Dutch-book arguments and inductive logic
Hi all. One thing I hope we can discuss in connection with Branden's fascinating talk is the Ramsey/Skyrms/Armendt interpretation of Dutch Book arguments. Their idea is that such arguments dramatize an INCONSISTENCY in a probabilistically incoherent agent's evaluations. Thus, they apparently see a very strong analogy between deductive and inductive logic. More generally: to what extent is inductive logic a generalization of deductive logic, and to what extent are they separate enterprises?
(I've moved this post of Alan Hajek's from the comment section of our welcome message to here. -WR)
2 comments:
Alan,
Good question! But I take part of Branden's point to be that inductive logic should be kept distinct from Bayesian belief-strength. The dutch-book arguments are supposed to show that ideally coherent belief-strength should satisfy the usual probability axioms. But perhaps a Bayesian inductive probability function should not just be the agent's belief function, but needs to be a separate function -- one that should ultimately inform belief-strength, but which cannot itself just be the belief-strength function (due to old evidence problems, for example).
So I'd like to divide your question:
1. To what extent is inductive logic a generalization of deductive logic, and to what extent are they separate enterprises? But this may depend on how one answers the following questions.
2. What is the relationship between inductive logic and Bayesian degree-or-belief?
3. What is it that the dutch book arguments show about degree-of-belief?
4. If probabilistic inductive logic differs from degree-of-belief, are the ductch book arguments relevant to probabilistic inductive logic? -- and more generally, what sorts of considerations ground or justify probabilistic inductive logic (e.g., if dutch book arguments don't)?
Jim
Alan,
Since it's been a few days and we haven't heard form others yet, I'll jump back in with my own view.
It seems to me that deductive logic has two features that split when we move to inductive logic and coherent belief-strength. Belief coherence is similar to consistency. And I do take dutch-book arguments for the Bayesian notion of belief coherence and utility to motivate a kind of (ideal) consistency constraint on belief strengths and preferences.
But deductive logic also has a notion of "truth promotion" represented by logical entailment. In deductive logic consistency and entailment are interdefinable. But the notion of truth promotion one wants from an inductive logic is not interdefinable with probabilistic coherence (it seem to me). From an inductive logic (i.e. a logic of confirmation) one at least hopes to have a logic whereby the truth of evidential premises makes it likely that those hypotheses that the logic says are "highly confirmed" will (on enough such evidence) very probably be the true ones. And I don't think that the usual account of Bayesian coherent belief-functions (together with the usual updating kinematics) yields this its own. I think that a separate notion of Bayesian confirmation (i.e. separate logic of confirmation functions) is needed to get that.
I think that there is such a (Bayesian) probabilistic logic of confirmation -- a logic of confirmation functions that can be shown to have this "probable truth yielding feature". Supposing I'm right about that, then the idea is that such confirmation functions may be used to impose additional constraints on belief-strengths (i.e. as represented by Bayesian belief functions) -- so that on a long enough evidence stream one has a good chance of ending up with a high degree-of-belief in true hypotheses.
Jim
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