Friday, February 28, 2014

Who's Afraid of Veridical Wool?


I have been taking an informal approach to the Liar paradox, for the following reasons. After much thought, I find self-descriptions like ‘this is false’ to be about as true as not. I am therefore beginning, with the following – previously posted – post, with the equally ancient paradoxes of vagueness. And my approach is informal because I find the precision of mathematical logic to be inapposite when there is no sharp division between something being the case and it not being the case. Although the literature on these paradoxes has become increasingly formal, following Bertrand Russell’s interest in Georg Cantor’s mathematics (at the start of the twentieth century), we do not need non-classical logic to resolve them, I think; rather, we need to focus on the context of classical logic, natural language, in which the paradoxes are expressed. Below, and temporally prior to, ‘Vagueness’, I have posted ‘Liar Paradox’ and ‘Cantor and Russell’.

It was via Russell that I came to consider the Liar paradox, having developed an interest in Cantor because of qualms about the fitness of the real number line as a model of actual continua, which developed as I did my MSc in Mathematics (at the end of the twentieth century). With this post I have come to the end of my journey; I am left wondering why our mathematics became set-theoretical, and then category-theoretical, and similarly, why our natural philosophy became the physicalism of Einstein et al, and then string-theoretical. How well, I wonder, will our democracies be able to regulate the biotechnical industries of this century? I have serious doubts, stemming from my research into physics, theoretical and empirical, and from the history of our regulation of financial industries (which are surely less complex). Still, in the absence of any interest in my research, I have been developing more aesthetic interests over on Google+ (see sidebar:)

Vagueness


It is, of course, when our words describe the world that they are true. So for example, ‘Telly is bald’ was a true description of Aristotelis Savalas when he was a baby. (As he himself said, “We’re all born bald, baby.”) Now, Telly did not go from being a bald baby to not being bald by growing just a few hairs, because ‘bald’ has not got so precise a definition. So if, as seems possible, Telly did not suddenly grow a lot of hair, then he will only gradually have stopped being bald. There could, possibly, have been times when ‘Telly is bald’ was true, later times when ‘Telly is bald’ was not true, and times in between when something else was the case – or could there? If ‘Telly is bald’ was neither true nor untrue at those intermediate times, then ‘Telly is bald’ was not true and was true, which is ruled out by the meaning of ‘not’.

So, such intermediate times seem to be logically impossible. And yet, we can hardly know a priori that Telly suddenly grew a lot of hair. And while we can introduce new terms that are less vague than ‘bald’ – e.g. 100 hairs or less and you are bald101, otherwise you are not – that would hardly solve our problem with ‘bald’. So let us assume, for the sake of argument, that Telly stopped being bald gradually: What was going on at the intermediate times? Well, some of those around Telly may have been thinking of him as bald, while others thought of him as not bald. And the vagueness of ‘bald’ gives us no reason to think that any of them were wrong. But, Telly was certainly not very bald at such times, and nor was he clearly not bald, so why not think of him as having been about as bald as not? Were ‘is bald’ about as true as not of Telly, ‘Telly is bald’ would not so much not be true as be only about as true as not, and it would not so much not be untrue as be about as true as not. So, that would solve our problem.

We reason best with descriptions that are either true or else not true, but the words of natural languages are a little vague,1 so the two classical truth-values, ‘true’ and ‘false’,2 meet at a place – in logical space – where descriptions are described as well by ‘not true’ as by ‘true’. For another example, imagine a rough table-top being gently sanded flatter and flatter. Eventually it becomes flat enough to count as flat, in the usual contexts. But sanding away just a few scratches would hardly have flattened it, so the borderline between flat and not flat is more like a pencil line than a mathematical line. Our table-top will, briefly, be only vaguely flat, or about as flat as not. ‘Flat’ is not, in that sense, well defined: It is a vague predicate, not a definite predicate. But, there is a sense in which it is defined perfectly well: There are such things as tables, which are flat by design; and there are, similarly, bald men. Precisely redefining ‘flat’ and ‘bald’ – and ‘man’ and ‘table’ – in order to avoid the problem of vagueness would lose us some of our ability to refer to reality. Indeed, we would lose rather a lot of that basic function of language, because most of our words are to some extent vague.

This also solves such puzzles as the Sorites: We might suppose, for example, that the truth-value of ‘the table-top is flat’ could not change with the sanding away of a single scratch. If so, then gently sanding a rough table-top for even a very long time could not make true ‘the table-top is flat’. But, while ‘not flat’ is contradicted by ‘flat’, it is not necessarily contradicted by ‘about as flat as not’. So as the table-top begins to be about as flat as not, we would not be wrong to call it ‘not flat’. Our calls could change from ‘not flat’ to ‘about as flat as not’ in the blink of an eye, with no sanding at all. (Our original supposition is less plausible when there is a borderline truth-value.)

There seems to be a ubiquitous vagueness in natural language, but it is not really a problem. It is surprising, but only because it is so unproblematic that it usually goes unnoticed. Our words are defined as precisely as our purposes have required them to be, and the slight vagueness means that we can always make them more precise. When ‘Telly is bald’ becomes problematic, for example, ‘Telly is getting hairy’ will be more straightforwardly true. ‘Getting hairy’ is hardly less vague than ‘bald’, but its borderlines are in different places. We can usually move the borderlines out of the way, even though we cannot remove all the vagueness. And since we do not have to do much with descriptions that are about as true as not – other than identify them as needing to be replaced with truer descriptions – hence we need only adjoin ‘about as true as not’ to the classical truth-values. Indeed, we should only do that: The precision of formal logic is inapposite when we have left behind the sharp division between something being the case and it not being the case. A more formal definition could only be an inaccurate – if deceptively precise – mathematical model of the most natural definition.

Following Aristotle, the classical definitions are as follows. To say of what is the case that it is the case, or of what is not the case that it is not the case, that is to speak truly. And to say of what is the case that it is not the case, or of what is not the case that it is the case, that is to speak falsely. So an adequate adjunct could be: To say of what is about as much the case as not that it is the case, or that it is not the case, that is to say something that is about as true as not. A description that is much truer than not will be true enough to count as true, by definition of ‘much’, while one that is not much truer than not will be about as true as not by definition of ‘about’. And if we need to make sharper distinctions, then we need to avoid borderline cases and use classical logic.3

Now, descriptions are normally of other things, but self-description is allowed – e.g. ‘this is in English’ is a true self-description – so consider this example: This description is true. Let us call that self-description ‘T’ (for Truth-teller). T says only that T is true, so it is certainly possible for T to be true; but another possibility is that T is false, because if T was false then it would follow from the meaning of T (that T is true) only that T was not true. And since there is no more to T than that – since T does nothing but describe itself (as true) – hence there is no reason why T should be true rather than not true, or false rather than not false. So it would make sense were T about as true as not.
Furthermore, some self-descriptions are paradoxical if they are not about as true as not (the post below concerns the Liar Paradox).

Notes

1. Bertrand Russell, ‘Vagueness’, The Australasian Journal of Psychology and Philosophy 1 (1923), 84–92, reprinted in Rosanna Keefe and Peter Smith (eds.), Vagueness: A Reader (Cambridge, MA: MIT Press, 1997), 61–68. For the state of the art, see Richard Dietz and Sebastiano Moruzzi (eds.), Cuts and Clouds: Vagueness, Its Nature and its Logic (New York and Oxford: Oxford University Press, 2010).

2. ‘“X is Y” is false’ just means that X is not Y, so in classical logic, where either X is Y or else X is not Y, ‘false’ and ‘not true’ are interchangeable.

3. A good introduction to the mathematics of classical logic is Stewart Shapiro, Classical Logic.

Liar Paradox


The Liar paradox concerns such assertions as this: The assertion that you are currently considering is not true. Let us call that assertion ‘L’. L says that L is not true, so if what L says is the case, then L is not true. But statements are true if what they say is the case, so L would also be true. Does it follow from that contradiction that what L says is not the case? But if it is not the case that L is not true, then L is true. And if any statement is true, then what it says is the case. So in short, L is true if, and only if, L is not true.

That is paradoxical because we expect L to be either true or else not true. But, if L was about as true as not, then it would follow – from the meaning of L (that L is not true) – only that L was about as untrue as not (about as true as not). And that is a general linguistic possibility (see Vagueness). Now, since L asserts that L is not true, L asserts that it is not true that L is not true – i.e. it asserts that L is true – as well as that L is not true. And that is worrying, because ‘L is true’ would be the negation of ‘L is not true’ were ‘L’ naming a classical proposition; but, classical logic would not apply to L were L about as true as not. And while it would certainly be an unusual fact about such self-referential denials – that as they deny that they are true they thereby assert that they are – it is not too odd. On the contrary, it would help us solve the main problem facing any resolution, the so-called ‘revenge’ problem:

Consider the following self-description (call it ‘R’): The description that you are now reading is not at all true, not even about as true as not. If R was about as true as not, then it would be false – not about as true as not – that R was not even about as true as not. But, R is the claim that it is not at all true that R is not at all true – i.e. that R is to some extent true – as well as that R is not at all true, so if R was about as true as not, then although it would be false that R was not even about as true as not, it would be true that R was to some extent true. R would appear to be, not so much false, as about as true as not. Or, would R rather seem to be both true and false? But R, like L, makes only one assertion – that it is itself untrue – the meaning of which includes it not being the case that it is not true.

The thought that L is both true and false does not necessarily contradict the present resolution, though. If a description is about as true as not, then it is about as true as not that it is true, and it is about as true as not that it is false. Furthermore, since most philosophers think that L is certainly not true (whatever else it is), hence the fact that some philosophers – e.g. Graham Priest – think that it is true (and false) just adds to the plausibility of its being about as true as not. Still, there is only some truth to Priest’s resolution,1 according to the present resolution. To see why, it may help to consider the following version of the paradox: Is the answer to this question ‘no’? Questions of the form ‘is X Y?’ want answers that are either ‘yes’ (X is Y) or else ‘no’ (X is not Y), but the answer to our question cannot be ‘yes’ (that would mean that it was ‘no’), and it cannot be ‘no’ (that would mean the answer was not ‘no’). It would be coherent to reply that the answer is to some extent ‘no’, because it is not just ‘no’, it is to some extent ‘yes’, because it is to some extent ‘no’. And it would be natural for us to shorten that to ‘yes and no’. But, that cannot mean that the answer is, and at the same time is not, ‘no’; it can only mean that the answer is to some extent ‘yes’ and is to some extent ‘no’.

There is also some truth to the resolution that sentences like ‘this description is not true’ cannot be used to make assertions: They cannot be used to make classically logical assertions. But, there is surely only some truth to this resolution. I can say ‘what I am now saying is not true’ and mean by those eight words that what I am thereby saying is not true. Neither the fact that I am thereby saying that it is not true that what I am saying is not true, nor my belief that what I am saying is only about as true as not, stops me using those eight ordinary words to assert that what I am saying with them is not true.

There is also some truth to the resolution that adds ‘neither true nor false’ to the classical truth-values (according to the present resolution), because when a description is about as true as not, it is neither true enough nor false enough for classical logic. But again, there is only some truth to that resolution. It is only about as true as not to say that it is not true, and only about as true as not to say that it is not false. A more sophisticated version replaces ‘true’ and ‘false’ with ‘certainly true’ and ‘certainly not true’, and then adjoins ‘possibly, but only possibly, true’ to those. But maybe those are more like belief-states than truth-values. A more formal approach models ‘true’ by 1 and ‘false’ by 0 – as in Boolean algebra – and then uses a continuum of numbers to bridge the gap – a so-called ‘fuzzy logic’2 – but again, those are more like probabilities than truth-values.

Still, the fuzzy logical resolution is not too odd: L being true insofar as it is not true does imply that L is as true as not, which is well modelled by a truth-value of 0.5. Nevertheless, if truth is not so much a matter of degree as a fundamentally black-and-white affair with an indistinctly grey boundary, then L being as true as not would not mean that L was exactly as true as not, so much as about as true as not. To see why, it may help to consider the following version of the paradox. According to Peter Eldridge-Smith,3 there is a possible world in which Pinocchio’s nose grows if, and only if, he is saying something that is not true, but no such world in which he says ‘my nose is not growing’ because his nose would then be growing if, and only if, it was not growing. Our world is quantum mechanical, though. So it is possible for objects to be in entangled states, and so it is logically possible for Pinocchio’s nose to be as much growing as not. And such states are most accurately described with probabilities. But even if Pinocchio’s nose was growing exactly as much as not, his ‘my nose is not growing’ would have to have the borderline truth-value of the language of his ‘my nose is not growing’.

Many resolutions of the Liar paradox have been investigated. But the explanatory power of the present resolution is only enhanced by those alternatives: If the present resolution is true, then as we have to some extent already seen, there is some truth to those alternatives, which goes some way towards explaining why each of them was suggested; and furthermore, most of them promise a way around a highly unattractive mathematical proof – a proof of the temporality of number (aka Cantor’s paradox) – which the present resolution does not. See post below, Cantor and Russell (posted prior to this:)

Notes

1. For Priest’s resolution, as one formal system amongst many, see §4.1.2 of J.C. Beall and Michael Glanzberg, Liar Paradox.

2. Petr Hajek, Fuzzy Logic.

3. Peter and Veronique Eldridge-Smith, ‘The Pinocchio Paradox’, Analysis 70 (2010), 212–215.

Cantor and Russell


Georg Cantor was a brilliant nineteenth century mathematician whose discoveries led to the foundation of mathematics becoming axiomatic set theory.1 Cantor’s paradox concerns the collection of all the sets. Now, a collection is just some things being referred to collectively, of course, and a set is basically a non-variable collection. (Sports clubs and political parties are variable collections, for example, while chess sets and sets of stamps are non-variable.) And numbers – non-negative whole numbers – are basically properties of sets. Cantor’s core result was an elegant proof that even infinite sets have more subsets than members (in the cardinal sense of ‘more’). It follows that if there was a set of all the other sets, then it would have more subsets than members, whence there would be more sets than there are sets – each subset being a set – which is impossible. And it follows that there is no set of all the other sets. That consequence is known as ‘Cantor’s paradox’; but, how paradoxical is it? Presumably {you} did not exist until you did, so why expect the collection of all the sets to be non-variable?

A more paradoxical consequence is, I think, that there is no set of all the numbers – for a proof of that consequence, see section 3 of my earlier Who's Afraid of Veridical Wool? – which is paradoxical because we naturally think of numbers as timeless, whence their collection should be non-variable. To see the problem more clearly, suppose that 0, 1, 2 and 3 exist, but that as yet 4 does not; the problem is: How could 2 exist, but not two twos? The existence of n (where ‘n’ stands for any whole number) amounts to the existence of the possibility of n objects, e.g. n tables (were physics to allow so many), and possibilities are, intuitively, timeless: For anything that exists, it was always possible for it to exist.

Nevertheless, if there are too many numbers for them all to exist as distinct numbers, perhaps they are forever emerging from a more indistinct coexistence. Possibilities are not necessarily timeless. You were always possible, for example, but that possibility would – had you never existed – have been the possibility of someone just like you. Looking back, there was always the possibility of you yourself, as well as that more general possibility; but had you not existed, there could have been no such distinction. Note that if the universe had bifurcated into two parallel universes, identical in all other respects, then the other person just like you would not have been you. And however many parallel universes there were, another would not appear to be logically impossible. So, it appears to be logically possible for apparently timeless possibilities – e.g. the possibility of you yourself – to emerge as distinct possibilities from more general possibilities.

Those taking numbers to be timeless also have some explaining to do: They need to find a plausible lacuna in the Cantorian proof that numbers are not timeless. But, what they have found is more paradoxes akin to the Liar. Cantor’s paradox concerned the set of all the other sets because the set of all the sets would have had to contain itself as one of its own members, and we do not normally think of collections like that. But as Russell thought about Cantor’s counter-intuitive mathematics, he considered the collection of all the sets that do not belong to themselves: If that collection was a set, then it would belong to itself if, and only if, it did not belong to itself. That is basically Russell’s paradox. Like Cantor’s, it is not obviously paradoxical – it just means that there is no such set – but Russell thought of sets as the definite extensions of definite predicates, and predicate versions of his paradox are more obviously paradoxical. E.g. consider W.V.O. Quine’s version: ‘Is not true of itself’ is true of itself if, and only if, it is not true of itself. That is paradoxical because we naturally assume that ‘is not true of itself’ will either be true of itself, or else it will not. But if predicate expressions can be about as true as not of themselves, then it would follow from the meaning of ‘is not true of itself’ that insofar as ‘is not true of itself’ is true of itself it is not true of itself, and that insofar as it is not true of itself it is not the case that it is not true of itself. And it would follow that ‘is not true of itself’ is about as true as not of itself.

Russell also thought of definite descriptions as names, and the English name for 111,777 – one hundred and eleven thousand, seven hundred and seventy seven – has nineteen syllables. According to Russell, 111,777 is the least integer not nameable in fewer than nineteen syllables, and Berry’s paradox is that ‘the least integer not nameable in fewer than nineteen syllables’ is a description of eighteen syllables.2 Again, that is not very paradoxical; we can always use a false description as a name – cf. ‘Little John’ – and ‘John’ can name anything in one syllable. But consider the following two sentences.3 The number denoted by ‘1’. The sum of the finite numbers denoted by these two sentences. The first sentence denotes 1, so if the second sentence denotes anything, then it denotes a finite number, say x, where 1 + x = x, and there is no such number. So if the second sentence denotes anything, then it does not denote anything. But it cannot simply fail to denote, because if it does not denote anything, then the sum of the finite numbers denoted by those two sentences is 1. Since the second sentence denotes 1 if, and only if, it denotes nothing, perhaps it denotes 1 as much as not. Cf. how ‘King Arthur’s Round Table’ began as a definite description and ended up referring more vaguely.

In stark contrast, the set-theoretic paradoxes – e.g. Cantor’s – do not have resolutions akin to the present resolution of the Liar paradox: How could a collection of numbers be as variable as not? (Collections of numbers are not like collections of noses, so it could not be like Pinocchio’s nose.) Those paradoxes do have a fuzzy logical resolution, via fuzzy sets, though. And those taking L to be true and false can find it true and false that some collections belong to themselves. And those taking natural Liar sentences to be nonsensical often have a formalist take on infinite number. And of course, if the set-theoretic paradoxes have the same underlying cause as the semantic paradoxes – as Russell thought – then they should all be resolved in similar ways. But, if there are two kinds of paradox here – as Ramsey thought – then the inability of the present approach to resolve the set-theoretic paradoxes would hardly count against it. On the contrary, that inability would amount to some evidence for it, by helping it to explain the attractions of the major alternatives, especially the formal ones: A non-classical logic would be very useful were one trying to fly in the face of a mathematical proof.

Notes

1. For a detailed history, 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 and Oxford: Princeton University Press, 2000).

2. Attributed to G.G. Berry by Bertrand Russell, ‘Mathematical Logic as based on the Theory of Types’, American Journal of Mathematics 30 (1908), 222–262.

3. Based on Keith Simmons, ‘Reference and Paradox’, in J.C. Beall (ed.) Liars and Heaps: New Essays on Paradox (Oxford: Clarendon Press, 2003), 230–252. Simmons’ version was more complicated, and omitted the crucial word ‘finite’.

Friday, November 29, 2013

Who's Afraid of Veridical Wool?

Do we need non-classical logic to resolve the Liar paradox? Or do we need to see that the natural context of classical logic – natural language – has a slight but ubiquitous vagueness? When something is as much the case as not, it is a borderline case; and similarly, self-descriptions like ‘this is false’ are about as true as not. And when there is no sharp division between something being the case and it not being the case, then the precision of any mathematical logic is inapposite.
......The modern literature on the Liar paradox is very formal, but following Russell there has been a related effort to resolve Cantor’s mathematical paradoxes, which may well explain that. My analysis of the Liar paradox has 4 sections: Vagueness (1,300 words), Liar Paradox (1,300), Set-Theoretic Paradox (1,200), Semantic Paradox (900), plus notes (700)...

......Who's Afraid of Veridical Wool?

..........................PDF

Monday, November 18, 2013

I think, so I'm iffy

"I deliberate, so the future is open," is, if you think about it, a pretty good description of a rational argument with one premise (a premise of which one can be certain). My making the effort to deliberate well (because I would blame myself if I did not) presupposes that there is, as yet, no fact of the matter of what I will be thinking.
......To make such an effort is to force the future away from a state that it would otherwise be in, of course. And for me to think of that state as already unreal would undermine my motivation to make such an effort. And of course, for me to make no such effort would be for me to care little for the quality of my thoughts, which would be irrational.
......That was a précis of my comments on a Prussian post, themselves inspired by Nicholas Denyer's 1981 defence of arguments like "I deliberate, so my will is free."

Wednesday, November 13, 2013

The Use of Brackets...

...to squirrel away, via RSPB blogs on 21 Oct and 8 Nov :)

Tuesday, July 16, 2013

All Men Are Men

Suppose that you are thinking of having a child:
......Your child will be like you, to some extent, and will to the same extent be like his or her father, just as half of your genes come from your father and half from your mother. So, your child's genes will be 50% your man's, 25% your father's, and 25% your mother's. And of course, what goes for you goes for your mother, and for hers, etc. So:
......Your child's genes will be 50% your man's, 25% your father's, 12.5% your maternal grandfather's, 6.25% your maternal grandmother's father's, 3.125% your maternal grandmother's maternal grandfather's, etc.; i.e. they will be 50% male genes + 25% male genes + 12.5% male genes + 6.25% male genes + 3.125% male genes + ... = 100% male genes.
......Everyone has a biological father and mother, and so we have a mathematical and, to some extent, empirical (and of course fallacious) argument that all men are men.

Monday, July 08, 2013

A peculiarity of the Liar paradox

Consider the following sentence: “The self-referential statement expressed by this sentence is not true.” Taking the phrase “this sentence” to refer, self-referentially, to that very sentence, the most obvious meaning of that sentence is that it is not the case (is not true) that the self-referential statement expressed by that sentence is not true. But that is just to say that the statement expressed by that sentence is true, which is the negation of the obvious meaning of that sentence.
......Since the statement expressed by that sentence is both that such and such is the case and that it is not the case, which is self-contradictory, it may well follow that the statement in question is false, as suggested by Dale Jacquette (2007: ‘Denying the Liar’, Polish Journal of Philosophy 1, 91–8). But other philosophers – amongst whom I would once (five years ago) have counted myself – think that because an assertion that such and such is the case is clearly different in meaning to an assertion that it is not the case, such Liar sentences do not express any proposition at all, but are rather meaningless nonsense.
......However, I argued recently that Liar statements are in fact as true as not, and that the Liar paradox is, in that sense, a typical semantic paradox (for details see my The Liar Paradox, and my On the Cause of the Unsatisfied Paradox, in the April and June issues of this year's The Reasoner respectively); whereas, the problem above seems to be unique to the Liar paradox, e.g. it does not arise with Yablo’s paradox, in which there is no self-reference. So, I am wondering how else we might address this part of the Liar paradox.
......Could the problem be due to substitution failure? Perhaps replacing “this sentence”, in the sentence in question, with a near-copy of the sentence itself – the only difference being that ‘that’ replaces ‘this – changed the proposition expressed by that sentence to its negation. Similar failures can occur with propositional attitude reports, e.g. consider the difference between “Lois believes that Clark is thirsty” and “Lois believes that Superman is thirsty”; for details see Jennifer Saul (2007: Simple Sentences, Substitution, and Intuitions, OUP). But then, Liar sentences are sentences of a very different kind; they need only involve a self-referential name, e.g. ‘L’, plus ‘is’, ‘not’ and ‘true’.
......Another possibility is that Liar statements are identical to their negations. As a rule, the negation of a proposition is a different proposition, of course; but, propositions are either true or else false, as a rule, whereas we are now looking at propositions that are as true as not. Now, an elementary part of language is the subject-predicate description, “S is P” (e.g. “that salmon is pink”), and so a simple model of truth might use strips of paper with “S is P” on one side of the strip and “S is not P” on the reverse side, for all S and P in some simple language: All the strips with non-fictional S get stuck onto the things of the world, with “S is P” uppermost if S is P and “S is not P” uppermost if S is not P. We might extend that model to include cases where S is as P as not by giving the strips a twist in the middle before sticking them down, and by including non-fictional strips with no worldly referent, such as “100 is a round number that is also a square.” And then we might think of our Liar sentence as being like a Möbius strip, the twist due to its being as true as not, and the joining of its ends being due to its being self-referential.
......In any case, this peculiarity of the Liar paradox gives us an easy answer to the Revenge problem for this resolution of the Liar paradox, which is as follows: If “what I am saying is not true” is as true as not, then what about “what I am now saying is not only not true, it is not even as true as not”? Were that about as true as not, what it said would seem false. But, what it said was that it was not at all true that what was said was not at all true, so it said not only that what was said was not at all true, but also that it was to some extent true. So if it was as true as not, then although it would indeed seem to have been false – false that what was said was not even as true as not – it should also, and to the same extent, appear true – true that what was said was to some extent true – whence it should seem to have been as true as not after all.

Wednesday, May 29, 2013

The Set-theoretical Paradoxes

I have another piece on semantic paradox in The Reasoner in June; but, what about the set-theoretical paradoxes? The seminal paradox of Bertrand Russell (1902: ‘Letter to Frege’, in 1967: Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 124–5), for example, concerns the class of classes that do not belong to themselves. (In this context, classes are extensions of predicates: all and only the things that satisfy the predicate belong to the class.) Some classes – e.g. the class of humans – do not belong to themselves – the class of humans is a class, not a human – and the class of all such classes is paradoxical: it belongs to itself if, and only if, it does not. Russell conceived this paradox when thinking about the set-theoretical paradoxes, because a class is a kind of set; but, Russell’s paradox can also be expressed directly in terms of predicates:
Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. (Ibid, 125)
Note that “is a human” is a predicate expression, not a human, so it does not describe itself; the question is, what about “does not describe itself”? It describes itself if, and only if, it does not, which is paradoxical (it is widely known as Grelling’s paradox). But, we might as well say that “does not describe itself” describes itself insofar as it does not, from which it follows that it describes itself as much as it does not. So the question arises, could the class of classes that do not belong to themselves belong to itself as much as not? Classes can be like that, e.g. the class of all men would be like that if some hominids had been about as human as not. If we call one such hominid ‘Strider’, then it was about as true as not that Strider was human. Were there no such hominids, then some human would have had non-human parents.

That – the fuzzy set – resolution of Russell’s paradox coheres rather well with my preferred resolutions of the semantic paradoxes – e.g. The Liar Paradox, from The Reasoner 7(4) – but, the set-theoretical paradoxes originally arose from the mathematics of Georg Cantor, and they concerned, not classes of classes, but numbers of numbers. And of course, numbers are far from fuzzy. You and I, for example, are two people, and there is surely no doubt that we know what is meant by ‘two’ (for all the uncertainty over Strider's personhood). So, let us begin with 0, 1, 2, 3 and so forth, the products of the process of adding 1 to the previous number, starting with 0. Rather trivially, the collection of all those numbers is all of them, referred to collectively; and while some collections – e.g. stamp collections – are variable, if a collection is non-variable then we can say that it is a set (as in “a set of stamps”). On this conception, a set is some particular number of logical objects. To include the numbers 0 and 1, and so make this conception more like the standard conception (and also simplify proofs), let us also include logical objects that can play the role of singletons – sets with a single element – and Ø, the empty set. So, given that we have a set {0, 1, 2 …}, it contains some definite number of numbers, say א (aleph) of them.
......Cantor showed that every set has more subsets than it has elements, in the cardinal sense of ‘more’. (Two sets have the same cardinal number of elements when the elements of each set can all be paired up with those of the other (cf. Hume's principle).)
Let S be any set, and let P (for ‘powerset’) be the set of all its subsets (including Ø and S). If S and P had the same cardinality, then there would be one-to-one mappings from S onto all of P, so let M be one such mapping. Let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it, then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. So, D is contradictory, and so there is no such M. So S and P do not have the same cardinality, and since P contains a singleton for each element of S, P is bigger than S.
So, {0, 1, 2 …} has beth-one subsets, where beth-one is bigger than aleph, and the set of all those sets has beth-two subsets, and so on. If that endless sequence of bigger and bigger sets is a non-variable sequence, then there is a union – a set of the elements – of all those sets, which is even bigger, with בω (beth-omega) sets. (Omega is the ordinal number of the sequence 1, 2, 3 and so forth.) And that union has בω + 1 subsets, and so on: for any such set there is the set of its subsets, and for any endless sequence of such sets there is, if it is a non-variable sequence, its union. In total, there is a sequence of sets – and a corresponding sequence of numbers, the sizes of those sets – which must be variable; were it not, we would have moved on from that ordered set of sets to its union, and thence to the subsets of that union (and so on). But of course, it is paradoxical that our total sequence of numbers is variable – is of necessity growing forever – because few of us think that numbers that do not already exist could suddenly appear. Suppose, for example, that the number 101 had not always existed; would that not mean that there was once a time when there were no such possibilities as, for example, the possibility of 101 Dalmatians? And note that this paradox cannot be resolved as Russell’s paradox was resolved above, because the idea of something being as variable as not is nonsensical.

Nevertheless, the intuition that numbers are atemporal is not unquestionable, because new possibilities can be constructed out of more general possibilities. You were always possible, for example, and yet the possibility of you in particular was only distinct from the more general possibility of people just like you once you existed (to be directly referred to). And it is not too odd to think of arithmetic as constructed from such logical concepts as those of possibility and class. E.g. the obvious meaning of “2 + 2 = 4” is that if we had two things of some kind, then if we got another two of that kind we would have four. So, it is conceivable that, while 101 Dalmatians were always possible, there was once a time when that possibility only existed as part of a more general possibility (of bigger numbers). Such constructivism can be defended atheistically – e.g. see George Lakoff and Rafael E. Núñez (2000: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, New York: Basic Books) – and theistically, e.g. see Paul Copan and William Lane Craig (2004: Creation out of Nothing: A Biblical, Philosophical, and Scientific Exploration, Grand Rapids, MI: Baker Academic).
......Whether we are atheists who believe that the human brain evolved in a finite world, or theists who entertain divine ineffability and infinitude, we would have such reasons to doubt that we could ever be justifiably sure about the nature of infinity, though. And another reason why we should keep an open mind about this is that, while it is clearly counter-intuitive to think of the finite cardinal numbers as temporal, it is, if you think about it, no less counter-intuitive to think of them as atemporal. E.g. the arithmetic of such numbers as א and ω is very different to that of the finite cardinal numbers, whence the theoretical behaviour of that many objects is counter-intuitive. Hilbert’s famous Hotel can be built upon Galileo’s paradox, for example. And the difference between cardinal and ordinal arithmetic gives rise to the counter-intuitive behaviour of my quasi-supertask (2003:Infinite Sequences, Finitist Consequence’, British Journal for the Philosophy of Science 54, 591–9). And the infinite set of the finite cardinal numbers covers the whole range of the finite (in units), and yet every one of those numbers is infinitely far from infinite, whence Lévy’s paradox. For more examples, see José Benardete (1964: Infinity: An Essay in Metaphysics, Oxford: Clarendon Press), and Peter Fletcher (2007: ‘Infinity’, in Dale Jacquette, Philosophy of Logic, Amsterdam: Elsevier, 523–585).
......Intuitively, numbers are timeless; but while it is certainly possible that there is a set of all the finite cardinal numbers, it is also possible that there is not. Both possibilities are counter-intuitive, so both can be supported in ways that would seem compelling were it not for that ‘both’. So, one might think that modern mathematics would have been based on results that follow, not just from one, but from both possibilities. However, such is not the case. Now, the ubiquity of the standard real number line might be explained by its being easy to use, simple and familiar, but there is a similar bias towards assuming that there is a non-variable collection {0, 1, 2 …} in such fundamental research areas as theoretical physics and pure mathematics, which is puzzling. For clues, see Ivor Grattan-Guinness (2000: The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel, Princeton University Press), and Peter Markie (2013:Rationalism vs. Empiricism’, in Edward N. Zalta, The Stanford Encyclopedia of Philosophy).

It might be objected that there is no real puzzle because {0, 1, 2 …} is not an informal set in pure mathematics, but is an axiomatic set defined by means of a formal logic. However, that would be to ignore, not to explain, the mystery. We know perfectly well what the cardinal numbers 0, 1, 2, 3 and so forth are; if we had some axioms that did not describe them, we would not throw those numbers away and start using those axioms instead, however nice their formal properties were. To do so would hardly be scientific.
......Perhaps I should add that we do not get a third kind of set-theoretical paradox from the axiomatic conception. Paradoxes arise when we have contradictory beliefs, and formal structures have no intrinsic meaning; formal axiomatic sets only give us mathematical models of set-theoretical paradoxes. So, while it is true that paradoxes can be avoided if we use formal sets, we did not really resolve the set-theoretical paradoxes by moving from naïve set theory to axiomatic set theory.