In classical probability theory, the probability of all excluded middle instances (A v ~A) is 1 and the probability of all contradictions (A & ~A) is 0. This makes sense, given (some other, even more basic assumptions and) the fact that one of the basic organizing principles of classical probability theory is that for every A, P(A) and P(~A) have to add up to 1.

Of course, given some non-classical assumptions about logic, all that falls apart...as Hartry Field points out somewhere (in one of his essays in "Deflationism and Paradox," I think?), the dialetheist should accept that for some A's, the sum of P(A) and P(~A) can be high as 2, and the truth-value gap theorist should accept that it can be as low as 0.* But what I'm concerned with, at the moment, is *classical* logic and *classical* probability theory. This is my worry:

Even if you make thoroughly classical assumptions about logic--no truth-value gaps and no true contradictions--and take those assumptions to be extremely well-justified given current evidence, and the current arguments for and against them, it still seems to make sense to be a *fallibilist* about these matters. To admit that we can't be entirely sure, that considerations sometimes brought to bear against these assumptions (from the paradoxes of self-reference to Aristotle's sea battle problem to issues related to vagueness and "partially-defined predicates" or "over-defined predicates" or whatever) have at least some weight, even if we ultimately find them unconvincing, that we should be open to new arguments and new evidence that might be brought to bear by future dialetheists or by future gap theorists. To, in other words, grant at least *some* rational credence to belief in truth-value gaps and/or true contradictions, even if we firmly reject these beliefs, much the same way we grant at least *some* rational credence, however slight, to Lorentzian absolutism about space and time, or to extreme skeptical possibilities in epistemology, or to belief in ghosts.

Now, if you are a frequency theorist about probability, there's no problem about fallibilistically believing orthodox things about logic on the one hand, and accepting classical probability theory on the other hand, since, if there are no truth-value gaps and true contradictions, the number of each that will show up over the course of any sort of hypothetical limiting frequency will be 0, and thus it will always be correct to say that P(A v ~A)=1 and P(A & ~A)=0. On the other hand, I see no way for the Bayesian epistemologist, who takes probability to be intimately linked to *justification*, not a calculus of frequencies but of rational credences or some such, to have their cake and eat it too on this issue. Unless I'm missing something, it looks to me like the Bayesian has to give up on classical logic, classical probability theory or logical fallibilism, that they can't hold onto all three, given with their view of probability.

Thoughts?

*Note that the sort of "truth-value gap" under discussion here--and this applies to the rest of the post--are exceptions not just to Bivalence but to the Law of the Excluded Middle.

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## 5 comments:

Sounds to me like a new instance of an old problem (look for the evil demon here ...); more precisely, it sounds like you are asking a question about "higher-order" degrees of belief or "credences about what my credences are". Suppose I assign .75 to A - the question you are asking isn't one about the probability of A, which is what standard confirmation is about, but a question about what my credence is that .75 is the right number. That's a different question than the question about my credence in or justification of A. And that seems to me to be where logical fallibilism gets to be relevant in this sense, not to the (first-order) assignments of values per se (I don't like "credences" - suggests a psychological notion; that's not what confirmation theory should be about).

But either you or I am missing something more central, I think. What, exactly, is the conflict between Bayesianism and frequency theory supposed to be? Yes, you have a couple of radical wingnuts who wants to eliminate probability in favor of credences, but it seems to me that the majority of Bayesians are either frequency or propensity theorists about probability (my hunch would be - mostly frequency theorists).

Well, no, I'm pretty sure I'm just talking about regular old probabilities, not meta-probabilities or anything like that, and saying that *if* you take them to be degrees of rational confirmation or anything epistemic like that, and you accept classical probability theory, that seems to rule out logical fallibilism by suggesting that we're rationally entitled to be absolutely certain that there aren't any gaps or gluts. This only becomes an issue of the probability that we gave gaps and gluts the correct probability assignment (0) *if* you start from the assumption that the probability calculus is about something like confirmation. That's an assumption that I'm very suspicious about--to put my cards on the table, given the fact that it seems impossible to hold onto Bayesian epistemology *and* classical logic *and* classical probability theory *and* logical fallibilism at the same time, the item on that list I'm most amenable to voting off the island is the first one.

My impression of the state of the debate about probability theory is very different from yours, and maybe you're aware of views that I don't know about, and you can help point me to better information than I have, but here's how things look to me right now:

My impression is that people who call themselves Bayesians are neither frequency theorists nor propensity theorists, but take probabilities to be something more like "logical relationships between theories and the evidence for them." Sure, some--perhaps even quite a few--might be pluralists, and think that the probability calculus equally well models degrees of rational confirmation or whatever on the one hand and frequencies on the other, that these are just different types or senses of probability, but when they have their Bayesian epistemology hats on, they're treating probabilities as something like rationally justifiable credences or degrees of confirmation or whatever.

When you say that the probability of a given coin that you're about to flip coming up heads is .5, the frequency theorist interprets that as meaning that, in the contrast class of relevantly similar coins being flipped in relevantly similar ways over the course of a hypothetical limiting frequency, half of them come up heads and half of them come up tails, while the Bayesian takes "the probability that the coin will come up heads is .5" to be a statement about epistemology--given the evidence you have about the coin, the most rationally justifiable way to apportion your degrees of belief that the coin will come up heads and that it won't would be 50/50. These seem to be substantially different explanations, neither of which entails the other, and as explications of what we mean we use probability-speak to talk about core cases like coin flips, they do seem to be in competition with one another. Moreover, it's frequently claimed that there are things that one of these explanations (the Bayesian one) can account for that the other (the frequentist one) cannot, namely "single case probabilities," like "Einstein's Special Theory of Relativity is probably true."

Of course, even a Bayesian who's willing to utterly concede the word "probability" to the frequentist, if they think that classical probability theory can be applied to juggling degrees of rational confirmation, such that the degree to which A and ~A are each rationally confirmed by the evidence will always add up to 1, still faces the problem I'm pushing here about logical fallibilism.

Well, I am not going to claim to be an expert on the current state of affairs w.r.t probability theory, but saying that "the probability calculus is about something like confirmation" sounds like saying that arithmetics is about sticks and stones or apples (period). If the probability calculus is about something, it is about numbers. The Bayesian claim is that we can use the probability calculus to model rational belief update w.r.t evidence, it is not a metaphysical question about what probabilities are. Calling people who believe in objective probabilities AND that the probability calculus is useful for modelling rational belief update "pluralists" seems something of a misnomer.

Most Bayesians (I suppose) subscribe to the Principal principle, namely the principle that (rational) "subjective" probabilities should reflect actual probabilites (if these latter are to be understood as frequencies or propensities is a separate matter). Thus, the reason that your credence that the coin should come up heads should be .5 is because the actual probability of it coming up heads is .5. My .5 degree of belief that the coin will come up heads and the actual probability that it will is thus connected by the Principal principle, but those two sides of it seems to be very different things.

As for the thought that logical fallibilism was (somehow) a higher-order thing, I am less certain. What I had in mind was that the possibility that our logic is wrong might also entail that our probability calculus is wrong (or at least not complete), and that in this sense, logical fallibilism reflects this kind of 'meta-concern' about the actual framework we are using. Of course, if our calculus is wrong, then that will affect the actual value I assign to a certain degree of belief, but it doesn't by that token change my actual (rational) credence in some claim p, just the way of modelling it. At least that seems to be part of what is going on here. On the other hand, you might take logical fallibilism to imply that no belief should ever reach 1, but always be 0.9999... (or you could take Williamson's line to confirmation theory and make the calculus non-monotonic - even if my credence that p is 1, it wouldn't follow that it couldn't change from 1 given sufficient misleading evidence or (simply) memory failure). These two questions are, however, entirely independent, and I am not sure that your argument works if you take care to keep them apart.

I failed to complete one of my sentences here:

"you might take logical fallibilism to imply that no belief should ever reach 1, but always be 0.999..." ... given the possibility that our logic is wrong - but I don't see why that would be a problem whatsoever for Bayesianism.

Hmm. Maybe I'm hanging out with the wrong people--phrases like "a probability is a logical relationship between a theory and the evidence for it" are things I hear a lot from people who are into Bayesian epistemology. Certainly when I took Tim McGrew's grad seminar on probability theory at Western Michigan lo these many years ago when I was getting my MA, he argued strenuously against both frequency and propensity theory and he certainly presented Bayesianism (of both "objective" and "subjective" varieties) as being in competition with those views. At any rate, it's certainly possible that all the folks I'm interacting with are on the periphery of things and the pluralism language is off-base. I have no idea.

The point is just this: if one takes the classical probability calculus to accurately model confirmation theory or our theory of (rationally justified) credences or whatever--and that's a *much* weaker claim that saying that probabilities *are* degrees of confirmation or rationally justified degrees of belief, or "logical relationships between hypotheses and the evidence for them" or whatever--then it seems to me that you have the following problem:

If you want to be a fallibilist about logic, you shouldn't regard the degree of certainty that's rationally justified about (A v ~A), or the degree to which it's confirmed or whatever, as being 1 and you shouldn't regard the degree of certainty that's rationally justified about (A & ~A), or the degree to which its confirmed or whatever as being 0. You should, maybe, as you say, regard the former as being .99999, the latter as being .00001. So far, so good. The problem is that, given some basic assumptions about how probabilities of claims relate to probabilities of conjunctions and disjunctions of them, these (fallibilistic) assignments seem inconsistent with a basic principle of classical probability theory, which is that the number assigned to A and that assigned to ~A should always add up to 1.

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