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Something to point out to someone who says this is that 'possibility' is not a constraint - EVERYTHING is possible. As far as I know (and correct me if i'm wrong cause that would be a major fuckup), you can't assign a probability of zero to anything. You can't seperate the possible things from the impossible ones, and then focus only on the possible. 'Possible', by itself, applies to everything, so you don't say anything by declaring something 'possible'. It's only when you start talking about degrees of possibility that the word has any meaning.
Definitional contradictions are impossible. For example, I can say that I will encounter a married bachelor, or a non-female vixen, with P=0. This doesn't actually say anything about the world; I could figure out that there are no married bachelors without leaving in my room, simply by knowing that bachelor is defined as "man who is not married." Mathematical truths (like 2+2=4) and non-contradiction (as mentioned deeper in this thread with the buttering of pancakes) are specific instances of definitional contradiction.
You're generally right, though. Truths that actually involve looking at the world, i.e. ones that are not inherently about language, cannot have P=0 or P=1.
Actually, certain truths about the self and "subjective" experience may also hit P=0 and P=1. It seems I can be certain of my existence, even if I can't be certain of what I am or what causes my existence. I can also be certain that there exists some X such that X exists. I also think I can be certain that it looks like there's a computer screen in front of me, that my knee feels slightly uncomfortable, and that I am literate. All of this seems to have P=1; the implied causes may be false - I may not actually have knees, for example - but I certainly do seem to be having the sense-experience. I don't think this contradicts your point in any practical sense, though.
Another failure mode of arguing from a definition is that you could be wrong about the definition.
I don't know if what I'm about to say is a nitpick, but I think it's relevant to the issue, so I'll say it anyway:
But words have histories behind them, and there is a reason why the term "bachelor" exists. The term "bachelor" carries connotations that go beyond simply "union(male,~married)". To borrow from an example from Hubert Dreyfus (and do forgive me for reading him), if I told you I was having a party and I wanted you to bring bachelors, would you consider bringing priests or gay men?
What's actually happening is that we believe "bachelor" has one meaning, while expecting people to imagine a different clump of conceptspace ("connotation") when we actually use it.
Only when you confine the issue into being a purely logcal one, with "bachelor", "man", etc. as suggestively-named LISP tokens can you identify purely logical (P = 0 or 1) truths. But at that point, you've destroyed the mutual information between those words and the outside world, including the usage of those terms in the outside world. And in that case, your statement is no longer about bachelors, but rather, about abstract logical relationships in Platonic space.
Part of the problem is that "possibility" has different meanings depending on context. In everyday speech, it seems to be used to indicate degrees of probability. When people declare a certain event "possible" in everyday speech, they usually mean that it has a low but nontrivial probability, given the everyday state of the world. In this sense, I might say that it's "impossible" for me to become a NFL player, even though in a philosophical discussion we would recognize that the probability that I could become a NFL player is greater than 0.
Problems occur people equivocate between different meanings of "possibility," or introduce a certain meaning into a type of discussion where it doesn't belong. For instance, it's "possible" that the Flying Spaghetti Monster exists and created the world, but this is not the kind of "possibility" that people deal with in everyday life.
Exceptions to "everything is possible" include logical contradictions (such as mathematical falsehoods).
I voted you up, but I'm genuinely confused here - does the concept of probability/possibility apply to a strict, axiomatic, isolated (yet human created and thus fallible) system like mathematics?
That's a very interesting question. Philosophers have been arguing about the concept of possibility (and its dual, necessity) for some time.
There's a sense of "necessary randomness", that Chaitin has written very extensively about.
http://www.umcs.maine.edu/~chaitin/
Corresponding to this notion, there are stochastic models of (generally-agreed) necessary truths. The best known is probably the "probability of n being prime":
http://primes.utm.edu/glossary/xpage/PrimeNumberThm.html
But there are plenty others - e.g. the 3n+1 problem
http://en.wikipedia.org/wiki/Collatz_conjecture
or the question of who wins (first or second player) an integer-parametrized family of combinatorial games.
More exotically, Neal Stephenson's "Anathem" and Greg Egan's short stories "Luminous" and "Dark Integers" explore the possibility that what we think of as "necessary truths" are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence. (Note: I think this might sound ridiculous to a logician, but moderately reasonable to a cosmologist.) It is quite difficult to tell the difference between a necessary truth and a contingent truth which has always been true.
More prosaically, we do make errors and (given things like cosmic rays and other low-level stochastic processes) it seems unlikely that any physical process could be absolutely free of errors. We might believe something to be impossible, but erroneously. Your answer to the question "Are there any necessary truths?" probably depends on your degree of Platonism.
Not all logical contradictions have to do with mathematics. For instance, it's impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered. Pancakes tend to have a very poor grasp of math.
Your confusion may have to do with epistemic v. metaphysical possibility. I can imagine that I am so deeply, profoundly confused about the universe that I could be mistaken about arithmetic; therefore, it's sort of epistemically possible that two and two be five. However, as it happens, it's not actually possible that two and two be five. Because I am part of a philosophy department in which David Lewis is practically worshiped, I'll put it this way: I can think about two and two making five, but there's no possible world in which that thought is reality.
It does have to do with mathematics, though. A buttered pancake is a pattern, an unbuttered pancake is a different pattern. Each pattern can be expressed as a series of bits, and the first series will not equal the second.
Are you 100% certain about that?
No. Do I need to explain epistemic possibility versus metaphysical possibility again?
I find your distinction between "epistemic" and "metaphysical" possibility to be fairly useless. Are you using a different brain when you consider "epistemic" possibility than when you consider "metaphysical" possibility? Does your concept of "metaphysical" possibility somehow not reside within your own mind, but rather "out there" somewhere in the realm of Platonic logic? That seems suspicious. And even if it were somehow "out there", how do you know you're not mistaken about what it is?
More to the point: can you name a single "metaphysical" possibility that does not reduce to an "epistemic" possibility when considered from the context of your own brain?
It is a metaphysical possibility that the universe is actually entirely random and only seems to us as if it were lawful. But it is not epistemically possible for me to ever be 100% certain that the universe is random, or that it is lawful.
So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map. The fact that I represent both concepts within my own brain doesn't seem to change this.
This would seem to make it useless to talk of metaphysical possibilities, seeing as there is no way to directly access the territory.
I don't think so. I cannot directly access reality, but it still seems very useful to me to speak of it. Even if only to have something for my beliefs to try to correspond to.