Saturday, October 15, 2016

Telling the Truth


The following sentence occurs in the preface of a book: “All the sentences in this book are true.” That sentence is saying that all the other sentences in that book are true, whence it is too. Were all the other sentences true, it would be silly to say that that one was false just because it was logically possible for it to be false (if it is false then not all the sentences are true), and so if the other sentences were all true, then that one would be too.
The Truth-teller is the self-referential bit of that sentence: “This sentence is true.”
The Truth-teller is saying only that what it is saying is true (which all sentences implicitly assert anyway), so it is not saying much, and so there is not much for it to be true or false about. It would be consistent for it to be true, and consistent for it to be false, but what could determine which it is? Maybe there is no fact of the matter, what it is. So, is that sentence neither true nor false? But then “This sentence is true” would clearly be false (and the fact that it would refutes the idea that it says nothing at all). Still, it is not saying much, and so it is not very true and not very false. It is therefore fairly false that it is true, and so what little it is saying – that it is true – is fairly false.
The Truth-teller is not saying much, but what it says is more false than true. It may therefore be the case that the Truth-teller is not very true and only about as false as not. To see why that might be possible, consider a man going bald: As he goes bald he will not, by the loss of just one or two hairs, become bald and so there might be an intermediate or overlap stage at which he is about as bald as not, when it would make sense for it to be about as true as not that he was bald, and about as false as not. There is a lot to be said for, and against, such a possibility; here it would be most apposite to look at the Truth-teller’s paradoxical companions.
The following sentence occurs in the preface of a work of fiction: “None of the sentences in this book are true.” If that sentence, say Sent, was true, then none of the sentences in that book would be true, and so Sent in particular would not be true; and that contradiction means, by reductio ad absurdum, that it is not the case that Sent is true. So either Sent is false – in which case at least one of the other sentences would have to be true – or else it is neither true nor false; and it could of course be the case that none of the other sentences are true, so Sent is neither true nor false. But that cannot be because Sent is meaningless, because we have just been reasoning logically with its meaning.
Sent does have two obvious meanings, though. As well as the one we have been working with (its literal meaning), it was clearly supposed to be saying that none of the other sentences in that book were true. And since it did express that latter meaning (which we might call its literary meaning) clearly enough for us to notice, hence it did also have that meaning. So, if none of the other sentences were true, then it would be true that none of them were true, and so in a sense Sent would be true (with the literary meaning) and not false; and it would false that there was not even that truth, and so in a sense Sent would not be true (applying the literal meaning) but would be false. It follows that Sent is false: If it is not false because one of the other sentences is true, then it is still false because it is in another sense true.
Working with only the literal meaning was paradoxical, though; it left us with something like a reductio of the logical presumption that sentences are either true or else not true. Logic, and language itself, suit veracity and falsity being black and white, but there is also vagueness; and we do want to think something veridical about paradoxical sentences. So, the actual reductio above should be replaced by: Insofar as Sent is true, Sent is not true. It follows that Sent, taken literally, is about as true as not. Still, it is more realistic to read it as being true if none of the other sentences are. So for simplicity, let us consider the paradoxical bit by itself: “This sentence is not true.”
That sentence (the Liar) is saying that what it is saying is not true. Unlike the Truth-teller, it is not just saying something that all sentences implicitly assert, so it does not seem to be saying nothing; and note that if it was saying nothing, then it would not be saying anything true, whence it would clearly be true. So, Liar is paradoxical; insofar as Liar is true, it is not true, and insofar as it is false it is true. It follows that Liar is about as true as not, and about as false as not: It is to some extent true, that it is not true, because it is only to some extent true; and it is not exactly false that it is not true, because it is to some extent false.
For a final complication: “This sentence is not true, and not even about as true as not.” If that sentence, say Even, is about as true as not, then Even is saying something false. Nevertheless, Even, by saying that Even is far from true, is thereby saying that it is far from true that Even is far from true. So although Even is saying something that is false, it is something that is also true, and about as true as not. Consequently Even can be about as true as not, and about as false as not.

Friday, September 09, 2016

Proof's Nearest Kin


Paradoxes are akin to proofs: We have a paradox when we have a very good argument for something that is beyond belief, a proof when we have a logical argument for something not too odd. Many a paradox is therefore a proof by reductio ad absurdum of the negation of its weakest premise. Georg Cantor’s famous paradox of the 1890s was exceptional, being a logical argument for a contradiction, but it thereby proved that human reasoning is not perfectly logical. In response the twentieth century saw a proliferation of formal logics, but as we develop such calculi with mathematical precision we might easily forget that some illogicality is unavoidable in our reasoning. To remind us, then, the following is the essence of Cantor’s paradox.
Consider any 3 things, e.g. a chair, a plate and a fork. There are clearly 3 different ways of making a pair from those 3 things (e.g. the chair and the fork are, collectively, a pair), each of which derives from, and is defined by, the presence of 2 particular things in our original collection of 3 things: Given those 2 things, we have that way of making a pair. Now, making a pair is just one way of making a selection. There are 8 different ways of making selections from our original collection because there are 23 ways of assigning the labels “In” and “Out” to 3 things (e.g. the chair has “In,” the plate has “Out,” and the fork has “In”). Given our original collection, there are also those 8 combinatorial possibilities, those 8 ways of making selections, ways that are entirely grounded in our original things and which are therefore distinguished from each other whether there is a selector who can make those selections or not. Let us call them possible selections, from our original collection, and say that they are collectively the selection collection for our original collection. In general, for any natural number n, if there is a collection of n things, then there is also a selection collection of 2n possible selections from it.
We can safely assume that there are 2 things (e.g. you and I are 2 people), so there is also the selection collection of 4 possible selections from those 2 things, and the selection-collection of 16 possible selections from those 4, and so on. All those possible selections are there already, intrinsically distinguished from each other, and so there are infinitely many things, which are certainly things in the weak sense that there are numbers of them, in the fairly weak sense of cardinal number. Two collections of things have the same cardinal number of things when there are 1-to-1 mappings from each collection onto the other. Cardinality is fairly weak, e.g. there are clearly fewer prime numbers than natural numbers in some stronger sense, but it is an equivalence relation – it is reflexive, symmetric and transitive – and so it partitions collections into equivalence classes. For any collection of things, T, there are possible selections from T – each corresponding to some combination of as many “In” and “Out” labels as there are things in T – even if T is infinite, and so there is a selection collection, S(T), of all the possible selections from T. And for the following reason (which is essentially Cantor’s diagonal argument) every selection collection is cardinally bigger than its original collection.
Suppose that S(T) has the same cardinality as T. There would then be 1-to-1 mappings from T onto all of S(T). So let M be one such mapping. We might then use M to specify D as follows: For each thing in T, if the possible selection that M maps that thing to includes that thing (in other words, if that thing has the label “In” in that possible selection) then D does not include it (i.e. it has the label “Out” in D), but otherwise D does; and there is nothing else in D. Since the only things in D are things in T, D should be in S(T); but according to its specification, D would differ from every possible selection that M maps the things in T to, and so D would differ from everything in S(T). D is therefore contradictory, and so there is no such M. Consequently S(T) does not have the same cardinality as T. And since S(T) contains at least one thing for each thing in T – e.g. the possible selection whose only “In” label is assigned to that thing – and cardinality is an equivalence relation, hence S(T) is bigger than T.
Given you and I, then, there is some infinitely big collection, say N, and so by the diagonal argument above with T = N there is also a bigger collection, S(N), and by the diagonal argument with T = S(N) there is the even bigger S(S(N)) = S2(N), and there is similarly S3(N), and so on. All the things in all those collections are collectively the union of those collections, U, which is bigger than each of those Sn(N), for natural numbers n, because it contains all the things in each S(Sn(N)). Furthermore S(U) is even bigger, and so on; so there is also the union, say V, of all the Sn(U) for natural numbers n. And again, S(V) is even bigger, and so on and so forth. Now, each of the possible selections that such endlessly reiterated selection collections and infinite unions would or could ever show to be there is already there, as a combinatorial possibility that is implicitly distinguished from all the others of that kind, and so they are collectively some collection, C, of all the possible selections of that kind. But by the diagonal argument, their selection collection, S(C), would contain even more things of that kind. That contradiction shows that we went wrong somewhere, but why should even highly evolved primates know where?

Monday, May 23, 2016

On the Famous Proof that the Natural Numbers are Indefinitely Extensible


There are clearly numbers of things in the world, e.g. you and I are two individuals. And to think of us in that way is essentially to think of us as the elements of a pair-set, {you, I}. So let us say that a natural set is any whole number of things. Note that {you, I} is also a subset, there being other people. And note that numbers and sets are themselves things in the sense that there are numbers and sets of them.

Do mathematicians discover facts about such natural numbers as 2? If so, then Georg Cantor’s famous paradox of the 1890s was essentially a mathematical proof that the natural numbers (i.e. 0, 1, 2, …) are ever-growing in number, because what Cantor did was to obtain a contradiction from the assumption that there is a set of them all. I will go through the mathematical steps of the proof below; but to begin with, Cantor’s proof was paradoxical because each whole number is essentially the logical possibility of that many things, and if something is ever going to be possible, then it was never logically impossible.

Cantor’s proof has therefore been taken to be a refutation of the assumption that mathematicians discover facts about such natural numbers: Most twentieth-century mathematicians defined the natural numbers, e.g. 0 is usually defined to be the empty set within an axiomatic set theory (usually ZFC), 1 to be {0} and so on. Axiomatic set theory is used because Cantor’s proof used a natural kind of set that has been blamed for the paradox (and called “naïve”). But such sets exist as clearly as natural numbers do and so, like the natural numbers, they should not be defined, but scientifically described.

That is because there is at least one way in which the natural numbers could, possibly, be ever-growing in number, because logical possibilities can become more fine-grained over time. Suppose that time is not so much like space that future events are already there (at future times), and consider any existing man. He was always possible, but it is only with hindsight that we can describe the logical possibility of his existing with such direct reference to him. Before he existed there was only the possibility of someone just like him, to whom we could not directly refer.

The logical possibility of n things, for each natural number n, could, then, have originally been part of the logical possibility of numbers of things, only becoming the logical possibility of things in that number when that number was created, perhaps as part of the analysis of the concept of a thing by some creative power that exists primarily in a world of spiritual stuff (and which may therefore appear triune to us), thereby creating an abstract realm of sortal natural kinds and their associated numbers prior to the physical objects and incarnate creatures of this world. Note that what matters here is only that that is a logical possibility (not even Richard Dawkins claims that God is impossible, only that He almost certainly does not exist).

Let us, then, assume that there is a natural set of all the natural numbers, N = {0, 1, 2, …}. Clearly the subset {0, 1, 2} is already part of N, as is every other subset; all the subsets of N are there implicitly, and so there is the set of all of them, P(N), the power set of N. Cantor showed that P(N) is cardinally bigger than N (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) by way of a diagonal argument; and both of those steps generalise: Given any set, there are implicitly all of its subsets, so that there is also its power set, to which the following diagonal argument applies.

Let S be any set, and let P(S) be its power set. If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S), so let us assume that they do and 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. 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. Consequently D is contradictory, and so there is no such m, and so S and P(S) do not have the same cardinality. And since P(S) contains a singleton for each element of S, hence P(S) is cardinally bigger than S.

So as well as N, there is also P(N), and P(P(N)) and so forth, an infinite sequence of power sets. Consequently there is also the set of all the elements of all of those sets, U, their union. U is cardinally bigger than each of those power sets because it contains all the elements of the power set of each of them. And of course, P(U) is cardinally bigger than U. And so on (there is another infinite sequence of power sets, then another union, and eventually an infinite sequence of unions that we can also take the union of, and so on).

There must be a set, T, of all that could possibly be found in that way (via power set and union), because all of it is already there to be found. But if T is a set, then P(T) contains cardinally more of precisely those sorts of elements; and that contradiction means that we went wrong somewhere. And from our assumption of N we made only logical moves, so it must have been that assumption that was false: The natural numbers are certainly ever-growing in number (in time that is not much like space).

Tuesday, May 10, 2016

What is Proof?

Over a hundred years ago, Cantor proved that the natural numbers are temporal: Assume, with Plato and against Aristotle, that they are not temporal, so that they all coexist, insofar as numbers do exist (the main thing is that we can count them, e.g. {1, 2, 3} are three numbers). Since they all exist atemporally, so do all their subsets (e.g. {1, 2, 3}), and so there is a set (an atemporal collection) of all the subsets of that set. By a simple diagonal argument (which you can google) that set of subsets is of larger cardinality than the original set. (Two sets have the same cardinality when the elements of one of them can be put into some one-to-one correspondence with the elements of the other.) And the set of all of the subsets of that set of subsets is of even larger cardinality, because the diagonal argument applies quite generally to any set and the set of all its subsets. (This gives us three equivalence classes of infinite sets, associated with three infinite cardinalities.) So, we can get cardinally bigger and bigger sets in that way. And when there is an endless sequence of such sets, the union of all of them will also be an atemporal collection, because each of those sets was implicit in the previous set, and it will have a cardinality larger than any of those sets, because each is followed in that sequence by sets of larger cardinality. And from that union we can again consider the set of all its subsets, and so on. Now, all these atemporal collections are implicit with the set of natural numbers, so they all exist (insofar as such things do exist) atemporally; but, Cantor proved that they cannot all exist atemporally: Suppose they do. Then there is a set of them all. But implicit in them is the collection of all of their subsets, which would be cardinally more of them, whereas we have assumed that we had the set of them all. So, we have assumed that the natural numbers are not temporal, and obtained a contradiction; that is a classical mathematical proof of the temporality of the natural numbers. However, most people assume that numbers are timeless, and so Cantor took himself to have proved that the totality of the numbers was indeed contradictory (akin to human reasoning being inferior to religious insight), while most of his peers replaced the natural numbers with axiomatic structures that had not been shown to be contradictory. Axiomatic set theory has been the foundation of mathematics for nearly a hundred years, but why do mathematicians throw numbers away (why take number-words to be referring to axiomatic sets) just because of an inconvenient proof? We expect others to accept the conclusions of our proofs, when we have proofs...

Monday, August 03, 2015

Thinking About Things


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.

Saturday, August 01, 2015

Having tired of writing, I spent the last year-and-a-half learning to photograph, mostly in the garden and around the village, and sharing my photos on Google+ while enjoying the other pictures on that magic magazine. I got a bit bored with photos too, recently, but am now dipping back into both writing and pictures, and I got rid of my old blogger profile (enigMan, a "Meaning"-full name) in case you were wondering...

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 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.