A Liar Proof

The Liar paradox is essentially a proof 
by reductio ad absurdum that it is not the case 
that assertions must be either true or else not true.

This assertion, which you are currently considering, is not true.
          Let that assertion – if it is an assertion – be called ‘L’.
          If L is an assertion that L is not true, then L is an assertion that it is not true that L is not true, and so it is also an assertion that L is true. Of course, L is not simply an assertion that L is true. Nor is it the conjunction of those two assertions, because it is wholly the assertion that L is not true, if it is an assertion, and only thereby an assertion that L is true. That is unusual, to say the least; but it is clear enough what is being asserted, what we are considering, and so L clearly is an assertion. And if L is as true as not – see below – then it is as true to say that L is true, as it is to say that it is not, and so there is that consistency.

          But, logic does seem to take L to a contradiction. By ‘logic’ I mean that which formal logics model mathematically. Formal axioms are abstracted from informal but rigorous arguments, arguments so rigorous that we regard them as proofs. Were such a proof to include a step that did not correspond to any axiom, we should have a reason to revise our formal logic; we should have no reason to reject the proof. So, if L is true – if it is true that L is not true (and that L is true) – then L is not true (and true). But L cannot be true and not true, of course; the ‘not true’ rules out its being true. And so if L must be either true or else not true, then it follows that L is not true. But if L is not true – if it is not true that L is not true (and that L is true) – then L is true (and not true); and L cannot be true and not true.

          So, logic takes L to a contradiction if – and as shown below, only if – we assume that assertions must be either true or else not true. The negation of that assumption is not logically impossible – see below – and so it is that assumption that logic is taking to a contradiction. That assumption is certainly very plausible, of course. To want the truth of a matter is to want things to be made clear. It is to want the vagueness to be eliminated. Nevertheless, there are a variety of abnormal situations where it would be highly implausible for the assumption in question to be true. And L is not a normal assertion. Suppose, for example, that @ is originally an apple, but that it has its molecules replaced, one by one, with molecules of beetroot. The question ‘what is @?’ is asked after each replacement, and the reply ‘it is an apple’ is always given. Originally that answer is correct: originally it is true that @ is an apple. But eventually it is incorrect. And so if the proposition that @ is an apple must be either true or else not, then an apple could (in theory) be turned into a non-apple – some mixture of apple and beetroot – by replacing just one of its original molecules with a molecule of beetroot. And that, of course, is highly implausible.

          What is surely possible, since far more plausible, is that @ is, at such a stage, no less an apple than apple/beetroot mix, that it is as much an apple as not, so that the assertion that @ is an apple is as true as not. That assertion could not be true without @ being an apple, nor not true without @ not being an apple (and we can rule out neither true nor not true, because that is just not true and true).

More precisely, @ is likely to move from being an apple to being as much an apple as not in some obscure way that is, to some extent, a matter of opinion. In between true and not true we may therefore expect to find states best described as ‘about as true as not, but a bit on the true side’, ‘about as true as not’ (a description that would naturally overlap with the other descriptions) and ‘about as true as not, but a bit on the untrue side’. For such abnormal situations, formalistic precision would be quite inappropriate, because the truth predicate is indeed suited to the elimination of vagueness. It is much better to say ‘it is as much an apple as not’ instead of ‘it is an apple’ when the former is true, the latter only as true as not.

          But we cannot express L better, we have to understand it as it is. Fortunately, if we do not assume that assertions must be either true or else not true, then from the definition of L it follows only that L is true insofar as L is not true, that L is as true as not. There is no contradiction, and so the Liar paradox is a disguised proof by reductio ad absurdum that it is not the case that assertions must be either true or else not true. Note that there is no ‘revenge’ problem with this resolution. E.g. consider the strengthened assertion R, that R is not even as true as not (which is thereby also an assertion that R is at least as true as not). If R is true then R is false (and true), if R is as true as not then R is false (and true) and if R is false then R is true (and false); but, if R is about as true as not, a bit on the untrue side, then it would be about as false as not to say that R was not even as true as not (and about as true as not to say that R was at least as true as not). Greater precision than that would be inappropriate for an assertion as unnatural as R.

Definitive Selections?

Are definitive selections too odd?
      When we think of some things, and various combinations of them, it seems clear that all those combinatorial possibilities are there already, awaiting our consideration. And yet I am asking you to imagine that when a Creator, some such brilliant mind, considers some things, all those possibilities are blurred together (although none so blurry that it cannot be picked out); or am I?
      I am suggesting that for selections of selections of ... of selections, from some original collections, each possible selection from those will be a particular possibility only as it is actually selected by our Creator, independently of whom no collections of things would exist, were there such a Creator (as there provably is). The possible selections that make S(N) bigger than N (to use the terminology in my Cantorian diagonal argument) are those endless sequences of ‘I’s and ‘O’s that are pseudorandom; to make them, infinitely many selections have to be made, each one of which involves some arbitrarily large finite number of selections. They might be made instantaneously by our Creator, of course; and if so, then typical selections from S(N) could be made arbitrarily quickly.
      What about S(S(N)), which contains more things than infinite space contains points? Well, a Creator might be able to do all of that instantaneously. And similarly for selections from U, and UU, and maybe UUU; but still, you see how our Creator would have to do much more, and much, much more, and so on and so forth, without end. It is therefore quite plausible that for selection-collections that it would take me far more than mere trillions of pages to describe, our Creator would be unable or unwilling (and thence unable) to make all such selections instantaneously. After all, it is logically impossible for all possible selections to be made instantaneously. To will an incremental development of such abstract mathematics, as a necessary aspect of the creation of any things, might be regarded as a price worth paying for some such creations. And it is also quite plausible that were the Creator unable to do something (even as a consequence of such a choice) then that thing really would be impossible, given that the very possibility of it derives from that Creator.
      Solid things are solid; but mathematical properties related rather abstractly to their individuality can be works in progress; why not? Modern mathematics has a weirder story to tell of such matters! It is relatively straightforward to think of Creation as dependent upon a Creator who transcends even its mathematics. So, it may not be too odd to think of a Creator creating number by definitively adding units: 1, 2, 3 and so forth; is that any weirder than a Creator creating something ex nihilo? Number is paradoxical, so that the ultimate totalities of numbers are indefinitely extensible, and so numbers just do pop into existence, somehow; and what more reasonable way than by their being constructed by a Creator? What would be very weird indeed would be their popping into existence all by themselves, what with them being essentially structural possibilities rather than concrete things. It makes some sense to think of us creating them, as we think about the world around us, but there is something very objective about numbers of things. And again, if it makes sense for us to do it, then how can it be too odd to think of a Creator doing it, in a Platonistic way?
      There will be better ways to think of definitive selection, I am sure; but, it is the case that such weaselly words are the norm nowadays. For example, how can simple brute matter (just atoms, in molecules of atoms, each just some electrons around a nucleus) have feelings, such sensitive feelings as we have? How is that possible? Am I asking for a description of a possible mechanism? Perhaps; but a common enough answer is: Well, it must be possible, because we have such feelings, in this physical universe; although I don't know how sensitive we humans really are, looking at our world! Such answers are accepted by many scientific people, as they "work" on possible mechanisms!

Logic

There are many logical paradoxes.
A famous example is the Liar paradox: “This is a lie.”
If that is a lie, then it is a lie that it is a lie, so it is not a lie.
But if it is not a lie, then what it says is false, so it is a lie.
Whereas, if it is not a lie, then it is not the case that it is a lie.
Contradiction! So, our logic gives us paradoxes. But, so what?
Even highly evolved apes would hardly have a perfect logic.

Most modern thinkers think of themselves as highly evolved apes, in a purely material world that just happens to exist. They/we think so because they/we have taken logical looks at the evidence; but, what happens to our image of ourselves as scientific if we can play fast and loose with logic? We want to be very careful in any choice to embrace illogicality in our thinking; we want, ironically, to make a very logical choice about any such thing.

In taking logical looks at the world, we may well have given low prior probabilities to the existence of a Creator, maybe following Richard Dawkins; but, what if there is a logical proof that there is a transcendent Creator? That would change everything! Thoughts that such a proof could not be possible are naturally based on those very low priors, and at the end of the day there is such a proof. Still, were we to simply refuse to countenance the possibility of a transcendent Creator, then any such proof would become just another logical paradox; and such simple refusals are not necessarily illogical:

I see a tree, so I know it is a tree; that is certainly rational. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind, but so what? I have been assuming that it is no such thing; and even now, after thinking of this particular possibility, I still have no idea how unlikely, or likely, it really is, and so I still cannot do any better than to continue to make that assumption. Making it makes my knowledge a sort of gamble, but such is human knowledge in the real world.

And yet, where do we draw the line? If we had a proof that the tree was really an alien quasi-stick-insect, then surely that assumption would then be illogical. What if you have a very good argument for something that I really do not like; can I take that dislike to trump your argument? Surely not. My dislike can of course motivate me to believe that there is probably a fatal flaw in your argument, but I really should be bothered by the excellence of your argument. Surely I should not just exhibit my dislike, and observe that to err is human. Surely we should all assume logic. Even if it is flawed, it is our logic, and so assuming it would just be the most human error; and maybe our logic is not that bad. Let us look again at the Liar paradox:

“The assertion you are currently considering is not true.”
Let that assertion be called “L” so that: L is true if, and only if, L is not true.
Were “true” a vague predicate, L would be true insofar as L was not true,
from which it would follow logically that L was as true as not.
It follows logically that if “true” could be a vague predicate,
then the Liar paradox is actually a proof by reductio ad absurdum
that it is a vague predicate: then, and only then, is there no contradiction.

Is it only then? That is, after all, why this is a paradox. You could say that the meaning of “true” rules out truth being vague; and of course, truth itself is not normally vague, far from it: to want the truth is to want things to be made clear. But Liar sentences are deliberately constructed to be paradoxical, when they are not simply mistakes that should be rewritten to make them clearer. And consider the following, which is similarly far removed from our normal uses of language:

      Consider “It is an apple”
as an answer to the question “What is A?”
      where A is originally an apple,
but has its molecules replaced, one by one,
      with molecules of beetroot.

Originally, “It is an apple” is a correct answer (or in other words, it is true that it is an apple), but eventually it is not. If that answer must be either correct or else not (if that proposition must be either true or else not true), then an apple can be turned into something else (presumably a mixture of apple and beetroot) by replacing just one of its original molecules with a molecule of beetroot, which certainly seems absurd. It is surely possible, since it does seem more plausible, that A is, at such a stage, no less an apple than it is apple/beetroot mix; that it is, at such a stage, as much an apple as not, so that the proposition that it is an apple is as true as not; what else could it be?

Apparently Timeless Possibilities

Apparently timeless possibilities could, possibly,
become more numerous over time, e.g. as follows:

You were always possible,
but had you never existed,
then that possibility would have been
the possibility of someone just like you.
      It could not have been
      the possibility of you in particular 
      were you not there to refer to.
Looking back now,
we can see that there was always
that possibility, of you in particular 
as well as the more general possibility,
even before you came into being.

Now, Presentism is logically possible,
and if Presentism is true then there may
originally have been no such distinction,
even though you were always possible.
      Under Presentism it could have been
      that you might not have existed.
The distinction could therefore have
arisen when you came into being.

It is therefore logically possible
for apparently timeless possibilities
to emerge as distinct possibilities
from more general possibilities.

The Signature of God

I think belief in God reasonable only if it is based on considerations available to all humans: not if it is claimed on the basis of a special message to oneself or to the group that one belongs.

Anthony Kenny ("Knowledge, Belief, and Faith," Philosophy 82, 381-97)
      So what better signature of the creator of homo sapiens than an elementary logical proof that there is a God? In my last post, I described the argument that given some things, cardinally more selections from them are possible.
      That post ended with a brief description of how that means that paradox arises: we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection; it would follow that they were all there already, that they are collectively some impossible collection of all those possible selections.
      Logic dictates that we have made some mistake; and this version of Cantor's paradox arises because we are considering combinatorially possible selections: that is why the sub-collections that define those selections were able to become so paradoxically numerous, why the paradoxical contradiction did not just show that there are not, after all, so many extra things, over and above the original things.
      My resolution begins by observing that apparently timeless possibilities could, possibly, become more numerous over time; it begins that way because if possible selections are always becoming more numerous, then we would never have all of them. A Constructive Creator could, possibly, make the definitive selections; and if that is the only logical possibility, then that is what has been shown.
      Note that serious mathematicians have taken Constructive mathematics seriously, and when constructed by a transcendent Creator the mathematics would be much more Platonic, and much more Millian. Consider, for an analogy, how God's commands could, just possibly, define ethics. And note that such creative possibilities are not that different to the Creating of mere things ex nihilo, if you think about it: how is such Creation even possible? For us, the laws of physics present immutable limits to what can be done; for a God, such laws are, metaphorically, a brushstroke.
      We live in a world of things, and numbers of things; and for us, numbers appear timeless. But logic does seem to say that such numbers are impossible. When we first think of the origin of things, we might think of things that could have been there forever, like numbers. But logic seems to say that there was originally stuff, not things; perhaps mental stuff, perhaps a God that is not exactly one thing. There would have been some possibility of things, and more arithmetic the more that God thought about that possibility.
      I should add a note about what sort of God is being shown to exist. The proof does not show that God could not have created a four-dimensional world in a Creative act above and beyond that temporal dimension. So this God might be what we call "timeless," and might know all about the future; or not. And either way, this God could always have known all of our textbook mathematics, if only because that is essentially axiomatic.

Cantorian Diagonal Argument

Cantor’s diagonal argument that there are more real numbers than natural numbers gets its name from its picture proof (as below), and it generalizes to show that powersets are always bigger than their original sets.

Cantor originally used collections instead of sets, but they gave him a paradox. Now, a collection of things is just those things being referred to collectively, so it is hard to see how that could have been the problem. But Cantor introduced the notion of a set, or consistent collection, and most modern mathematicians use axiomatic sets, for added confidence, and define numbers from them.
      Nevertheless, there really are numbers of things, so I take the logical essentials of Cantor’s paradox, and find a lacuna. My version of the diagonal argument shows that given some things, cardinally more selections from them are possible. As is the case with Cantor’s diagonal argument, it is best to begin with small collections, and build up from there, so that the general case can be more easily understood, by comparison with those simpler cases.

There are clearly three things:
      clearly there are
Given those three words, we can select a couple, e.g. ‘clearly’ and ‘there.’ There are three ways of making a pair – three different pairs that could be made – from those three words: {‘clearly,’ ‘there’}, {‘there,’ ‘are’} and {‘clearly,’ ‘are’}.
      Each of those ways of making a pair derives from, and is therefore defined by, the presence of two particular things in the original collection. Given those two things, there is that way of making a pair, whether or not anyone would or could make it. Because the original things were distinct, those pairs are distinct, and so each can count as one thing, as when we think of those three ways of making a pair. Possible selections are, in that sense, things.
      In a similar sense, collections are things, e.g. those are three collections of words. But given some things, thinking of just some of them as yet another thing can easily seem like the weak link in a chain of reasoning that leads to a contradiction. (After all, the collection of one thing is just that thing.) It is clearer that, given some things, a way of making a pair from them is indeed another thing. (A way of selecting just one thing is a way of selecting.)

Making a pair is just one way of making a selection. There are, in total, 2 to the power of 3, or 2^3= 8 possible selections from a collection of 3 things (cf. its powerset). It is an elementary result in combinatorics that there are 2^3 ways of assigning 2 labels, say ‘In’ and ‘Out,’ to 3 things, e.g.
      for the pair that is ‘clearly’ and ‘there,’
      ‘clearly’ has the label ‘In,’ as does ‘there,’
      while ‘are’ has ‘Out’ because it is not in that sub-collection.
In general, for any collection of things, T, there is a selection-collection, S(T), of all the combinatorially possible selections from T, each corresponding to some combination of as many ‘In’s and ‘Out’s as there are things in T.
      There is the selection-collection of 2^8 = 256 possible selections from the aforementioned 8, for example; and there are, similarly, a further 2^256 possible selections from that collection, and so on.
      Each of those possibilities is distinguished and defined by the presence of pre-existing things in the original collection, and so they are all implicitly there already, with our original 3 things.
      There is therefore an infinitely big collection of combinatorial possibilities, say N (from which further selections might possibly be taken, giving us S(N); and so on).

Two collections of things have the same cardinal number of things when there are one-to-one mappings from each collection onto all of the other. Cardinality therefore captures some of the intuitive sense of there being as many things in one collection as there are in another. Whether cardinal numbers are rightly called ‘numbers’ or not does not matter here; for the purposes of this proof, the main thing about cardinality is that it is an equivalence relation: it is reflexive, symmetric and transitive. Cardinality therefore partitions collections into equivalence classes. In particular, S(N) is not in the same class as N, for the following reason.

For simplicity, the things in N will be given the names
‘1’ (e.g. naming {‘clearly,’ ‘there,’ ‘are’}),
‘2’ (e.g. naming {‘clearly,’ ‘there’}),
‘3’ and so forth. To get the things in S(N) we associate the things in N with either ‘In’ or else ‘Out,’ and so each thing in S(N) can be named by an infinite sequence of ‘I’s and ‘O’s.
      If S(N) had the same cardinality as N, then we could associate each combinatorially possible sequence of ‘I’s and ‘O’s with one of the names of the things in N, and thereby list all the combinatorially possible sequences of ‘I’s and ‘O’s.
      E.g. the element of S(N) whose name is the sequence I, I, O, O, … might be associated with the element of N named by ‘1,’ and so on:

 N         S(N)

 1           I           I         O         O         …

 2          O         O         I          O         …

 3           I          O        O         O         …

 4           I          O         I           I         ...

…         …         …        …         …         ...

If we can name an element of S(N) that is not in that list (for any such list), then that will show that N and S(N) are not the same size; and we can specify one that is different from each of those by specifying that:
its 1st label differs from the 1st label of the element associated with 1,
its 2nd label differs from the 2nd label of the element associated with 2,
and so on (e.g. O, I, I, O, ...). And that diagonal argument generalizes to show that every selection-collection, S(T), is cardinally bigger than its original collection, T, as follows.

We begin by supposing, counterfactually, that S(T) has the same cardinality as T, i.e. that there are one-to-one mappings from T onto all of S(T). Let M be one such mapping.
      We use M to specify a collection 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 (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. Consequently there is no such M; S(T) does not have the same cardinality as T. And since S(T) has at least one element for each thing in T – e.g. the selection of that thing – hence S(T) is bigger than T.

So, given any 3 things, there is an infinitely big collection, N, and an even bigger collection, S(N), and the even bigger S(S(N)) = S^2(N), and similarly S^3(N) and so on.
      All the things in all those collections are collectively the union, U, of those collections; so, there is also U. U is bigger than each of those S^n(N), for natural numbers n, because it contains all the things in each S(S^n(N)). Furthermore S(U) is even bigger, and so on. So there is also the union, say V, of all the S^n(U) for natural numbers n. S(V) is even bigger, and so on; and so forth, past W, say.
      There will be a union, UU, of U, V, W, …, and thence a union, UV, of all the S^n(UU), and similarly UW, and so on, through VU, VV, VW, ..., and so forth. There will be a union, UUU, of UU, VV, WW, …, and a union of U, UU, UUU, …, and so on, and so forth.

Paradox arises because we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection. It follows that they were all there already, that they are collectively some collection, C, of all those possible selections.
      But, there cannot be any such collection, because its elements, simply by existing, would define those of S(C); and by the diagonal argument, S(C) would contain even more things of that very kind. And we cannot – by the meaning of ‘not’ – have both that C does contain all such possible selections and that C does not contain them all.

      [Lacuna]

an inconvenient Proof ?

Do I have a proof that there is a transcendent Creator?

Well, the essence of Cantor's paradox is a logical argument for a contradiction, with an obscure lacuna: there need be no contradiction if arithmetic (that is Millian, or ordinary, common or garden arithmetic) is constructed forever. Were it not for that lacuna, we might have to throw logic away and replace it with some formal logic (symbolic calculi called "logics") or other, whilst being unable to choose logically between them. But, there is that lacuna, and so we do have a proof of the existence of a transcendent constructor of arithmetic; and of course, Millian arithmetic could only be constructed by the Creator of all other things.
      Note that a purely logical existence proof would be the appropriate signature of the Creator of homo sapiens. And consider some other kinds of proof, by way of comparison; here are a couple of examples:

Suppose there was a serious crime. Fortunately you have a suspect, and a good case against him. The defense says that your case means nothing, because you are only human, that to err is human. She goes on to detail how nice your suspect is. You point out that she should therefore doubt that opinion of him, since she is human, whilst you need not entertain any such doubts because it is not your argument; indeed, you have convicted lots of criminals on less evidence, so you ask her if they should all go free? Of course not, she says, it is only your case against this man that is thrown into doubt by our common humanity, because he is so very nice indeed! She has simply ignored your observation about her own humanity; maybe she erred in doing so! But what should we conclude from an assumption that we cannot trust our conclusions?!
      Let us suppose that your case is exceptionally water-tight: there is lots of physical evidence, and everyone else has cast-iron alibis, while your suspect has no alibi at all; and this crime is just the sort of thing that he would do. There really is no reasonable way that your suspect is innocent. He even bragged about his guilt to you in private. Since your case is so water-tight, hence all her talk about your humanity is just that: talk. It is, if anything, further evidence of his guilt, that she feels that she has to resort to such meaningless talk.

Or, suppose I say that 2 + 2 = 4. Someone says that he would prefer it to be 5, and tries to show that it really can be 5 by saying that: "If we measure two lengths and put them together, then we could find that two point three five units plus two point three five units equals four point seven units; and if we round all those measurements to the nearest integer, so that it is arithmetic, then we get two plus two equals five." Even so, there are such proofs as this (due to Frege): 2 + 2 = 2 + (1 + 1) = (2 + 1) + 1 = 3 + 1 = 4 (via definitions and associativity). And note that he only wants 5 because it really is bigger than 4 = 2 + 2. My interlocutor retorts that we have to have axiomatic arithmetic, on pain of paradox (e.g. Cantor's paradox), and that he likes those axioms that let him have 5. Could some paraconsistent logic not give arithmetical axioms the power to give him his 5 as well as us our 4, he wonders; but no, that is not really logic, and axioms that give him 5 are not arithmetical. My interlocutor will not give up though, and he has lots of friends. Even so.

The Death of Logic

A hundred years ago (more or less) logic died.

It was either logic and a transcendent Creator,
or neither... And atheism was in the ascendant
a hundred years ago (while the Creator shown, logically, to exist
was not that of the embattled religions of those war-faring days).

Prima facie, though, logic took off at that time: it was formalized,
and we now have lots of formal logics, within Analytic Philosophy.
But, what's so logical about reacting to the Liar paradox by redefining "truth"?
And what's so logical about having each ordinal, but not having every ordinal?

We may begin with physics: nowadays we have String Theory.
Suppose we get a really good String Theory, say "S," one day.
There's no guarantee that we won't need a better theory later,
so why would we use S to redefine all our physical entities?
If the description of electrons in S was E, for example, then
we could replace "electron" with "E,"  but why should we?
A very good reason why not is that electrons are electrons.
(A better theory of them won't be any sort of theory of E.)

And mathematics is a subset of the properties of possible objects:
one object and another object is (one plus one) objects, and so on.

But for a hundred years, science has replaced "1" with "{{}}."
Did not the logician Frege refute Mill's description of "1"?

It turns out that he did not; and he could not have,
because 1 is, basically, Mill's 1.

Set Theory mimics mathematics,
so for applications it hardly matters; but,
do the best mathematicians really believe
that 1 is nothing like Mill's 1, is really {{}}?
We all learn what 1 really is at an early age.
{{}} was chosen following Cantor's paradox,
but it also followed that logic had to be replaced.
Logically, there was that paradoxical proof; and
while the Liar paradox is nowadays interpreted
as another reason to replace truth, and its logic,
with something formal, that is not really scientific:
science pursues truth, and logic takes truths to truths
(where to say, of what is, that it is, is to speak the truth).