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?