That’s not really it, though, because I think the “0 and 1 are not probabilities” claim is really about degrees of belief in non-mathematical propositions. In its most-reasonable-to-me form, it says something like “Even if you have an argument that statement S is true with probability 1, you should believe Pr[S] < 1, because your argument could be wrong”. And there’s… really not a lot I could say in response to that. Except I would note that the value 1 isn’t really special here.
But there’s a lot of things that go together with this idea that I do disagree with. In very many senses, even non-mathematical propositions do end up having probabilities of 0 or 1. For instance:
Any time we deal with (even theoretical) infinities (this one is important because here we get events with probability 0 that can actually happen)
Tautologies (duh)
Conditional probabilities (nobody really disagrees with this, but I think lots of probabilities we think are unconditional aren’t)
Any belief that I can never be talked out of (given how the human mind works, probably most beliefs we have are like this actually)
Plus in practice accepting “0 and 1 are not probabilities” rhetorically or otherwise just means that you stop writing 1 and start writing 1-epsilon. Whose belief is it really that doesn’t affect anything?
even non-mathematical propositions do end up having probabilities of 0 or 1 … Tautologies
Tautologies are true for mathematical reasons and there is little difference—as far as probability assessment goes—between “P ∨ ∽P” and “Yding Skovhøj is the highest peak of Egypt or Yding Skovhøj is not the highest peak of Egypt”. Thus, tautologies (and pseudologies, or how do we call their false counterparts) don’t really make a category distinct from mathematical statements.
Conditional probabilities
I am not sure what you mean here. Of course there are conditional probabilities of form “if X, then X”, but they already belong to the tautology group.
Regarding mathematical statements it’s nevertheless important to notice that there are two meanings of “probability”. First, there is what I would call “idealised” or “mathematical probability”, formally defined inside a mathematical theory. One typically defines probability as a measure over some abstract space and usually is able to prove that there exist sets of probability 1 or 0. This is, more or less, the sort of probability relevant to the probabilistic method you have linked to. Second, there is the “psychological” probability which has the intuitive meaning of “degree of belief”, where 1 and 0 refer to absolute certainty. This is, more or less, the sort of probability spoken about in “1 and 0 aren’t probabilities”.
These two kinds of probabilities may correspond to each other more or less closely, but aren’t the same: having a formal proof of a proposition isn’t the same as being absolutely certain about it; people make mistakes when checking proofs.
Plus in practice accepting “0 and 1 are not probabilities” rhetorically or otherwise just means that you stop writing 1 and start writing 1-epsilon. Whose belief is it really that doesn’t affect anything?
If believing P doesn’t affect anything, then naturally believing non-P doesn’t affect anything either. So, if you agree that “1 and 0 aren’t probabilities” is an inconsequential belief, does it mean that your answer to my original question is “yes”?
If believing P doesn’t affect anything, then naturally believing non-P doesn’t affect anything either. So, if you agree that “1 and 0 aren’t probabilities” is an inconsequential belief, does it mean that your answer to my original question is “yes”?
Saying that my belief in P is inconsequential implies that actually I am acting as if I believed not-P, even though I profess a belief in P. I argue that, conversely, many people who profess a belief in not-P act as if they actually believe P.
The point is that “acting as if one believes P” and “acting as if one believes not-P” can sometimes be the same actings. This is what I meant by “inconsequential”. I want to know whether, in your opinion, this is such a situation; that is, whether there is some imaginable behaviour (other than professing the belief) which would make sense if one believed that “1 is not a probability” but would not make sense if one believed otherwise.
Any belief that I can never be talked out of (given how the human mind works, probably most beliefs we have are like this actually)
I suspect that with enough resources you could be talked out of any of your beliefs. Oh, sure, it would take a lot of time, planning, and manpower (and probably some people you approve of having the beliefs we’d want to indoctrinate you with). You’re not actually 100% certain that you’re 100% certain that 0 and 1 are probabilities.
The trouble with thinking 0 or 1 is a probability is that it is exactly equivalent to having an infinite amount of evidence, which is impossible by the laws of thermodynamics; minds exist within physics.
Furthermore, a feeling of absolute certainty isn’t even a number, much less a probability.
I suspect that with enough resources you could be talked out of any of your beliefs.
At some point you have to ask: who is this “me” that can have any arbitrary collection of beliefs?
(And yes, incidentally, I don’t assign 100% probability to the fact that I assign 100% probability to the statement “0 and 1 are probabilities.” I think I could be persuaded, not to have a lower confidence in the 0-1 statement, but to believe that my confidence in it is lower than it is. This is sort of hard to think about, though.)
Well, on a somewhat trivial note, I (plan to) make my living proving that certain things have probabilities distinct from 0, so if 0 and 1 weren’t probabilities to begin with I’d be out of a job.
That’s not really it, though, because I think the “0 and 1 are not probabilities” claim is really about degrees of belief in non-mathematical propositions. In its most-reasonable-to-me form, it says something like “Even if you have an argument that statement S is true with probability 1, you should believe Pr[S] < 1, because your argument could be wrong”. And there’s… really not a lot I could say in response to that. Except I would note that the value 1 isn’t really special here.
But there’s a lot of things that go together with this idea that I do disagree with. In very many senses, even non-mathematical propositions do end up having probabilities of 0 or 1. For instance:
Any time we deal with (even theoretical) infinities (this one is important because here we get events with probability 0 that can actually happen)
Tautologies (duh)
Conditional probabilities (nobody really disagrees with this, but I think lots of probabilities we think are unconditional aren’t)
Any belief that I can never be talked out of (given how the human mind works, probably most beliefs we have are like this actually)
Plus in practice accepting “0 and 1 are not probabilities” rhetorically or otherwise just means that you stop writing 1 and start writing 1-epsilon. Whose belief is it really that doesn’t affect anything?
Tautologies are true for mathematical reasons and there is little difference—as far as probability assessment goes—between “P ∨ ∽P” and “Yding Skovhøj is the highest peak of Egypt or Yding Skovhøj is not the highest peak of Egypt”. Thus, tautologies (and pseudologies, or how do we call their false counterparts) don’t really make a category distinct from mathematical statements.
I am not sure what you mean here. Of course there are conditional probabilities of form “if X, then X”, but they already belong to the tautology group.
Regarding mathematical statements it’s nevertheless important to notice that there are two meanings of “probability”. First, there is what I would call “idealised” or “mathematical probability”, formally defined inside a mathematical theory. One typically defines probability as a measure over some abstract space and usually is able to prove that there exist sets of probability 1 or 0. This is, more or less, the sort of probability relevant to the probabilistic method you have linked to. Second, there is the “psychological” probability which has the intuitive meaning of “degree of belief”, where 1 and 0 refer to absolute certainty. This is, more or less, the sort of probability spoken about in “1 and 0 aren’t probabilities”.
These two kinds of probabilities may correspond to each other more or less closely, but aren’t the same: having a formal proof of a proposition isn’t the same as being absolutely certain about it; people make mistakes when checking proofs.
If believing P doesn’t affect anything, then naturally believing non-P doesn’t affect anything either. So, if you agree that “1 and 0 aren’t probabilities” is an inconsequential belief, does it mean that your answer to my original question is “yes”?
Saying that my belief in P is inconsequential implies that actually I am acting as if I believed not-P, even though I profess a belief in P. I argue that, conversely, many people who profess a belief in not-P act as if they actually believe P.
The point is that “acting as if one believes P” and “acting as if one believes not-P” can sometimes be the same actings. This is what I meant by “inconsequential”. I want to know whether, in your opinion, this is such a situation; that is, whether there is some imaginable behaviour (other than professing the belief) which would make sense if one believed that “1 is not a probability” but would not make sense if one believed otherwise.
I suspect that with enough resources you could be talked out of any of your beliefs. Oh, sure, it would take a lot of time, planning, and manpower (and probably some people you approve of having the beliefs we’d want to indoctrinate you with). You’re not actually 100% certain that you’re 100% certain that 0 and 1 are probabilities.
The trouble with thinking 0 or 1 is a probability is that it is exactly equivalent to having an infinite amount of evidence, which is impossible by the laws of thermodynamics; minds exist within physics.
Furthermore, a feeling of absolute certainty isn’t even a number, much less a probability.
At some point you have to ask: who is this “me” that can have any arbitrary collection of beliefs?
(And yes, incidentally, I don’t assign 100% probability to the fact that I assign 100% probability to the statement “0 and 1 are probabilities.” I think I could be persuaded, not to have a lower confidence in the 0-1 statement, but to believe that my confidence in it is lower than it is. This is sort of hard to think about, though.)