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.

The Irony Age continued

My previous post was quite brief, so here are a few more thoughts on the two paragraphs quoted therein. The first paragraph was about estimating the danger posed by the LHC to the planet (if not the universe), and the big argument for safety is that cosmic rays produce such collisions all the time. Whatever a collider might produce, it’s very likely that such has already been produced on the moon, for example, lots and lots of times. And of course, the moon’s still there. That argument doesn’t seem to depend upon the niceties of particle physics. But cosmic rays spread out from the sun. So they are most concentrated near the sun. What if some merging of products of collisions is most likely nearest the sun? Such events might not occur on the moon, but might occur in the most concentrated beams of our biggest colliders. So how likely is it that such an event causes tiny ripples on the sun? The problem is that such an event would destroy the earth. And our most popular theories have little to say about such questions (and did fail to predict dark matter).

Furthermore, suppose we could estimate the answer at no more than one in a billion. Would that be safe? We are talking about the possible destruction, not only of less than ten billion people, but of all possible future human beings. What figure should be given to that? So we also need some way of determining just how safe a one-in-a-billion chance of destroying the human race really is (as Sample noted). Since that problem is so intractable (cf. the St. Petersburg Paradox), surely the main thing here is that we do have better things to do, things associated with more mundane risks (and more immediate benefits). Even theoretical physicists have plenty of other puzzles to solve. A competitive Academia may encourage them to excel at the language game of string theory, but surely the most puzzling thing in theoretical physics (given the materialism there) is the absence of anything at the fundamental level that could conceivably give rise to awareness when the fundamental particles are parts of complicated biochemical systems (the elephant in the room in which we debate synthetic biology).

Or they could address their big methodological problem, which is the question of what they should be thinking they’re doing. The language of science is mathematics, but standard mathematics is heavily influenced by Formalism, which encourages the move from general interest in a new type of theory, to the adoption of the presuppositions of that type of theory. We all know what is meant by ‘1 + 1 = 2’, but standard mathematicians will tell you that it means that {0, {0}} follows {0} in the von Neumann series (where those are ZFC sets, and ‘0’ denotes the empty set). They will say that that is just the language game that is modern mathematics. But mathematics is not a game, but the language of science; and that word ‘language’ is being used metaphorically. The literal languages of science are our natural languages, which include mathematical terminology when one is doing science. It is a philosophical question, what those terms refer to, if anything; but the meaning of mathematical statements is clearly akin to logic (not a made-up, formal logic, but the logic that all scientists should apply).

The Irony Age

One problem that continues to plague discussions over the safety of particle colliders, though the issue is relevant to other areas of cutting-edge science, such as synthetic biology and genetics, is that it is impossible to have what could sensibly be called an informed public debate on the issues. The people who understand the issues best work in the field under debate, so the accusation of vested interests cannot be avoided. Ironically, the most high-profile opponents to a new technology are often so badly informed they are quickly dismissed as crackpots, and rightly so. The result is an illusion of public debate. Ill-informed opponents do a disservice to people with genuine interest and concern by squandering the opportunity for an even-handed discussion of the risks.

That paragraph is from p.193 of Ian Sample’s Massive (Virgin Books, 2011), and it reminded me of the following, from the bottom of p. 640 of R. W. Hamming’s ‘Mathematics on a Distant Planet’, American Mathematical Monthly 105 (1998), 640–650, the gist of which was that we should have based our mathematics on known truths, not—as was increasingly done throughout the twentieth century—on axioms taken to lie beyond truth and falsity.

I need to mention a few things in my life that have shaped my opinions. The first occurred at Los Alamos during WWII when we were designing atomic bombs. Shortly before the first field test (you realise that no small scale experiment can be done—either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, “It is the probability that the test bomb will ignite the whole atmosphere.” I decided I would check it myself! The next day when he came for the answers I remarked to him, “The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen—after all, there could be no experiments at the needed energy levels.” He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, “What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?” I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, “Never mind, Hamming, no one will ever blame you.” Yes, we risked all the life we knew of in the known universe on some mathematics. Mathematics is not merely an idle art form, it is an essential part of our society.

Academics are of course free to pursue whatever interests them; and a hundred years ago, set theory interested many pure mathematicians. But academics will only be successful if their interests are those of their peers (or industry), so there’s some irony there. Any young mathematician bothered by set theory would have been unlikely to have gone into pure mathematics. Perhaps physicists uninterested in string theory are unlikely to choose theoretical physics; but certainly, a slight bias can become the rule, over a century or so. And a rule will tend to exclude other possibilities (even in the absence of corruption), making it impossible to estimate costs and benefits properly. (To be continued.)

Russell, Cantor, Curry, and Simmons’ paradox

This post is the fourth part of Vaguely True Liars.

The paradoxes of infinity being tangential here, I’ll just touch upon Russell’s paradox, which concerns collections (although it was originally formulated in terms of predicates). The obvious collections – e.g. the class of all chairs – don’t include themselves, as members (as one of the collected things), but some might, e.g. the collection of all the non-chairs would not itself be a chair. So, let the collection of all the non-self-membered collections be called ‘R’. One member of R is the class of all chairs; but is R a member of itself, or not? If it is then it’s self-membered, so it shouldn’t be; but if it isn’t then it’s eligible to be, so it ought to be. That’s Russell’s paradox. The obvious resolution – akin to my first suggestion for Grelling’s paradox – is to consider, not R, but the collection of all the other non-self-membered collections.

If I’m right about Grelling’s paradox, predicates won’t always be associated with classes, so it would be reasonable to resolve these two paradoxes differently. But of course, things are not quite that simple. We should, for example, replace ‘non-self-membered’ with ‘well-founded’ if we want to rule out such possibilities as a collection V that contains U and all the other non-self-membered collections, where U contains V and all those others. And then there’s Cantor’s paradox: If there was a (well-founded) collection of all the other (well-founded) collections, then there would be more sub-collections than there are collections, via Cantor’s diagonal argument, which shows that any collection – even an infinitely big one – has more sub-collections than it has members (at least if it’s a well-founded and non-variable collection) [i]. However, a sub-collection (of some collection) is just another collection (of some of the original collection’s members), and so Cantor’s argument is that any collection of all the others would be bigger than itself. For such reasons, Russell’s paradox may well belong with the paradoxes of infinity [ii].

So let’s look next at a paradox due, in its essentials, to Haskell Curry. Suppose I say ‘if what I’m now saying is true, then pigs fly’. That’s clearly making the same type of assertion as my earlier Liar statement. But if we generalise ‘pigs fly’ to ‘the Pope’s next assertion is true’, we get the following argument for papal infallibility. Suppose I say ‘if what I’m now saying is true, then the Pope’s next assertion is true’. And suppose, just suppose, that what I said was true. What I said was that if what I said was true, then so was whatever the Pope then said. So we are supposing that if we suppose that what I said was true – as we are doing – then we are also supposing that what the Pope said next was true. So we are, in effect, supposing (just supposing) that what the Pope said was true. To recap, we supposed that what I said was true, and thereby supposed that what the Pope said was true. But that means that what I said – that if what I said was true, then so was what the Pope said – was true, and hence that what the Pope said next was true.

But of course, the Pope might, for all we know, have said that pigs fly, so the argument was certainly fallacious. It would be interesting to consider why it was; but what’s most apposite here is how unlike the Liar paradox it was. The most obvious difference is that the word ‘not’ did not appear in it (although ‘not’ could be introduced via the material conditional) [iii]. Furthermore, by deriving fact from mere possibility, Curry’s paradox can seem more like the Ontological Argument, than the Liar paradox. Now, I suppose that how we resolve my particular example should depend upon how we treat future contingents, as well as how we treat self-reference. Regarding the former, note that what I said would have been like my Liar statement (or possibly true), had the Pope next said something false (respectively true).

But in view of the latter, Simmons’ paradox (which I recently considered in the post that that link links to) is more apposite, because that paradox can be resolved by noting that reference is in general a matter of degree. So to sum up, it may well be that a natural kind of paradox arises from our tendency to ignore – to see past – the imprecision of description. As we use language, we naturally focus upon apposite elements of truth and falsity – much as we see the picture, not the pixels – and so we tend to overlook such possibilities as Liars being vaguely true. An obvious danger is that, by considering too few possibilities, we misconstrue the evidence for our favourite logic. Indeed, since standard set theory was developed in response to such paradoxes of infinity as Cantor’s, there’s some danger of the language of modern science having been built on shaky foundations [iv].

[i] For the mathematical details, see any introduction to set theory. For a mathematician’s view of the philosophical details, see Peter Fletcher, ‘Infinity’, in Dale Jacquette (ed.), Philosophy of Logic (Amsterdam: Elsevier, 2007), 523–585.

[ii] For more on the kind of paradox that includes Russell’s and Cantor’s (and the Burali-Forti), see Stewart Shapiro and Crispin Wright, ‘All Things Indefinitely Extensible’, in Agustin Rayo and Gabriel Uzquiano (eds.), Absolute Generality (Oxford: Clarendon Press, 2006), 255–304.

[iii] For some discussion, see Graham Priest, Beyond the Limits of Thought, second edition (Oxford: Clarendon Press, 2002), 168–9, 278.

[iv] For historical details, see Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton University Press, 2000).

My ‘revenge’ paradox (and Yablo’s paradox)

This post is the third part of Vaguely True Liars.

In my previous post I resolved a Liar utterance; can something similar be said of the other Liar sentences? Since there are many such sentences, let’s just see if we can resolve the trickiest. In general, the most difficult sentences for any putative resolution are those that threaten so-called ‘revenge’ paradoxes, which is when the terminology of the resolution – in our case, ‘vaguely’ – is used to make a new Liar sentence that resists resolution along the same lines. So suppose I said ‘what I’m now saying is not even vaguely true’. Something that’s not even vaguely true is, prima facie, something untrue. And if this new statement is asserting its own untruth then, like my previous utterance, it should be vaguely true. But then what I said – that it wasn’t vaguely true – would seem to have been false, not just vaguely untrue. And it would then, if false, seem to be true, rather paradoxically. So this does seem to be my ‘revenge’ statement.

Was it asserting its own untruth? Something that’s not even vaguely true (in our sense) is something that’s at least a little less true than not. And that’s compatible with it being more vaguely true, in the sense of more roughly as true as not. (It’s also compatible with it being false, of course.) Now, because of that compatibility, my ‘revenge’ statement seems true if more vaguely true, as well as false if vaguely true. Is there something in between the vaguely and the more vaguely true? Well, what about the vaguely more vaguely true? That’s a clumsy turn of phrase, but it is describing an unusual statement. And perhaps we could more loosely say that my statement was vaguely true, and then qualify that by adding that, more precisely, it’s vaguely more vaguely true than the vaguely true it refers to. (How vague the latter was would depend upon what counted as true when I uttered my ‘revenge’ statement.)

That’s a bit obscure, but such obscurity would at least explain how this resolution could have been overlooked, were it correct. And we can get some clarification by glancing at the related problem of higher-order vagueness [i]. That problem might, for example, arise with vaguely bluish colours – those that are roughly as blue as not (in some context) – were we tempted to regard them as neither blue nor not blue (in that context). If they weren’t blue and yet were blue, in the same context, then they would be contradictory, so we should resist that temptation. However, being so tempted we might take them to be, for example, neither definitely blue nor definitely not blue. And then we would face a ‘revenge’ problem, via the question of what happens between the definitely blue and the vaguely bluish colours. Were these colours neither definitely blue nor vaguely bluish, and the latter not definitely blue (nor definitely not blue), then these colours would be neither definitely blue nor not definitely blue, in the same context.

That’s a problem of higher-order vagueness. But such problems don’t affect the common-sense approach that I’ve been taking. While I don’t want to call vaguely bluish colours ‘blue’ or ‘not blue’, that’s only because calling them either would be only vaguely true, not because it would be false. Blue shades smoothly into blue-green, in reality. And a colour that’s roughly as blue-green as it is blue might be described quite accurately as ‘blue’ (since it’s a faintly greenish blue). Similarly, a good description of my ‘revenge’ statement could be ‘vaguely true’ (qualified as required). After all, the term ‘vaguely’ is an especially imprecise term, and highly context-sensitive, being generally used to gesture beyond the other adjectives in use, or made apposite by such use.

Things can be made more complicated, of course. E.g. a more awkward ‘revenge’ paradox might be based on Yablo’s paradox [ii]. Imagine a place where ‘no previous utterance here was even vaguely true’ was said by someone, once a year – every year, throughout the infinite past – with nothing else ever being said there. Now, it seems to me that all those utterances would have been (vaguely more) vaguely true. But it would certainly be more awkward to justify that view in this case. Still, suppose my approach failed here; that might only mean that this paradox was more like the paradoxes of infinity than the Liar. Certainly, this paradox concerns an infinite set of semantically ungrounded sentences, rather than self-description (or circular description [iii]). And the paradoxes of infinity are not our topic. (To be continued.)

[i] That problems of higher-order vagueness are akin to ‘revenge’ paradoxes was noted by Mark Colyvan, ‘Vagueness and Truth’, in Heather Dyke (ed.), From Truth to Reality: New Essays in Logic and Metaphysics (Abingdon: Routledge, 2009), 29–42.

[ii] Stephen Yablo, ‘Paradox without Self-Reference’, Analysis 53 (1993), 251–2.

[iii] A simple example of a circular Liar is the following pair of sentences. The next sentence is true. The previous sentence was false. I regard them both as vaguely true, when read or said in the obvious way.

Liar statements are about as true as not

This post is the second part of Vaguely True Liars.

Things are usually described well enough, for some obvious purpose. E.g., if something is obviously blue, then we might call it ‘blue’ when referring to it. If, given a different object, some other description sprang to mind, well, maybe the object wasn’t blue. Or perhaps it was blue, but something else about it was more apposite. Another possibility is that it was as blue as not, however, because colours don’t divide into those that are blue and those that aren’t. On the two sides of any such line, between the blue and the other colours of some spectrum, would be colours that were indistinguishable; but of course, colours that appear identical will both be blue enough to count as blue if one is. So there’s no such dividing line; rather, there are colours that are vaguely bluish. Intuitively, ‘that’s blue’ said of such colours would be vaguely true. It would not be true enough to count as true, but being roughly as true as not, nor would it be more than vaguely false.

There are two basic logical possibilities, i.e. true, or not. Statements are true insofar as they describe how things are, as opposed to how they aren’t. And when a description isn’t true enough, we can usually replace it with a more detailed description. E.g. we can replace ‘that’s blue’, when it’s vaguely true, with ‘that’s vaguely bluish’. And we don’t always have to make things so explicit, because we naturally focus upon the pertinent elements of truth in what’s being said (or perhaps upon some obvious falsity). Indeed, that may well be why we have the concept of truth (and that of negation) [i]. Perhaps it’s also why we fall for the Liar paradox.

Suppose I say ‘what I’m now saying isn’t true’. If what I said was true, then as I said, what I said wasn’t true. Does it follow that what I said wasn’t true? The paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. So you may well wonder if I really said anything, with my Liar utterance. But if not, then surely you would have found my utterance incomprehensible, rather than paradoxical, and so I think that the meaning of my utterance must have been fairly clear. It seems to me that I was saying that what I was saying wasn’t a good enough description of itself for it to count as simply true. Now, since it was nothing if not self-contradictory, it wasn’t describing itself very well. But therefore it seems to have been describing itself quite well after all. Still, perhaps its self-description was almost good enough to count as simply (or absolutely) true, but its self-contradictory nature meant that it fell just short enough to avoid paradox.

Much as ‘is heterological’ had to be as heterological as not, my Liar utterance seems forced to be about as true as not. I say ‘about’ in view of the underlying imprecision of natural language (and presuming more accuracy could lead to a ‘revenge’ paradox that would take us back to this position anyway). The paradoxical reasoning rules out the non-vague extremes, but it being vaguely true that what I said was not true implies only that what I said was vaguely untrue – vaguely false (since I was making an assertion) – which coheres well enough with it being vaguely true for there to be no more contradiction. And this resolution also explains why my utterance seemed true when thought of as false, and vice versa. By analogy, if you were given something blue-green, for example, you might wonder whether it was really more green than blue. But if it’s roughly as blue as not, then it would look bluer as you postulated it amongst – and hence saw it in your mind’s eye against – various shades of green. The contrast would enhance its bluishness. And if you thence thought of it as possibly blue, it would similarly seem not to be.

You may be wondering what exactly the element of truth would have been, were my statement vaguely true. Well, it would also have been vaguely false, so it would have been vaguely true that it was false. So we might say that the element of truth was that there was an element of falsity (and vice versa) [ii]. But more precisely, I’m suggesting that my statement wasn’t describing itself very well, that it was neither true enough to count as simply true, nor sufficiently false to be less than vaguely true. It may well have seemed untrue (if true) and then true (if untrue), but that was while those two inaccurate descriptions were being each other’s context. My statement had only the one context of its utterance. And it must have been about as true as not, if the alternatives are paradoxical. (To be continued.)

[i] The more usual reason given for why we have the concept of truth is that it allows such sweeping claims as ‘everything the Pope said was true’. For more on that reason, see John Collins, ‘Compendious Assertion and Natural Language (Generalized) Quantification: A Problem for Deflationary Truth’, in Cory D. Wright and Nikolaj J. L. L. Pedersen (eds.), New Waves in Truth (Basingstoke: Palgrave Macmillan, 2010), 81–96.

[ii] Elements of truth and falsity are often propositions. But it might be argued that Liar sentences express no proposition, in their paradoxical contexts; and doubts about emphasising propositions have, for example, been raised by W. V. Quine, Philosophy of Logic (Englewood Cliffs: Prentice-Hall, 1970), 8–13.

Introduction (via Grelling’s paradox)

This post is the first part of Vaguely True Liars.

To begin very simply, ‘is long’ is not long, not for a predicate expression. Kurt Grelling called it ‘heterological’. Heterological expressions don’t apply to themselves. Grelling asked, is ‘is heterological’ heterological? It is if it doesn’t apply to itself – that’s what ‘heterological’ means – but if it is, then it applies to itself – that’s what ‘applies’ means – and so it isn’t heterological. It is if it isn’t, and it isn’t if it is; that’s Grelling’s paradox [i]. Could we arbitrarily include ‘heterological’ in, or else exclude it from, the range of its application? But then ‘heterological’ would not, in that particular instance, mean what it should, intuitively, mean. Furthermore, Grelling’s paradox is of a kind with the Liar paradox (our main topic), and taking such an approach to the Liar would amount to denying the universal relevance of truth (or its coherence) [ii].

The approach I take to such self-descriptive paradoxes begins by noticing that descriptive accuracy is in general a matter of degree. I would say, for example, that because ‘is pretty’ makes us think of prettiness, it’s vaguely pretty, but only vaguely pretty because it’s only a word (outside of calligraphy or song). It’s a matter of opinion, of course; but what about ‘is a bit long’, or ‘is a little lengthy’, or ‘is only slightly lengthy’? Anyway, if descriptive accuracy is a matter of degree, in general, then I should have said that a predicate expression is heterological insofar as it doesn’t apply to itself. And if that is – or should be – what ‘heterological’ means, then ‘is heterological’ is heterological only insofar as it isn’t. That is, it’s as heterological as not. So we might say that it’s only vaguely heterological.

And descriptive accuracy is likely to be a matter of degree, in general (in various context-sensitive ways). Our words are unlikely to be much better defined than our purposes have required them to be, and so we should expect a ubiquitous – since ordinarily unobtrusive – imprecision throughout our natural languages. Such imprecision is not usually important – that’s why it’s there – and even when it does matter, we can always clarify what we mean, because it has enabled our languages to be the versatile tools that they needed to be. Now, descriptions can become more accurate by becoming more detailed, so long as they remain true. Can they also become more accurate by becoming truer?

If you said ‘that’s blue’ of a fading print, for example, would your words become truer as the print became bluer? Certainly, a description can become truer by becoming more detailed, as when we bring out the element of truth from some half-truth. So perhaps we should say that, in general, our words are true insofar as they describe the world. E.g. ‘snow is white’ is true insofar as snow is white. The famous biconditional – ‘snow is white’ is true if, and only if, snow is white – is either a special case of that, or else it presumes that the elements of truth and falsity can always be effectively isolated. (Clearly ‘snow is white’ is usually true enough, but snow can also be faintly blue, discoloured, transparent, or sparkling all the colours of the rainbow.) Now, the idea that there are degrees of truth is nothing new. That it’s only common sense is shown by such common phrases as ‘true enough’ and ‘very true’; and it has been formally explored by the so-called ‘fuzzy’ logicians [iii]. But I want to emphasise its prima facie plausibility here, because if statements can be, not just true or not, but also vaguely true – about as true as not – then the Liar paradox is easily resolved.

A simple Liar statement is ‘this is false’, which is false if true, but if false then not false (a statement is false when its negation is true). Since statements might be neither true nor false, a better example may be ‘this is not true’. And since sentences can mean different things in different contexts, an utterance of ‘what I’m now saying isn’t true’ is what we shall examine below. However, whereas modern introductions to the Liar paradox tend to be rather formal [iv], I shall be quite informal. Philosophers are in the business of clarifying things, and imprecision can of course lead us astray, so it’s unsurprising that many modern philosophers are fond of formal precision. But formal languages must get their meanings from natural ones. And an obscure formalism might also lead us astray. E.g., the use of biconditionals to define truth might go too easily unquestioned on a page of mathematical symbols; and for another example, see Curry’s paradox (below). So, let’s get back to informal alethic basics. (To be continued.)

[i] For a class of paradoxes that includes Grelling’s, see Thomas Bolander, ‘Self-Reference’, in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.

[ii] Alfred Tarski took that approach to formal languages (taking natural languages to be inconsistent). For details, see Wilfrid Hodges, ‘Tarski’s Truth Definitions’, in Zalta, op. cit.

[iii] Petr Hajek, ‘Fuzzy Logic’, in Zalta, op. cit.

[iv] J. C. Beall and Michael Glanzberg, ‘Liar Paradox’, in Zalta, op. cit.

Vaguely True Liars

My next 4 posts explore, informally and briefly, the possibility that Liar utterances are about as true as not. Abstract:

I suggest that Liar statements – e.g. ‘this is false’ – are about as true as not. In other words, they are vaguely true, and vaguely false. And truth does seem to come in degrees (e.g. if a colour is about as blue as not, calling it ‘blue’ would be about as true as not). I also suggest that ‘this is not even vaguely true’ can be called ‘vaguely true’, even though it may then seem false, not just vaguely false. That’s because it is, more precisely, vaguely more vaguely true (similarly, a colour that’s roughly as blue-green as it is blue would usually be called ‘blue’ becaue it is a faintly greenish blue). I give similar resolutions to the paradoxes of Kurt Grelling and Keith Simmons, while I regard as of a different kind those of Bertrand Russell and Haskell Curry.

Links to the 4 posts:

......Introduction (via Grelling’s paradox)

......Liar statements are about as true as not

......My ‘revenge’ paradox (and Yablo’s paradox)

......Russell, Cantor, Curry, and Simmons’ paradox

An application of the above is a common sense refutation of the Divine Liar argument (against omniscience), see my Liars, Divine Liars, and Semantics revisited (in April’s issue of The Reasoner).

Curry's paradoxes

The Liar paradox concerns utterances such as ‘what I’m saying isn’t true,’ which is, if true, not true, and which seems true if not. Another way of saying the same thing would seem to be to say ‘if what I’m saying is true, then pigs fly.’ Yet that utterance is paradoxical in a way so different that not only has it a different name – Curry’s paradox – it’s debatable whether its resolution should even resemble that of the Liar.
......Suppose I say ‘if what I’m saying is true, then P,’ where ‘P’ stands for any proposition. For simplicity, let’s say that C is the assertion that C implies P. Suppose, just suppose, that C is true. We are thereby supposing that C implies P. So we would also have P. That much is simple enough. Given C, and that if C then P, we get P. And yet that much is too much. If it’s true that by supposing C we also get P, then C really is true, and hence P is true, even though P could be asserting anything at all (even that pigs fly).
......That seems quite unlike the Liar. E.g. there was no ‘not’ in the previous paragraph: We didn’t consider C being either true or else not, and find both possibilities inadequate; nor did we see C seeming to say that it wasn’t something that C did indeed seem not to be. Rather, just by wondering what C was – in particular, whether C might be true – we seemed to get P. And so by deriving actual being from mere possibility, Curry’s paradox seems to be more like the Ontological Argument than the Liar.
......On the other hand, if it follows from the meaning of ‘not’ that either A or not-A, where ‘A’ stands for any proposition (e.g. that it’s raining), then it seems that if A implies B (e.g. that I’m carrying an umbrella), then either B or not-A (either I’m carrying an umbrella or it’s not raining). And conversely, if it’s the case that either not-A or B, then if it’s A (if it isn’t not-A), it must be B. So in short, C could seem to be asserting that either not-C or P. And then P effectively disappears if it’s false, leaving C asserting not-C.

Simmons' paradox

The following, which is akin to the Liar paradox, is a paradox of self-reference by Keith Simmons, described on p. 231 of his ‘Reference and Paradox’ in JC Beall (ed.) Liars and Heaps (Oxford 2003):

Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.

Here is what I think about those denotations:
......The first expression, ‘pi’, referred to 3.14159..., and the second to 6. So if the third expression does denote a number, say N, then N = N + 9.14159... Given that by ‘number’ we mean finite number, it seems that the third expression can’t denote a number. So those three expressions denote only pi and six. But then the sum of the numbers denoted by those three expressions is 9.14159..., and so the third sentence does seem to denote a number after all. Or rather, it does because it doesn’t; and furthermore, it seems to denote 9.14159..., but therefore it seems to denote 18.283..., or rather, 27 and a bit, etc.
......I have, however, been implicitly assuming that reference is an all-or-nothing affair. Usually we can – and indeed, should – take it to be so, but is it so in general? Imagine, for example, a man staggering through a desert. He sees a mirage, which he takes to be a pool, and as it happens there is a pool, just where he takes one to be, but it’s obscured from his view by the mirage. As he staggers towards it, he’s constantly thinking ‘that pool looks cool’. As he nears the pool, its image gradually replaces the illusory one, without him noticing, so that the referent of ‘that pool’ gradually changes to the pool. And so at some point he may have referred only vaguely to it.
......Such a case would of course be exceptional, but so are scenarios designed to be paradoxical. And it does at least seem possible that the third expression of Simmons’ paradox referred only vaguely to 9.14159... It would also have referred, even more vaguely, to 18.283 (and so on), but while that’s even odder, it too seems possible. And if the alternative is paradoxical, vague reference may not be too odd. And such a resolution would cohere with ‘this is not true’ being vaguely true (about as true as not), and ‘is heterological’ being as heterological as not (see my previous posts this year).

God is Timeless

When people say that God is timeless, they may mean many things. They may mean that He is above and beyond the mundane world, like the truths of mathematics, for example. But they would not then be disagreeing with Open Theism:
......Under Open Theism, God is certainly above and beyond His creation, a bit like a dreamer and the dream he finds himself within. And one of the few coherent philosophies of mathematics is Open Theistic Constructivism, in which God, being omnipotent, creates the truths of mathematics—from the basic concepts of a thing and of possibility (the latter grounded in His omnipotence)—doing so endlessly because such is the nature of the former concept, according to the resolution of Cantor’s paradox that takes it to be showing that cardinal numbers are collectively indefinitely extensible (one of the few coherent resolutions).
......Of course, they may instead mean that God is not changeable. But even that isn’t incompatible with Open Theism, under which God cannot change his essential properties. Nor can you, of course. You can’t become me, for example. You might change by becoming in a manner of speaking a different person (i.e. your character might change, for better or for worse), but under Open Theism God’s character remains perfect. So the only disagreement with Open Theism could be that such philosophers are denying that God could choose to cause any real change in anything; and not only is that clearly not what most believers mean when they say that God is timeless (is rather more like an absurd denial of the reality of change), such philosophers would be denying God’s omnipotence:
......The deliberate creation of anything contingent surely requires several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change (not necessarily one that takes place within spacetime).

Only the unfit evolve

From Philosophy Now, I learn 'that people and organisations that miss their goals disastrously perform better in the long run.'

Professor Desai, who led the study, said “knowledge gained from success was often fleeting while knowledge from failure often stuck around for years.” [...] He says failure causes a company to search for solutions and it puts the executives in a more open mindset. He doesn’t recommend seeking out failure in order to learn.

...which makes me think of the merchant bankers. Will they perform better now? Only if we make them (despite our politics being dominated by the short-term), I think. You know, their performance was always chaotic, even in the good old days, as the new maths of chaos showed us in the 80's. But they took past success to be indicate a propensity for success; ironically, they were bad at applied math.
......I find that ironic because it was pointed out to me, when I was reading physics (in the 80's), that investment banking was the sort of 'good job' that a physics degree qualified one for. (That was what made physics such a good degree.) At the time, I wondered whether we really would've been a better society had more women wanted to play such games. (Margaret Thatcher, with her Chemistry degree, hardly thought of investment banking as a waste of a physics degree.)
......What can you do? The young are always with us. And we can hardly want to dilute democracy, but, why not give more experienced voters additional votes? Perhaps we shouldn't take the vote away from those who've shown themselves to be very selfish (unless we've locked them up and thrown away the key), but why not give an extra vote to all those who haven't (yet) shown themselves up; indeed, why not give extra votes to those who deserve honours? Wouldn't that be fairer? (It would certainly be safer.)

Philosophers' Carnival #121

Welcome to Philosophers' Carnival #121. Not so much a one-to-one as a party (a rather Platonic party). The bouncer's been busy (arguably too busy, or not busy enough). And there's some of the hard stuff later on; so, enter...


......the Hallway (epistemology)

Keith DeRose imagines 'Nico at the Zoo with Zebras' (at Certain Doubts), an example of how knowledge attributions can sometimes be true even when they’re about insensitive beliefs.

John Wilkins compares Elliott Sober's 'Modus Darwin and the *real* modus darvinii' of 'affinity, explained by common ancestry' (at Evolving Thoughts), showing that the former should've been the latter.

Maryann Spikes thinks of 'Atheism and agnosticism (really, apisticism) as belief' (at Ichthus77), and also thinks that you can only be apistic if you don't claim to be.


......the Games-room (logic and language)

Ben Nelson wonders about 'Trust as a truth-maker' (at Talking Philosophy). Of course, "X trusts me" is made true by X trusting me, but Nelson takes a broader (even deeper) look at this kind of thing.

I ask 'Is 'pretty' pretty?' (It is, and it isn't:)

Matt asks 'How slippery is the slope?' (at The Consternation of Philosophy), and concludes that 'the slippery slope fallacy is a slippery beast, and is perhaps best not thought of as a fallacy at all.'


......the Dining-room (metaphysics)

Edward Feser asks 'Why are (some) physicists so bad at philosophy?' (at Edward Feser), and...

Eric Steinhart wonders 'Why Materialism is Unscientific' (at Camels with Hammers), both in response to astrophysicist Ethan Siegel asking 'Can You Get Something For Nothing?' (both answer No).

Jeremy Stangroom wonders, 'A First Unmoved Mover?' (at Talking Philosophy). He shows Copleston and Russell describing the Atomists differently; not because either was bad at the history of philosophy, but because good answers to the question Why should there be something rather than nothing? do include Because the metaphysically necessary being is perfectly good as well as the (now) obvious No reason.


......the Living-room (mind)

Kenny Pearce knows that 'Sometimes it's Rational to act Arbitrarily' (at Kenny Pearce). 'In ordinary cases it is irrational to take a certain course of action when you know there is a better one available to you,' but what if you are asked to choose any natural number of dollars? (Sobel thinks that choosing anything would still be irrational; why would he think that?)

Constantine Sandis entertains 'Enchanting Causes' (at Flickers of Freedom), and so 'tests our intuitions about what sort of desire makes an action intentional.'

Joel considers 'Killing a Vegan: Degrees of Subjectivity' (at Florida Student Philosophy Blog), arguing that chickens (as opposed to Vegans) may not feel phenomenological pain, because they don’t have the 'I' concept, or the neurological ability to do much more than react physically.


......the Kitchen (moral philosophy)

Robin Hanson asks 'What Virtue Privacy?' (at Overcoming Bias), and by discussing Thomas Nagel's 'Concealment and Exposture' argues 'that humans had huge heads to subtly evade social norms while pretending to enforce them.'

Tim Dean considers 'Morality, Health and Sam Harris' (at Ockham's Beard), arguing that Harris's moral realism makes naturalism harder to defend, and suggesting that we could just say that 'Being animals, we pursue health. And being social, we pursue morality.'

James Gray defends moral realism against Hume, by considering 'Intrinsic Values & Beliefs About Reality' (at Ethical Realism).

Antti Kauppinen explains 'How the Experience Machine Works' (at Experimental Philosophy), before objecting to Felipe de Brigard’s recent ex-phi objection to Robert Nozick’s result.

Richard looks at 'Natural Agents and Status-Quo Bias' (at Philosophy, et cetera), questioning Carolina Sartorio, who argued (via Trolleys) 'that we need stronger reasons to justify interfering in a process (e.g. deflecting a trolley) than to justify abstaining from such involvement.'

Clayton Littlejohn has 'Ethical Intuitions (II): Cosmic Coincidence' (at Think Tonk), the second in a series of posts on moral epistemology (some empirical arguments having been considered in 'Ethical Intuitions (Part I)'): A version of intuitionism on which moral properties supervene upon natural properties is defended against Matthew Bedke.

Jussi Suikkanen conjoins 'Deliberative Contractualism and the Conditional Fallacy' (at PEA Soup), arguing that the former (by Nicholas Southwood) commits the latter.

Thom Brooks announces 'Thom Brooks on "Punishment: Political, Not Moral"' (at The Brooks Blog). British Hegelians make Alan Brudner's retributivism more attractive, apparently.

Chris Bateman considers 'A Categorical Imperative for the Other' (at Only a Game), suggesting that 3 formulations of Immanuel Kant's categorical imperative are more easily seen to be equivalent if they (or something like them) are derived from Emmanuel Levinas' concept of the Other.

Anders Sandberg wonders how much 'Intolerance we ought to encourage?' (at Practical Ethics). 'At the very least we can make it a social rule that just as we frown at racist, sexist or homophobic statements we frown at pseudoscience or deceptive evidence.'


......the Backyard (other)

Paul Newall examines different views of 'Astrology and its problems: Popper, Kuhn and Feyerabend' (at The Kindly Ones), and suggests that 'the philosophical problem for astrology is not that it can always explain failures (Popper) or that it does not attempt to solve problems (Kuhn) but instead that it has stagnated (Feyerabend).'

Kieran Healy looks at 'Gender divides in Philosophy and other disciplines' (at Crooked Timber). More than 70% of US PhDs in psychology were awarded to women, and it's 60% in sociology. Still, it's only 40% in political science, and 30% in philosophy.

Brian Leiter also looks at 'Women in Philosophy in the US' (at Leiter Reports), and finds that the proportion teaching philosophy is only about 20%. That's pretty good, given our context (it's more than 5 times the proportion of female bloggers here).

Gary Williams has some 'Thoughts on Cordelia Fine's new book Delusions of Gender' (at Minds and Brains); e.g. 'Maybe 1000 years in the future there will be an equal amount of male and female physicists, philosophers, and computer scientists,' because our brains are (equally) plastic.

James Warren reveals 'Rejection letters of the ancient philosophers' (at Kenodoxia). Bitchin'


.........is there 'More on philosophy and society'? No, because the party's over (hopefully before the fighting starts). Almost all-male, and the kitchen the most popular place; what a party. But if this carnival bored (or annoyed) you, or if your entry bounced (for no good reason), the solution is to host a carnival, which you should also do if you liked this one, of course: No hosts = no carnivals.
......And whenever you find yourself reading an interesting post, of a philosophical nature, you should submit it, because no posts = no carnivals. Carnival #122 will be at Ichthus77

Is "pretty" pretty?

Are any words pretty? Maybe not (outside of calligraphy or song). But I’m reluctant to say that “pretty” isn’t pretty because it is not too odd-looking (as words go) and it does make us think of prettiness. Still, I am reluctant to say that it is pretty, and so it seems to me that “pretty” is about as pretty as not. Perhaps you think that “pretty” is pretty. Or perhaps you think that “pretty” is not pretty. But if about as many people think it is as think it is not, and lots of people have no strong opinion either way, then a good case could be made for “pretty” being about as pretty as not.
...... Descriptive accuracy is, in general, a matter of degree. E.g. “is long” is not a long English predicate, while “is so far from being short that, not only is it not short, it is rather long for a predicate written in ordinary English” is quite a long English predicate, and so it seems fairly plausible that there might be some predicate that means the same as “is long” and which is about as long as not. For just one more example, “is boring” seems a little interesting, now that I come to think about it, although it does not hold my interest for long.
...... Words are heterological if they do not describe themselves, and so it seems to me that “pretty” is about as heterological as not. And if words are heterological insofar as they don’t describe themselves very well, then the fact that if “heterological” is heterological then it is not heterological, and the fact that if it is not then it is, those facts show that “heterological” is about as heterological as not. Those two facts are called the Grelling-Nelson paradox, and they are a version of Russell’s paradox.

The most significant version of Russell’s paradox concerned collections. When we refer to some things collectively, we are referring to their collection. E.g. “all the words in this post” refers us to all of the words in this post (including those words). Some collections include themselves (e.g. the collection of all the collections would include itself), but most do not (e.g. the collection of all the words in this post is not itself a word in this post). Russell’s paradox is that the collection of all the collections that do not include themselves would include itself if it did not include itself, and would not include itself if it did. Mathematicians tend to think that Russell’s paradox shows that we should only use well-defined kinds of collections. But if we could not talk about things that we were finding it hard to talk about, we would never learn anything.

A similar resolution of the Grelling-Nelson paradox would be that it shows that the word heterological is not a well-defined word. Perhaps Grelling and Nelson (who were mathematicians) should not have introduced that word. But what is wrong with saying that “heterological” is about as heterological as not?

Classical Logic: how is it correct?

According to Stewart Shapiro, Classical Logic is first-order (and hence formal) predicate logic, which he describes in some detail (in that SEP entry), having first noted that:

Formal languages, deductive systems, and model-theoretic semantics are mathematical objects and, as such, the logician is interested in their mathematical properties and relations. Soundness, completeness, and most of the other results reported below [in that SEP entry] are typical examples. Philosophically, logic is the study of correct reasoning. Reasoning is an epistemic, mental activity. This raises questions concerning the philosophical relevance of the mathematical aspects of logic. How do deducibility and validity, as properties of formal languages--sets of strings on a fixed alphabet--relate to correct reasoning? What do the mathematical results reported below [in that SEP entry] have to do with the original philosophical issue?

Shapiro goes on to list some possibilities; e.g. perhaps "the components of a logic provide the underlying deep structure of correct reasoning."
......Another possibility is that "because natural languages are vague and ambiguous, they should be replaced by formal languages," or rather (since formal languages are all defined using natural languages) "regimented, cleaned up for serious scientific and metaphysical work." Now, scientists often do define their own scientific terms; but how could that process apply to logic? Informal logic must be good enough for us to work out the correct formal language to use, if there is one (otherwise our justifications would become circular). "Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects." Such a view makes sense, in view of the many logics studied by logicians; but what, then, can we say about logic in the sense of correct reasoning? We use such a logic when we use any mathematical model scientifically; we must reason correctly about the model. Is classical logic a good model of informal logic?
......Despite the various logics that logicians work on, most arguments are presented in a classical logical style (even those about other logics). So presumably we do think that such components cut our reasoning pretty close to its joints, so to speak. Is classical logic correct? Some philosophers say so (e.g. Alexander Pruss recently gave that as a reason for rejecting open future views, in comments on this post of his), but I wonder how it is. It's hard for me to specify my worries without having already resolved them; but let's look at some simple logical arguments, such as might be used to introduce classical logic, and see how they may fail to be examples of classical logic (without going too far into the related problems of metaphor and vagueness).
......Grass is green, and all flesh is grass, so, is all flesh green? That's clearly invalid, the second premise being metaphorical; but did we use classical logic to work that out? And suppose I was holding something that wasn't green; could I deduce that it wasn't grass? Again no, because it's not true that all grass is always green. But what if all Blurps were always green; could I deduce that I wasn't holding a Blurp? I don't see why not. And surely I could put that argument in classical logical form. And yet if "is green" is a classical predicate, then not only what I'm holding, but anything and everything is either green or else not green (LEM); whereas, in reality, there's clearly a shading from green, through greenish and vaguely greenish colours, to those that aren't green. Is it that classical logic is a good model, but that real (informal, or natural-linguistic logic) must be slightly different? If so, what's the point of all that maths that Shapiro introduces (rather well)? I'm not saying there's no point to it. (I'm hoping it's not turning philosophy into a pseudo-science.) I'm rather asking whoever's reading this post, what do you think about classical logic?

Reasoning badly from Yablo’s paradox

Paradoxes can be hard to resolve, so it can be hard to reason well from them. A nice example is a recent argument that the past is finite, by Laureano Luna (2011: ‘Reasoning from paradox’, The Reasoner 5(2), 22–23). I shall vary the details. Suppose there’s a place where, once a year, every year, someone says “No previous utterance here was true,” with nothing else ever having been said there. Details can be varied, so long as we would, were the past infinite, have an infinite sequence of similar utterances. Indeed, it’s because we can vary the details that the following contradiction seems to follow from supposing the past to be infinite (rather than from, say, supposing that language need not begin with evident truths).
......Each utterance in that place concerned the past, so it seems that each utterance should be either true—were none of the previous utterances there true—or else false—were at least one of them true. But none of them can be either, on pain of Yablo’s paradox: Were no utterance there true then, via what each said, each would be true (and not true); but were any of them true, then since none of the earlier ones would have been true, its immediate predecessor would also, via what it said, have been true (and not true).
......But even if such sequences of utterances are impossible, the past might be infinite. One possibility is that simply infinite sequences, e.g. the natural numbers, are indefinitely extensible (are Potential Infinite) in the sense that while there’s always a next element, e.g. a bigger number, there’s no complete collection of them all. Standard mathematics assumes that such isn’t the case, but we’ve yet to discover that it isn’t. And if it is, then although we naturally think of past years as stretching back in time forever, the past couldn’t be the whole of such an infinite sequence, and so our infinite sequence of utterances would’ve been impossible too. And yet the past might, even so, be infinite. E.g. there might have been, before the Big Bang, some infinitely slow process, which took an infinite time to complete (and before which there might have been something else, possibly with no beginning); such a process has an infinite duration in the sense that we might go any natural number of years back into it and not reach its beginning, and also in the sense that were it the unit of time, all the time since the Big Bang would be relatively infinitesimal.
......Another possibility is that truth, or descriptive accuracy, is essentially a matter of degree. We might take an utterance of “No previous utterance here was true” to be asserting that none of those previous utterances described the past well enough for it to be classed as true. Yablo’s paradox would then be ruling out every possibility except the possibility that all those utterances described the past only vaguely, that they were all vaguely true. (Does it seem that they would then have been failing to describe the past well enough to be classed as true? If so, note that the suggestion is that either classification—true or not—would be less accurate than that of vaguely true.) So, our contradiction may well have been due to our having used, in effect, a rather artificial language. So we seem to have shown only that either the language of those utterances doesn’t allow such sequences of sentences, or something else (e.g. maybe the past must be finite, or maybe the natural numbers are indefinitely extensible).

Philosophers' Carnivals: Now & Next

Philosophers' Carnivals "showcase the best philosophical posts from a wide range of weblogs," as it says on the carnival's homepage. From today, carnival #120 is at nicomachus.net. And carnival #121 will be here in 3 weeks time, so if you find yourself reading something nicely philosophical, posted between now and then, please consider submitting it, via the online submission form, even if you wrote it yourself: "Don't be shy, we want to hear from you, that's the whole point of this project! Your post doesn't need to be anything earth-shattering - it just needs to be something that other philosophically-minded people might enjoy reading."
......As for what you can submit, there are No Rules, except: "No self-help, mysticism, marketing spam, etc." Of course, marketing spammers are unlikely to have bothered reading as far as this, so telling them not to bother submitting seems pointless. And I wouldn't rule out what some academic philosophers like to call 'mysticism', e.g. Mathematical Platonism, Substance Dualism, Open Theism and so forth (since such is just realistic metaphysics). Nor shall I reject whatever formalized craziness such academics work on instead, of course (since I should be unbiased in my hosting). Indeed, since the number of the carnival will be 121 (which sounds like "one-to-one") there's even some hope for self-helpers (and Continental Philosophers) whose positive thinking has carried them thus far, because insofar as their posts describe how the ideal of the Socratic Dialogue relates to their brand of self-help (or Derrida) I shall look upon them kindly.
......Here's a cautionary tale about rule-following: Many years ago, a port on the east coast was industrializing. To its north and south were two large estates, the country houses of two progressive squires, who built factories and docks in the port, and cheap housing for their workers there. Peasants to the west of the port flocked there, to earn more and to be free from their old-fashioned and relatively oppressive squire. As his peasants deserted his lands, that squire soon found himself with cashflow problems, and eventually he was reduced to opening his mansion to the public. He even built an inhumane zoo in its overgrown grounds; but things got no better. He got more and more depressed. One day he became quite deranged, and smashed up his zoo. Then he climbed onto the back of a huge hippopotamus and rode it towards the port. Now, the two rich squires heard of him crashing through their workers' slums, but they were unable to stop him because he had the law on his side, the law which states that the squire on the hippopotamus is evil to the slums of the squires on the other two sides.

Liars, Divine Liars and Semantics revisited

Divine Liar arguments aim to show that there’s no omniscient being—that no one knows all that’s true—in the following way. Suppose I say “No omniscient being knows that what I’m now saying is true.” If (as I believe) no one is omniscient, then no omniscient being exists, to know anything. So in that case, what I said was true. What I said was therefore an assertion, whether it was true or not. And if it wasn’t true—if it’s not the case that no omniscient being knows that what I said was true—then some omniscient being knows that what I said was true, despite it not being true, which is impossible (knowledge being of truths). So I asserted a truth; and so either that was a truth that some omniscient being doesn’t know, which is also impossible, or else there’s no such being.
 ......However, resolutions of the Liar Paradox might show that such arguments are invalid, e.g. according to Daniel J. Hill (2007: ‘The Divine Liar Resurfaces’, The Reasoner 1(5), 11–12) and my earlier article (2008: ‘Liars, Divine Liars and Semantics’, The Reasoner 2(12), 4–5). So, suppose I say “What I’m now saying isn’t true.” If what I said was true then, as I said, what I said wasn’t true. Does it follow that what I said wasn’t true? The paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. The resolution defended earlier by me (2008) takes my utterance to have been meaningless, so that I didn’t really say anything. But we may then wonder how it was that it seemed so clear what my utterance would have meant had it been true; and my Divine Liar utterance was even more obviously meaningful. Another popular resolution would regard my Liar utterance as equivocal, with the word ‘true’ naming many different predicates in Hill’s (2007) Tarskian hierarchy. But formal languages can only be defined via natural language; and my informal Divine Liar utterance wasn’t obviously that equivocal.
 ......Questions of truth are essentially questions of how well our words are describing the world. So insofar as my Liar utterance wasn’t meaningless, it was asserting that it wasn’t describing itself very well, not well enough for it to have been true. And since it was nothing if not self-contradictory, it certainly wasn’t describing itself very well. But therefore, in view of what it was asserting, it seems to have been describing itself quite well after all. Was it describing itself well enough for it to count as true? I’m reluctant to call it ‘true’ as follows. If it was true because it wasn’t, then it was true and not true, but surely something’s only not some way if it’s not the case that it is. Nor do I want to say that it was neither true nor not true, as that’s just to say that it was not true and also true. Nevertheless, my utterance wasn’t describing itself very well, and was therefore describing itself quite well; so perhaps it was only partially true. If so then calling it either ‘true’ or ‘not true’ would both be inaccurate, would both be only partially true.
 ......We naturally focus upon whatever truth we can find in what people say, or upon an obvious untruth. And things are usually described accurately enough for some obvious purpose, or not accurately enough. But would it be unrealistic to think of truth (descriptive accuracy) as a matter of degree? The classic example is that of Vann McGee (1991: Truth, Vagueness, and Paradox, Hackett, 217): If “Harry is bald” is true insofar as Harry is bald, ‘true’ should be at least as vague as ‘bald’. And quite generally, why should we believe that our words are much better defined than our purposes have required them to be? Maybe natural language has a ubiquitous—since usually unobtrusive—vagueness. (That would explain why the discovery of a contradiction so naturally triggers an attempt to clarify our terminology.) And in particular, the Liar Paradox might be revealing this ordinarily obscure vagueness of ‘true’. That’s because if my Liar utterance was only partially true, then it would follow from what I said only that it was also partially not true, which clearly coheres with it being only partially true. There’s no inconsistency—no more paradox—and it seems that much the same could be said of any Liar sentences.
 ......And if that is how the Liar Paradox should be resolved, then my Divine Liar utterance would have been only partially true if there is an omniscient being. My Divine Liar argument was therefore fallacious, because arguments should have premises that are unequivocally true enough to count as true under all relevant hypotheses. But if you asked an omniscient being whether my Divine Liar utterance was true, she might say that it contained an element of truth. That might be a more informative—more true and less misleading—answer than a simple ‘yes’ or ‘no’.
 ......Similarly, the best answer to the question “Is this colour blue or not?” could be to say that it’s vaguely bluish. Ordinary objects are almost always either blue or not, but colours don’t really divide into those that are blue and those that aren’t. On the two sides of any such line, between the blue and the other colours of some spectrum, would be colours that were indistinguishable. So there’s no such division; and so there’s some colour of which, rather than saying that it’s blue, or that it isn’t, we ought to say that it’s bluish. Note that such a colour might look blue against a background of colours that weren’t blue, or even if you just wondered whether it belonged to that class of colours, and so postulated it amongst them (cf. what we find paradoxical about the Liar Paradox).
 ......Incidentally, some formal work on ‘true’ as a vague predicate is well described as Fuzzy Logic.

Liars Are Fairly True

Suppose I say “what I’m saying isn’t true.” If what I said was true, then as I said, what I said wasn’t true. Does it follow that my words weren’t true? The famous paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. A fairly popular resolution takes my words to have been meaningless, so that I didn’t say anything. But if my words had been meaningless, you could hardly have known what they would have meant had they been true. Is our ordinary conception of truth shown by such Liar-style sentences to be deficient? Let’s see why not.
......To begin with, such sentences are in some ways like Truth-teller-style sentences. If I said “what I’m saying is true,” for example, what would I be saying? Not much. Questions of truth are essentially questions of how well our words describe the world, and “this is a good description” isn’t much of a description. Still, it might not be too bad a self-description, precisely because there isn’t much to describe. If someone saying “what I’m saying is true” intended to be speaking the truth, should we deny that she was telling the truth? It may be hard to say, but therefore it might be that such sentences are not so much vacuous as vague. Since “what I’m saying isn’t true” also addresses nothing but its own descriptive power, might it also be, in its own way, rather vague? Consider the following analogy.
......If I said of some colour, “I wouldn’t say that it’s blue,” I might not be saying that it wasn’t blue, because colours don’t divide into those that are blue and those that aren’t. To see that, consider a spectrum: On the two sides of any such line, between the blue and the other colours, there would be colours that were indistinguishable. So there’s no such division; so there’s some colour of which, rather than saying it was blue, or that it wasn’t, I’d prefer to say, more precisely, that it was bluish but not very blue. (Since the perception of colour is subjective, you might say it was blue, or that it wasn’t.) Our perception of colour is also context-sensitive, e.g. it’s affected by surrounding colours, and by our preconceptions. So if I wondered if our colour really was blue, I might thereby see it as not blue, while if I then wondered if it was therefore not blue, it might seem pretty blue (even to me).
......And similarly, it’s when “what I’m saying isn’t true” has been thought of as definitely not true that it seems most clearly to be true. More precisely, while those words aren’t giving us a very good description of their own meaning—they’re self-contradictory—we therefore have a description that isn’t too bad, insofar as it’s saying that it’s not a very good description. In short, they’re rather nonsensical (and false), but therefore fairly true (and false). And that’s basically how Liar-style sentences are compatible with our ordinary conception of truth. We need a bit more clarification, but it should soon become clear that while we can always be more precise, there’s no threat to truth here.
......What is truth, if not a sufficiently accurate description? Usually we describe things accurately enough for some obvious purpose, or else we don’t, so we tend to assume that truth is black-or-white. But it’s really a matter of degree, in a context-sensitive way. E.g. the table at which I’m writing this is flat enough for that purpose, so “this table is flat” is true enough, but might be false were I writing about geometry. And in general, our words tend not to be much better defined than our purposes have required them to be. So natural language has a ubiquitous—since ordinarily unobtrusive—vagueness (whence the way to resolve paradoxes, and uncover other fallacies, usually involves clarifying some terms). Of course, the words of “what I’m saying isn’t true” have clear enough meanings, so there’s no simple equivocation to discover. But it should help us to resolve the paradox if we don’t demand anything too unrealistic. (Similarly, we shouldn’t demand that colours be either blue or else not blue.)
......Liar-style sentences present themselves as misrepresenting themselves, so their meaning is self-undermining. And they can be read (or heard) in two basic ways—each a necessary part of the other’s context—because their meaning self-undermines in a loopy sort of way. Insofar as Liar-style sentences are true they’re also false, and they need concern nothing but their own truth, so they can certainly be read as nonsensical. But they’re not just senseless, and hence not at all true, because insofar as they’re not true they’re easily read as true. So they also have that sense. But they can’t be nothing but partly true and hence partly false, because that would leave nothing for them to be true or false about.
......This resolution—that Liar-style sentences are fairly true, in that loopy way (they’re fairly true because they’re rather nonsensical, and they’re rather nonsensical because insofar as they’re true they’re also false)—is a strengthened version of the resolution that takes them to be nonsensical. So for those who believe that an omniscient being is logically possible, it allows a similar reply to Divine-Liar-style sentences. E.g. the problem with “no omniscient being knows this” is that it can’t be true if there’s an omniscient being, but if it isn’t true then, since no one could then know it, it would seem to be true. My new reply is that if it’s only fairly true (in this loopy way) then no epistemically perfect being would have to know it, except to know it for what it is. And note that “no omniscient being knows any of this” is simply false, e.g. such a being would know those words. (Similarly, “what I’m saying isn’t at all true” is fairly false.)