Objective probabilities

Suppose:

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. B_W(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: P_L,W(p)=P(e(r) entails p|B_W(r)).

If the language L is somehow natural for describing the members of W, then it makes sense to think of P_L,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 P_L,W(p)=0.

What is particularly nice about P_L,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 P_L,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 P_L,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(B_W(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.