The inferential method that solves the problems with frequentism — and, more importantly, follows deductively from the axioms of probability theory — is Bayesian inference.
You seem to be conflating Bayesian inference with Bayes Theorem. Bayesian inference is a method, not a proposition, so cannot be the conclusion of a deductive argument. Perhaps the conclusion you have in mind is something like “We should use Bayesian inference for...” or “Bayesian inference is the best method for...”. But such propositions cannot follow from mathematical axioms alone.
Moreover, the fact that Bayes Theorem follows from certain axioms of probability doesn’t automatically show that it’s true. Axiomatic systems have no relevance to the real world unless we have established (whether explicitly or implicitly) some mapping of the language of that system onto the real world. Unless we’ve done that, the word “probability” as used in Bayes Theorem is just a symbol without relevance to the world, and to say that Bayes Theorem is “true” is merely to say that it is a valid statement in the language of that axiomatic system.
In practice, we are liable to take the word “probability” (as used in the mathematical axioms of probability) as having the same meaning as “probability” (as we previously used that word). That meaning has some relevance to the real world. But if we do that, we cannot simply take the axioms (and consequently Bayes Theorem) as automatically true. We must consider whether they are true given our meaning of the word “probability”. But “probability” is a notoriously tricky word, with multiple “interpretations” (i.e. meanings). We may have good reason to think that the axioms of probability (and hence Bayes Theorem) are true for one meaning of “probability” (e.g. frequentist). But it doesn’t automatically follow that they are also true for other meanings of “probability” (e.g. Bayesian).
I’m not denying that Bayesian inference is a valuable method, or that it has some sort of justification. But justifying it is not nearly so straightforward as your comment suggests, Luke.
[Re-post with correction]
Hi Luke,
I’ve questioned your metaethical views before (in your “desirist” days) and I think you’re making similar mistakes now as then. But rather than rehash old criticisms I’d like to make a different point.
Since you claim to be taking a scientific or naturalized approach to philosophy I would expect you to offer evidence in support of your position. Yet I see nothing here specifically identified as evidence, and very little that could be construed as evidence. I don’t see how your approach here is significantly different from the intuition-based philosophical approaches that you’ve criticised elsewhere.
How do you know this? Where’s the evidence? I don’t doubt that some people say, “Stealing is wrong because it’s against the will of God”. But where’s the evidence that they use “Stealing is wrong” to mean “Stealing is against the will of God”?
How do you know? And this seems to contradict your claim above that some people use “Stealing is wrong” to mean “stealing is against the will of God”. That’s not about what we want. (I say that moral terms are primarily about obligations, not wants.)