Saturday, March 17, 2018

The Liar Proof

I show that 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. For definiteness, let L be an assertion that L is not true, such as the assertion expressed by the following sentence: This assertion, which you are currently considering, is not true.
          Since L is an assertion that L is not true, L is an assertion that it is not true that L is not true, and so it is an assertion that L is true, as well as not true. L is not simply an assertion that L is true, of course; nor is it a conjunction of those two assertions: L is wholly the assertion that L is not true, it is just that an aspect of that is that L is thereby an assertion that L is true. That is unusual, to say the least. One might wonder if L is indeed an assertion. But it is clear enough what is being asserted, clear enough for us to know what we are considering. And if L is as true as not – as shown below – then it is as true to say that L is true, as it is to say that it is not. 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 proofs. So, were some such proof to seem good to us, then had it used something other than an axiom, we might have a reason to formulate a new axiom, or to reformulate our axioms, but we would have no reason to reject the original proof. Let me briefly revise how logic seems to take L to a contradiction. 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 – 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. And as I argue next, the negation of that assumption is not logically impossible. It follows that it is that assumption that logic is taking to a contradiction. Now, that assumption is certainly very plausible. To want the truth of a matter is to want things to be made clear. It is to want vagueness to be eliminated. Truth stands opposed to vagueness. 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). Of course, @ 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. We might say, for example, ‘it is as much an apple as not’ instead of ‘it is an apple’ precisely because the former is true, the latter only as true as not.
          But, we cannot express L differently, 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, which means 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 really would be inappropriate for an assertion as unnatural as R.

Sunday, March 11, 2018

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!

Friday, March 09, 2018


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?

Monday, March 05, 2018

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.