The Adams Thesis for a conditional → says that P(A→B)=P(B|A). There are lots of theorems, most notably due to Lewis, that say that this can't be right, but they all make additional assumptions. On the other hand, van Fraassen has a paper arguing that any countable probability space can be embedded in a probability space that has a conditional → which satisfies the Adams Thesis and a whole bunch of axioms of conditional logic. The proof in the paper appears incomplete to me (it is not shown that all necessary conditions for the choice of [A,B] are met). Anyway, over the last couple of days I've been working on this, and I think I have a proof (written, but needing proofreading) of a generalization of van Fraassen's thesis that drops the countability assumptions (but uses the Axiom of Choice).
The conditional logic one can have along with the Adams Thesis is surprisingly strong. In my construction, for each A, the function CA(B)=(A→B) is a boolean algebra homomorphism. Thus, we have Weakening, Conjunction of Consequents, Would=Might, and the Conditional Law of Excluded Middle. The main plausible axioms that we don't get are Weak Transitivity and Disjunction of Antecedents (can't get in the former case; don't know about the latter).
The proof isn't that hard once one sees just how to do it, but it ends up using the Maharam Classification Theorem, the von Neumann-Maharam Lifting Theorem and oodles of Choice, so it's not elementary.
16 comments:
What were Lewis' additional assumption(s)?
By "the Conditional Law of Excluded Middle" are you referring to the subjunctive conditional law of excluded middle (from now on, SCLEM)? If so, then is it also the case that if we accept your construction and the Adams Thesis then we must deny one of the suppositions from your paper arguing against the SCLEM, which is a work in progress on your papers page?
If so, that would be an interesting result worth noting.
It is unclear whether the speech act behind this post is, "Hey, does anybody want to review a proof?"
Anyway, I am not the right guy to do it. But it sounds interesting.
I don't think philosopher like Alex Pruss has problem to find reviewers for his work :)
Actually, such highly technical proofs are, I expect, very hard to find reviewers for. I feel sorry for the referees. Unless they find a mathematician working in measure theory, the referee will have to learn about measure algebras and their classification.
I am historian, so I can read (not review) only non-technical papers :(
I must admit that it is surprise for me that it's hard to find reviewer in such huge academic community in USA.
This is only tangentially related to the comments above: I have often wondered about the relationship between mathematical talent and philosophical talent. I am sometimes attracted to the view that there is just one type of talent here. And that tackling mathematical problems requires more of that one kind of talent than tackling philosophical problems. Other times, though, when I have philosophical conversations with mathematicians or observe attempts to do philosophy by theoretical physicists, it seems to me that some people with great skill at mathematics are quite bad at philosophy. So this makes me think there are importantly different talents here. Since you are both a mathematician and a philosopher, I wonder whether you have any thoughts about this topic.
Milos:
When the work is specialized enough, there will only be about a dozen people interested in it. :-|
Scott:
The big philosophical talent is seeing when to use modus ponens and when to use modus tollens, once all the logical interconnections have been laid out. There is no analogue in mathematics: there once all the interconnections have been laid out, you're done.
From what I can see, Lewis' triviality was excluded by van Frassen by excluding the "implication of an implication" construct:
P( X -> ( Y -> Z ) ) (e.g. "If it rains enough, then if I plant it, it will grow")
which Lewis flattened via boolean logic substitutions into P(Z). Lewis assumed that construct was allowed (I'd say that van Frassen was actually the one with the extra assumption by excluding it). Van Fraassen managed to allow some others with 3 terms.
Are you allowing Lewis' 2-level implication, or is that what you are calling weak transitivity?
What is crucial to Lewis is the assumption that not only P(A→B)=P(B|A) but also P(A→B|K)=P(B|AK).
Ah, I see, thanks.
Actually, on reflection, the Adams Thesis (with uncontroversial axioms) implies almost-SCLEM: P(A&rrar;B or A→~B) = 1,
whenever P(A)>0.
For A→B and A→not-B will be incompatible if A is possible, so P(A&rrar;B or A→not-B) = P(A→B)+P(A→~B)=P(B|A)+P(~B|A)=1.
So, yes, reasons to deny SCLEM will be reasons to deny the Adams Thesis.
Hajek claimed to have proven the above Lewis assumption as a lemma:
see http://philosophy.anu.edu.au/sites/default/files/documents/Triviality%20Pursuit.Topoi_.final_.pdf
on page 15.
He does assume that a conditioned probability (P sub B of A) can be used wherever a non-conditioned probability P(A) can be used.
Yeah, the latter assumption is also problematic.
The theorem I announced in this blog post is forthcoming in a paper just accepted by the Notre Dame Journal of Formal Logic.
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