P(~J∨X∣D)≥P(J∣D). — unenlightened
Not sure why you think this.
Here's a diagram for what I said, which was
:
That seems pretty straightforward: we start with the yellow and green bits and pick up the blue as well. The blue might be empty, but we still know yellow+green+blue ≥ yellow+green.
Now here's yours, which was
:
Your claim is that orange+blue+green≥yellow+green. Maybe, maybe not. Depends on whether orange+blue≥yellow, doesn't it? And our hypothesis was that yellow+green is pretty big: Smith has strong evidence for his belief.
Two more points. Another interpretation of Smith's belief would be:
which looks like this:
That adds in the light blue bit. I don't think there's any reason to do this though, because all of Smith's reasoning is relative to
, his reasons for believing Jones owns a Ford. Adding the light blue bit doesn't change the argument anyway. It's just a bigger version of the "≥" we've already got.
Now what about
? Are the reasons Smith has for believing Jones owns a Ford reasons to believe Brown is in Barcelona? Well, they're not reasons to believe he isn't: there's no reason to think that Jones having always owned a car and always a Ford and now driving a Ford, etc., precludes Brown from being in Barcelona. So there's no reason to think
and
are disjoint. But it doesn't give you much to go on, so when I assigned a prior to
, I made it tiny, and that seems reasonable to me.
((Apologies for the crumminess of the diagrams.))