I am very much drawn to the objective causal tendency interpretation of chances. What makes a quantum die have chance 1/6 of giving any of its six results is that there is an equal causal tendency towards each result.
However, objective tendency interpretations have a serious problem: not every conditional chance fact is an objective tendency. After all, if P(A|B) represents an objective causal tendency of the system in state B to have state A, to avoid causal circularity, we don’t want to say that P(B|A) represents an objective causal tendency of the system in state A to have state B.
There is a solution to this: a more complex objective tendency interpretation somewhat in the spirit of David Lewis’s best-fit interpretation. Specifically:
- the conditional chance of A on B is r if and only if Q(A|B)=r for every probability function Q such that (a) Q satisfies the axioms of probability and (b) Q(C|D)=q whenever r is the degree of tendency of the system in state D to have state C.
There are variants of this depending on the choice of formalism and axioms for Q (e.g., one can make Q be a classical countably additive probability, or a Popper function, etc.). One can presumably even extend this to handle lower and upper chances of nonmeasurable events.
4 comments:
I don’t understand (b). No doubt a blind spot. Can I read “… whenever the system in state D has objective causal tendency q to move to state C”, or something similar?
On the substance: This sort of approach seems reasonable for quantum dice (i.e. for ‘simple’ states of a fundamentally indeterministic theory). But the chanciness of ordinary dice is different. The initial macrostate includes many microstates, some of which are almost certain to result in outcome ‘1’, some to result in ‘2’, etc.
Ian:
Yeah, I corrected it.
I think the uncertainty of ordinary dice is epistemic rather than chancy.
Looks cool! I don't understand this, but I'd love to see what Nevin Climenhaga thinks of it!
Basically, the idea (and there are other ways of implementing it) is that we start with objective tendency transitions and then see what other probabilistic claims can be derived.
I should probably have specified a third condition: (c) if A is metaphysically necessary, then P(A)=1.
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