The no-no Paradox

Roy Sorensen (2001: Vagueness and Contradiction, Oxford: Clarendon Press, 169) considered the following pair of sentences:
......The neighbouring italicized sentence is not true.
......The neighbouring italicized sentence is not true.
While it is logically possible that one of those sentences – or rather, one of the claims made by those sentences – is true and the other false, those two tokens of that sentence-type should have the same truth-value because there are no significant contextual differences between them. It is therefore plausible that each is as true as not. For each token, were the claim expressed by it as true as not, the other claim would be as untrue as not, which clearly coheres with it too being as true as not.

The Unsatisfied Paradox

In this month's issue of The Reasoner (page 185), Peter Eldridge-Smith gave the following informal description of his Unsatisfied paradox:

My favourite predicate just happens to be 'does not satisfy my favourite predicate'. Crete satisfies 'does not satisfy my favourite predicate' iff Crete does not satisfy my favourite predicate. Therefore, Crete satisfies my favourite predicate iff Crete does not satisfy my favourite predicate.

And not just Crete, there is no thing that satisfies Peter's favourite predicate, and no thing that fails to satisfy it without it also not being the case that it fails to satisfy it. Nevertheless, Peter's favourite predicate could be as true as not of Crete, or anything else. Predicates can do that, e.g. 'is blue' is as true as not of an object that is as blue as not, and some predicates apply equally to all things, e.g. 'is a thing' is true of all things.

The Pinocchio Paradox

Suppose that Pinocchio's nose grows if, and only if, he says something that is not true, and that he says "My nose is growing". Then his nose is growing if and only if it is not growing. (This paradox originated with Veronique Eldridge-Smith.) According to Peter Eldridge-Smith:

The Pinocchio scenario is not going to arise in our world, so it is not a pragmatic issue. It seems though that there could be a logically possible world in which Pinocchio’s nose grows if and only if he is saying something not true. However, there cannot be such a logically possible world wherein he makes the statement ‘My nose is growing’.

In the world in which Pinocchio's nose grows and shrinks in such a way, suppose that he says, of various uniformly coloured objects, that they are blue. What happens if the object is as blue as not? (There must be such colours, because otherwise some colour that is blue is the same colour as some colour that is not blue.) Well, whatever happens, that could also be what happens when he says "My nose is growing". It is, for example, possible that Pinocchio's nose is in a quantum-mechanically entangled state, as much growing as not. That seems to be a logically possible world.

Curry's paradox

Consider C, which says that if C then B, where B says that black is white. Suppose that C is true. Then it is true that C implies B. So if C is true, then B is true. But therefore C is true. So B is true. That in a nutshell is Curry's paradox. Every step in that argument that black is white appears to be a logical step.
......But, ordinary modal logic operates in a space of descriptions that are either true or else false. So perhaps we should not have supposed, to begin with, that C is true, because C is true only insofar as it is not true (because insofar as it is true we get a contradiction) and so it is as true as not. That is a consistent possibility because it is as true as not that a contradiction follows from a statement that is as true as not (since it would follow from a statement that was false). And it resolves the paradox because ordinary logic breaks down with propositions that are only as true as not (e.g. see the Sorites paradox, higher-order vagueness, and the revenge problem for this resolution of the Liar paradox).
......Curry's paradox shows that we should not even suppose the truth of some conditionals that are as true as not. The grammatical structure of C is therefore less logical than that of, say, A, the claim that A is not true, even though you could paraphrase A as the claim that if A is true then pigs fly. We suppose, for the sake of argument, the truth of claims like A when we reason about the Liar paradox, but we should not do that with the paraphrase. Cf. the peculiarity of the Liar, that A claims that A is true, since it claims that it is not true that A is not true.

Revenge revisited

The obvious resolution of the Liar paradox is that such self-descriptions as “this claim is not true” are as true as not. That resolution is obvious because it follows immediately from the problem, which is that insofar as they are not true, such claims are true. That is how the problem should be expressed, rather than as paradoxical biconditionals, if truth might be a matter of degree. And if a uniformly coloured object is as blue as not, then prima facie “the object is blue” is as true as not, so that is a prima facie logical possibility. Indeed, since the Liar paradox is a paradox, its less obvious resolutions are all very implausible, and so it is itself an indication that truth is in general a matter of degree.
......But, there is a fly in the ointment. What about “this claim is not even as true as not”? Were that claim as true as not, it would be false that it was not even as true as not. That is the revenge problem for this resolution. The problem is that insofar as the revenge claim is at least as true as not, it is false. But it follows that, while it is a bit more false than true, it is still about as true as not. So where is the problem? The revenge claim is almost as true as not. Now, if we use “vaguely true” to mean about as true as not, then we have another revenge claim, “this claim is not even vaguely true”. Insofar as that claim is vaguely true, it is false. It follows that it is as false as it is vaguely true, though. So this claim is just a bit more vaguely true. Again, where is the fly?
......It is trapped in the bottle of ointment, loudly buzzing. The buzz is that it is clearly not the case that “this claim is not true” is true. Were it true, it would be as it says it is, it would not be true. So it is not true. Even if it is as true as not, it is still not true enough to count as simply true. And furthermore, if “this claim is not even vaguely true” is vaguely true, then it what it claims is simply not true. How could it help to talk of the more vaguely true when “vaguely true” is such a vague term? Or so the buzzing goes. But, it is precisely because “vaguely true” is vague that there is no contradiction. The buzzing is understandable, but it does not spoil the ointment. It is understandable because we tend to imagine that things are black-and-white when we try to think logically.
......Consider the uniformly coloured object that is, in some contexts, as blue as not. Of course, blue is not normally a matter of degree. Oxford blue is deeper, not bluer, than Cambridge blue. “Oxford blue is blue” is not truer than “Cambridge blue is blue”. But, as greenish blue shades into bluish green there is bound to be a colour that is about as blue as not. There would otherwise be a blue colour that was the same colour as a colour that was not blue. And of course, a blue-green object will in some contexts look blue, e.g. when surrounded by red balls. But our object is in a context where it is as blue as not, so it does not look very blue. So I do not want to say that it is blue. If anything, I want to say that it is not the case that it is blue. But tempted as I am to say that it is not blue, the fact is that “the object is not blue” is, like “the object is blue”, only as true as not.

Simmons' Paradox

Keith Simmons told this story about ten years ago:

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.

The question is, what is X when X = pi + 6 + X, and the obvious answer is “infinity”. More precise possibilities include omega, aleph-null, beth-null, aleph-one et cetera; and because there are lots of possible answers, the “the” at the start of the third expression is a little deceptive. It is a bit like asking for the name of the king of France: the obvious reply is “which king of France?”
......What if “numbers” (in that expression) was replaced with “finite numbers” though? If the third expression with “finite” added does not denote anything (much as the expression “the present king of France” denotes nobody) then the sum of the numbers denoted by expressions on the board would be pi + 6, so the third expression should then denote pi + 6, whence it should denote pi + 6 + pi + 6, and so forth. If the third expression denotes anything, then it denotes something else, and if it denotes nothing then it denotes something. We seem to be stuck with a logical impossibility if the third expression is a denoting expression. Should we conclude that it is not a denoting expression?
......The problem is that the third expression is obviously a denoting expression. The light on the horizon is that reference is in general a matter of degree. To why, it may help to imagine a man staggering through a desert and seeing a mirage, which he takes to be a pool. Coincidentally there is a pool just where he takes one to be, but it is obscured from his view by the mirage. As he staggers on he is constantly thinking “that pool looks cool” rather obsessively. And as he nears the pool its image gradually replaces the illusory one without him noticing. So, the referent of “that pool” gradually changes to the pool, and so he will at some point have referred to the pool only as much as not.
......What did “that pool” denote when the man was referring to the pool only as much as not? I would say that it denoted the pool, but only vaguely. And similarly, the third expression with “finite” refers us to pi + 6 insofar as it does not, so it refers us to pi + 6 as much as not. It denotes pi + 6, but only vaguely.

Why was the Big Bang not a Black Hole?

A documentary about what happened before the Big Bang was repeated on the BBC last night, and it got me wondering: why did anything happen after it? Why was the Big Bang not a Super-Duper-Massive Black Hole? There was all this matter, all the matter in the universe, in this tiny, tiny space; so why the explosion? Why an inflationary explosion? And why is the universe still accelerating? Did dark energy make it all happen? Dark energy sounds like a physics of the gaps!
......I had already been wondering why an amount of antimatter equal to the observable matter of the universe would not be in the form of an uncollapsed standing wave (like electron shells around atomic nuclei). The popular theory of where all the antimatter went is that there was originally a lot more extra matter and an equal amount of antimatter which annihilated each other. But that would just create a lot of heat and light, none of which could escape a Black Hole. But, were the antimatter in a standing wave, then the uncollapsed antimatter suffusing the primordial atom would make it effectively massless, so there would be no Black Hole, while the repulsive force between the matter and the antimatter would cause an explosive expansion. Furthermore, the appearance of dark matter would be explained; while the standing wave would enforce a certain uniformity, much as the inflationary period is supposed to have done.
......I have not heard of any such theory, so that thought is not even philosophy of physics, but listening to the physicists in that documentary made me wonder whether there might be such a theory. The things they were saying were pretty off the wall (according to each other).

Is the ontological argument a logical proof?

Given that a being than which none greater can be imagined does exist in the mind, as an idea,
and given that a being that exists in reality is greater than a being that only exists in the mind,
does it follow that a being than which none greater can be imagined actually exists in reality?

Or does it only follow that the idea of a being than which none greater can be imagined
is the idea of a God that exists in reality?
First things first:

Does a being than which none greater can be imagined exist as an idea in your mind?

Presumably you would need to know what "greater" means in this context. Does it mean more powerful, wealthier, more popular, bigger, more tasteful, more politically correct, morally better, better looking, all of those and more, or just some of them? Perhaps the main thing here is that if there is a God in reality then we would be unable to imagine any greater being. Still, it is conceivable that God could imagine a being greater than Himself, because if there is a God then He may well be capable of feats much greater than any of us could imagine.
Second things second:

Is a being that exists in reality greater than a similar being that only exists in the mind?

Suppose that there had been a real Robin Hood, instead of our fictional Robin Hood. We might never have heard of that real Robin. And Robin Hood's greatness is presumably his power to inspire. So it is quite plausible that that hypothetically real Robin would not be greater than the fictional Robin that we know and love. Still, the main thing here is that a real God would presumably be greater than any fictional God.
So thirdly:

Could an ontological argument be a logical proof?

Surprisingly the answer is "yes": a very well-defined ontological argument could be a formal-logical proof that something called "God" has something called "existence" (and nowadays "logic" does usually means formal logic).

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.)