Saturday, May 9, 2009

Objective probabilities


  1. W is a set of possible worlds (or maybe situations?)
  2. L is a first-order language suitable for talking about what is going on at a member of W, and with a finite symbol-set
  3. S is the set of strings, of finite or countably infinite length, but with a starting point (i.e., "ababababab..." is acceptable, but "...ababababab" is not) in the symbol-set of L
  4. e(s) is the proposition expressed by a sentence s of L
  5. BW(s) is the claim that s is a member of S such that e(s) is true at exactly one member of W
  6. r is a random variable whose values range over the members of S and have the following property: P(r=s)=(n+1)−(l(s)+1), where n is the number of symbols in the symbol-set of L and l(s) is the length of s; thus, r simply chooses a random string in S, letter by letter, with an equal likelihood of any particular letter or of ending the string there.

Then it seems we can define a probability of a first-order[note 1] proposition p relative to the worlds in W as follows: PL,W(p)=P(e(r) entails p|BW(r)).

If the language L is somehow natural for describing the members of W, then it makes sense to think of PL,W as defining a natural probability measure for what goes on in members of W. If p is W-impossible, i.e., if p holds at no member of W, then PL,W(p)=0.

What is particularly nice about PL,W is that it favors worlds with simpler laws. Thus, it is a probability measure particularly well-suited to making scientific inferences.

A serious technical difficulty with the above definition is that PL,W(p) will not be defined for all p, but only for those p for which the set of sentences r such that e(r) entails p is measurable. One can avoid this difficulty by restricting the ps for which PL,W(p) is defined, or by replacing the Axiom of Choice with the axiom that all subsets of the reals are measurable.

A second technical difficulty is that P(BW(r)) might be zero. This difficulty will be avoided if we have at least one finitely simple world, where a world w is finitely simple if and only if there is a finite sentence r such e(r) is true at w and only at w. I suspect (again, I haven't written out the proof) that in that case we get the following interesting theorem: With probability one, we are in a finitely simple world. This suggests that the measure P might be useful for inductive purposes—it seems to be a measure that prefers simpler worlds.


Trent Dougherty said...

I recall us talking about this in less formal terms before. I still have to try to get my head fully around 6, but the two things which first struck me to ask were whether there is a problem about *what* set of worlds W is and whether you are open to any charges of smuggling in a principle of Indifference into 6--"equal likelihood." I ask the latter question not because it seems to me that it does, but only because, as a friend of Indifference, I have noticed that its foes accuse people of smuggling it in all the time, and seem to believe that all attempts at objective probability are bound to smuggle it in somewhere.

Regarding the former worry, I worry that relativising to non-maximal sets of worlds will be open to the same objections as raised against Carnap for relativising to languages. And of course there are the usual problems with maximal sets.

I don't say these problems can't be addressed satisfactorally, only that they are what struck me on first read, based on the kinds of objections I regularly encounter in the literature and in conversation.

Alexander R Pruss said...

How about we choose a maximal set of worlds with the properties that (a) every possible proposition expressible in L is true at some world in the set, and (b) for every world in the set there is a sentence that is true at precisely that world? I guess we need a postulate that a maximal such set exists.

Of course, the maximal set need not be unique. So we may need to allow some indeterminacy in the probabilities. Perhaps then we replace restrictive normative claims like "x should believe p only if p has probability at least q" with quantified claims like "x should believe p only if for every maximal set of worlds, p has probability at least q relative to that set", and permissive normative claims use existential quantification.