Imagine a rocket taking us away from the earth, a thousand miles away in the first half minute, another thousand in the next quarter, another in the next eighth, and the next sixteenth and so on, so that by the end of the minute every finite distance from the earth will have been surpassed: “At the end of the minute we find ourselves an infinite distance from the earth,” according to José Benardete (1964: Infinity: An Essay in Metaphysics, Oxford: Clarendon Press, p. 149). Accelerating beyond the speed of light is unrealistic, but not logically impossible: Benardete’s reasoning, about the nature of space, is not bad reasoning. So it is interesting that space as we naturally conceive of it does, as follows, need some such speed limit.
The meaning of ‘space’ comes, in the first instance, from our experiences of such spaces as those inside rooms, and those of the surrounding landscapes, all parts of an apparently boundless space: We can easily imagine going further and further in any direction, from any conceivable place in space. And if we try to conceive of space as having a boundary, then we naturally wonder what is on the other side of that boundary; the thought of a boundary to space is essentially the same as the thought of an impenetrable object occupying the space beyond that boundary. And it is similarly easy to imagine an object going faster and faster. But where does that get us?
Looking back towards the earth, from spatial infinity, we look through space that must have been traversed instantaneously at the end of that minute, because this space cannot be in that endless sequence of thousands of miles (if it was there, then it would be further away than any finite distance). Now, that sequence is a continuous stretch of space, with the earth at one end and some sort of endlessness facing us, and how could that endlessness possibly connect with the rest of the continuous stretch of space between us and earth? Clearly there can be no dividing line, because this sequence of elements of constant length cannot tend to one, and no final thousand miles was traversed. So, we cannot be an infinite distance from the earth.
The infinite speed that we would have ended up with is relatively reasonable, because over those thousands of miles we would have covered an infinite distance in a finite time, so that our average speed was already in that sense infinite. But it is certainly strange that averaging over speeds that were so very far from infinite should have resulted in one that was infinite. And in reality there is a light speed limit in space; and note that the necessity for some such speed limit would be compatible with similar spaces having higher speed limits. So, is such a speed limit indicated, or is there a better resolution?
Many believe that the rocket would instead vanish at the end of the minute. Vanishing at spatial infinity is not like vanishing into thin air, it is more like disappearing into the distance, cohering with intuitions about the unimportance of distant things. And if we centre coordinate axes on the earth, then the rocket will indeed have gone beyond their finite measures by the end of the minute, so that were we to think of space as all and only what is measured by such axes (as we may learn to do in mathematics) then it would indeed have vanished. But, if we centred coordinate axes on the rocket then it would instead be the earth that vanished; and it is of course absurd that the earth should vanish because it fired a rocket into outer space.
It might be objected that the rocket, not the earth, is moving away, and so the rocket, not the earth, should vanish. The spaces most familiar to us (rooms, roads, fields) do contain things that tend to be at rest (relative to the walls, the buildings, the ground) unless forced to move, so such spaces do seem to be stationary. Such was the ancient view of space; but, those spaces are parts of the surface of the earth, which spins and revolves around the Sun, relative to the Sun, and in that more modern view our familiar spaces are moving relative to the Sun. Space is neither stationary nor moving; rather, it is a space (an absence) in which objects move relative to each other.
Reference frames moving with constant speeds are privileged, so it might be objected that it is the rocket, not the earth, which is accelerating away. But in fact the earth is constantly accelerating around the Sun, which is similarly accelerating, while the rocket could have constant speeds almost all of the time. Furthermore, consider two identical objects moving apart in an otherwise empty space: Would they both vanish? But then, what if one of them had instead stopped and turned and followed the other at finite distances? Would both still vanish? That would only make sense if space was stationary. Would neither vanish? That would mean that the vanishing of one of them could depend on the motion of the other, after all.
So our problem cannot be solved by having Benardete’s rocket vanish (or teleport) at spatial infinity. We are therefore left with a paradox: Space must, but also cannot, contain infinite distances, if objects can go at any speed; there is this informal logical need for a physical speed limit. In view of the elementary nature of the above, I am tempted to speculate that Einstein, given only nineteenth century physics, may have noticed such a need. But in any case, the space in which we evolved does seem to be such that the objects within it cannot accelerate beyond the speed of light. So it seems reasonable to conclude that when we think about logically possible objects we should assume some such speed limit, in the interests of coherence.

## Monday, August 03, 2015

## Saturday, August 01, 2015

## Thursday, May 01, 2014

### Vagueness and Objectivity

Vagueness is a well known problem
in logic. Imagine, for example, a rough table-top being gently sanded flatter
and flatter. Eventually it will become flat (i.e. flat enough to count as flat
in some apposite context). However, since ‘flat’ is not so precisely defined
that sanding away a few scratches could be enough to flatten the table-top,
hence after each bit of sanding the table-top will still not be flat, from
which it follows that it will never be flat. That contradiction is a problem
that cannot be solved just by redefining ‘flat’ more precisely, because all the
terms of natural languages are, in such ways, at least a little vague, and it
is within such languages that we all reason. So, there is a borderline, between the table-top
being flat and it not being flat, that is more like a pencil line than a
mathematical line – there are borderline cases of flatness – but, there is no
region between the table-top being flat and it not being flat where it is
neither flat nor not flat, because in such a region the table-top would not be
flat and yet would be flat (which the meaning of ‘not’ rules out).
Nevertheless, it is logically possible for the table-top to be about as flat as
not. At such times it would not so much be false as

*only about as false as not*to say that it was not flat, and similarly, a little later, to say that it was flat (which resolves our logical problem). At such times, we might be more likely to say that the table-top was getting flat, since that would be true. We reason best with descriptions that are either true or else not true (false, in classical logic). Of course, ‘getting flat’ is no less vague than ‘flat’, but its borderlines are in different places; and in general, while we cannot remove all the imprecision from our languages, we can always move the borderlines out of the way of our logical language-use. Our words are defined as precisely as our purposes have required them to be, with the two classical truth-values – ‘true’ and ‘false’ – meeting at a place where descriptions are described as well by ‘not true’ as by ‘true’. We do not have to do much with such descriptions, other than identify them as needing to be replaced with truer descriptions, and so we need only add the following definition to the classical definitions of ‘true’ and ‘false’: 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 than that, then we need to avoid borderline cases and use classical logic. We do not need a formal definition of ‘as true as not’ (in some non-classical logic), because mathematical precision is inapposite when the sharp distinction between something being the case and it not being the case is absent. It would, in particular, be wrong to model the idea that self-referential claims like ‘this claim is not true’ are*about as true as not*as such claims having truth-values of 0.5, as the fuzzy logicians do. Now, while there are similar resolutions of the other semantic paradoxes (see other posts of mine), the set-theoretic paradoxes have no such resolutions: Sets are essentially non-variable collections and it makes no sense to think of a collection as being about as variable as not. That distinction, between semantic and set-theoretic paradoxes, originates with Frank Ramsey, who was a mathematical constructivist; and quite a few mathematicians believe that the set-theoretic paradoxes show that there are too many numbers – too many possible sizes of sets – for them all to exist as distinct numbers. But, such constructivism seems to clash with the objectivity of arithmetic: How could 2 exist but not, say, 4? Four is just two twos. So, most mathematicians think that the set-theoretic paradoxes should be showing something else, which may have motivated formalising the borderline truth-value in a mathematics that would then apply, instead of classical logic, to those paradoxes. But in fact, although the existence of whole numbers,*n*, is essentially the possibility of sets of*n*objects, and although such possibilities are intuitively timeless, such possibilities can emerge as distinct possibilities from more general possibilities. To see that, consider how the possibility of*you*would have been, had you never existed, the possibility of*someone just like you*: Looking back now, there was always the possibility of*you yourself*, as well as that more general possibility; but, there could have been no such distinction had you never existed. It is, then, logically possible for distinct numbers to emerge in an unending stream from some more indistinct coexistence – as possibilities inherent in the concept of*a thing*– and so a coherent story can be told of 1 + 1 = 2 existing – via the concept of*another thing of the same kind*– and 2 + 1 = 3 existing, along with the question of what 2 + 2 is, and only then 2 + 2 = 2 + 1 + 1 = 3 + 1 = 4 existing. Note that such a story might be more plausible were the small natural numbers replaced by large transfinite numbers. Furthermore, if the concepts involved were divine conceptions, then such arithmetic would be as objective as anything. So the main reason why the set-theoretic paradoxes are paradoxical is the prevailing atheism within science (which is all but a*reductio ad absurdum*of atheism).## 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 bald_{101}, 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:

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

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

......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?

## Monday, November 18, 2013

### I think, so I'm iffy

"

......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

......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 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

Subscribe to:
Posts (Atom)