Wednesday, April 06, 2011

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).

Tuesday, April 05, 2011

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.)

Monday, April 04, 2011

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).

Sunday, April 03, 2011

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.

Saturday, April 02, 2011

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.

Friday, April 01, 2011

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.