Saturday, October 27, 2012

Curry's paradox

Consider C, which says that if C then B, where B says that black is white. Suppose that C is true. Then it is true that C implies B. So if C is true, then B is true. But therefore C is true. So B is true. That in a nutshell is Curry's paradox. Every step in that argument that black is white appears to be a logical step.
......But, ordinary modal logic operates in a space of descriptions that are either true or else false. So perhaps we should not have supposed, to begin with, that C is true, because C is true only insofar as it is not true (because insofar as it is true we get a contradiction) and so it is as true as not. That is a consistent possibility because it is as true as not that a contradiction follows from a statement that is as true as not (since it would follow from a statement that was false). And it resolves the paradox because ordinary logic breaks down with propositions that are only as true as not (e.g. see the Sorites paradox, higher-order vagueness, and the revenge problem for this resolution of the Liar paradox).
......So, maybe Curry's paradox is showing us that we should not even suppose the truth of some conditionals that are as true as not. Of course, C does just that, with its "if C". But that just means that its grammatical structure might be less logical than that of, say, A, the claim that A is not true (which basically says that if A is true then pigs fly). That might explain how all the steps of our fallacious argument could have been logical, while the Liar paradox needs us to presuppose bivalency.

Sunday, October 21, 2012

Revenge revisited

The obvious resolution of the Liar paradox is that such self-descriptions as “this claim is not true” are as true as not. That resolution is obvious because it follows immediately from the problem, which is that insofar as they are not true, such claims are true. That is how the problem should be expressed, rather than as paradoxical biconditionals, if truth might be a matter of degree. And if a uniformly coloured object is as blue as not, then prima facie “the object is blue” is as true as not, so that is a prima facie logical possibility. Indeed, since the Liar paradox is a paradox, its less obvious resolutions are all very implausible, and so it is itself an indication that truth is in general a matter of degree.
......But, there is a fly in the ointment. What about “this claim is not even as true as not”? Were that claim as true as not, it would be false that it was not even as true as not. That is the revenge problem for this resolution. The problem is that insofar as the revenge claim is at least as true as not, it is false. But it follows that, while it is a bit more false than true, it is still about as true as not. So where is the problem? The revenge claim is almost as true as not. Now, if we use “vaguely true” to mean about as true as not, then we have another revenge claim, “this claim is not even vaguely true”. Insofar as that claim is vaguely true, it is false. It follows that it is as false as it is vaguely true, though. So this claim is just a bit more vaguely true. Again, where is the fly?
......It is trapped in the bottle of ointment, loudly buzzing. The buzz is that it is clearly not the case that “this claim is not true” is true. Were it true, it would be as it says it is, it would not be true. So it is not true. Even if it is as true as not, it is still not true enough to count as simply true. And furthermore, if “this claim is not even vaguely true” is vaguely true, then it what it claims is simply not true. How could it help to talk of the more vaguely true when “vaguely true” is such a vague term? Or so the buzzing goes. But, it is precisely because “vaguely true” is vague that there is no contradiction. The buzzing is understandable, but it does not spoil the ointment. It is understandable because we tend to imagine that things are black-and-white when we try to think logically.
......Consider the uniformly coloured object that is, in some contexts, as blue as not. Of course, blue is not normally a matter of degree. Oxford blue is deeper, not bluer, than Cambridge blue. “Oxford blue is blue” is not truer than “Cambridge blue is blue”. But, as greenish blue shades into bluish green there is bound to be a colour that is about as blue as not. There would otherwise be a blue colour that was the same colour as a colour that was not blue. And of course, a blue-green object will in some contexts look blue, e.g. when surrounded by red balls. But our object is in a context where it is as blue as not, so it does not look very blue. So I do not want to say that it is blue. If anything, I want to say that it is not the case that it is blue. But tempted as I am to say that it is not blue, the fact is that “the object is not blue” is, like “the object is blue”, only as true as not.

Saturday, October 20, 2012

Simmons' Paradox

Keith Simmons told this story about ten years ago:
Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.
The question is, what is X when X = pi + 6 + X, and the obvious answer is “infinity”. Possible answers are omega, aleph-null, aleph-one, and so forth. There are lots of possible answers, so the “the” at the start of the third expression is a little deceptive. It is like asking for the name of the king of France; the obvious reply is, which king of France?
......But, what if “numbers” in the third expression was replaced with “finite numbers”? The expression “the present king of France” denotes nobody. But, if the third expression with “finite” does not denote anything, then the sum of the numbers denoted by expressions on the board would be pi + 6, so the third expression ought to denote pi + 6. And then it should denote twice that, whence it should denote thrice, and so forth. If it denotes anything, it does it inconsistently. So, it denotes nothing consistently if it denotes anything, but if it denotes nothing then it denotes pi + 6.
......The light on the horizon is that reference is in general a matter of degree. To why, imagine a man staggering through a desert and seeing a mirage, which he takes to be a pool. Coincidentally there is a pool just where he takes one to be, but it is obscured from his view by the mirage. As he staggers on he is constantly thinking “that pool looks cool” rather obsessively. And as he nears the pool its image gradually replaces the illusory one without him noticing. The referent of “that pool” gradually changes to the pool, and so he will at some point have referred to the pool only as much as not.
......What did “that pool” denote when the man was referring to the pool only as much as not? I would say that it denoted the pool, but only vaguely. And similarly, the third expression with “finite” refers us to pi + 6 insofar as it does not, so it refers us to pi + 6 as much as not. It denotes pi + 6, but only vaguely.

Friday, October 12, 2012

Why was the Big Bang not a Black Hole?

A documentary about what happened before the Big Bang was repeated on the BBC last night, and it got me wondering why the Big Bang was not a Black Hole, a Super-duper-massive Black Hole. There was all this matter, all the matter in the universe, in this tiny, tiny space. So why the explosion? And why an inflationary explosion? And why is the universe still accelerating? Did dark energy make it all happen? That does sound like a physics of the gaps.
......I had already been wondering why an amount of antimatter equal to the observable matter of the universe would not be in the form of an uncollapsed standing wave (like electron shells around atomic nuclei). The popular theory of where all the antimatter went is that there was originally a lot more extra matter and an equal amount of antimatter which annihilated each other. But that would just create a lot of heat and light, none of which could escape a Black Hole. But, were the antimatter in a standing wave, then the uncollapsed antimatter suffusing the primordial atom would make it effectively massless, so there would be no Black Hole, while the repulsive force between the matter and the antimatter would cause an explosive expansion. Furthermore, the appearance of dark matter would be explained; while the standing wave would enforce a certain uniformity, much as the inflationary period is supposed to have done.
......I have not heard of any such theory, so that thought is not even philosophy of physics, but listening to the physicists in that documentary made me wonder whether there might be such a theory. The things they were saying were pretty off the wall (according to each other).