaguely untrue). That is a little complicated, so note that we might, more loosely, call either description ‘vaguely true’. To see that more clearly, let us glance at the analogous problem of higher-order vagueness.
......One problem with vagueness is that we cannot, without contraction, think of the vaguely blue colours as neither blue nor (in the same context) not blue. Some philosophers therefore think of them as neither definitely blue nor definitely not blue. But then they face a problem of higher-order vagueness, the question of what happens between the definitely blue and the vaguely blue colours. We want a gap there, rather than a line at which the definitely blue looks just like the vaguely blue. But we cannot, without contradiction, think of the colours in that gap as neither definitely blue nor vaguely blue, because then they would be neither definitely blue nor (in the same context) not definitely blue.
......Nevertheless, while we would usually avoid calling vaguely blue colours ‘blue’ or ‘not blue’, that is because calling them either would be only vaguely true, not because it would be false. Blue shades smoothly into green, via blue-green; and a pretty good description of the blueness of any blue-green colour might be ‘vaguely blue’, even though a better description could, for some of them (in some contexts), be ‘green’. And similarly, ‘vaguely true’ would be a fairly good description of any of the self-descriptions that give rise to Liar paradoxes (especially in view of the vagueness of ‘vaguely’), for all that it can be misleadingly inaccurate when the self-descriptions use ‘vaguely true’ themselves.
......Now, as well as various variants of the Liar paradox, there are also various other paradoxes that should, intuitively, have very similar resolutions. We have already met Yablo’s paradox, which concerns an infinite set of descriptions, each asserting the untruth of all the earlier ones. And that paradox does have a common-sense resolution. If all those descriptions were vaguely true then, from any of them being vaguely true, it need only follow that all of those before it were vaguely untrue. So that is a coherent possibility; and if it is the only one (within the bounds of common sense), then the descriptions of Yablo’s paradox are (probably) vaguely true, more or less.
......It appears, then, that we find such descriptions paradoxical because of a natural tendency to ignore descriptive imprecision. That tendency helps us to focus on the most apposite elements of truth and falsity in what is being said. So it is usually useful. We just have to take care with self-descriptions like ‘this is not an accurate description of itself’. Our next (and final) paradox is very similar to that Liar paradox, since it concerns the predicate expression ‘does not describe itself accurately’. (Regarding the other paradoxes of self-reference, the question of how similar they are to the Liar depends on how they should be resolved, so they would take us too far afield.)
......The expression ‘is long’ is not long, not for a predicate expression. So it does not, as a rule, describe itself accurately. By contrast, ‘is short’ does. The question is, does ‘does not describe itself accurately’ describe itself accurately, or not? If it does – if it is described by ‘does not describe itself accurately’ – then it does not describe itself accurately. But therefore, since it only fails to accurately describe expressions that are describing themselves accurately, it does describe itself accurately.
......To resolve this paradox we need only assume that descriptive accuracy might be a matter of degree. And for convenience, let us say that an expression is heterological when, and insofar as, it does not describe itself accurately. It follows that ‘is heterological’ is heterological insofar as it is not. So it is as heterological as not. In other words, the expression ‘does not describe itself accurately’ is vaguely heterological. And it follows that descriptive accuracy is, in general, a matter of degree. (Incidentally, Kurt Grelling and Leonard Nelson introduced the term ‘heterological’, along with this paradox, in 1908.)
......This may therefore be a good place to stop, and review the common-sense resolution of the Liar (and Yablo’s) paradox. Descriptions are true when, and insofar as, they are accurate. So the self-description ‘this is not true’ is true insofar as it is not true, and so it is as true as not. Indeed, all such descriptions are vaguely true (about as true as not), more or less. That resolution is simple, and intuitive. But it is hard to find it in the literature. And because it tends to be overlooked, the reasons for its neglect are also obscure. So let me close with one possibility.
......Logicians often use ‘1’ to signify truth, and ‘0’ for falsity, and the so-called fuzzy logicians use the number ½ to model half-truths. Fuzzy logic developed out of fuzzy set theory, and the most influential paradoxes of self-reference were those of set theory, as axiomatic set theory became the standard foundation of mathematics. Fuzzy logic is a mathematical logic, not common sense, but the former tends to be more attractive to analytic philosophers, who may well have preferred the precision of ½ to such vague words as ‘vaguely true’. Nevertheless, our words are unlikely to be much better defined than our purposes have required them to be, and even formal terms must ultimately derive their meanings from natural language.