I have two arguments which I want to combine. Depending on the way I do it I get different results.
Argument #1
P1) If a person is A, then it's likely that that person is also B.
P2) This person is A.
C) This person is likely to be B.
Here, "likely" is interpreted as having a probability greater than 0.5. Using conditional probability we can write P1 as
Pr(B|A) > 0.5
This argument is deductively valid. If our premises are true we can confidently say that the person is more likely to be B than not.
Argument #2
P1) If a person is C, then it's likely that that person is also B.
P2) This person is C.
C) This person is likely to be B.
Just like with the first argument we can write P1 as
Pr(B|C) > 0.5
And again, taking this argument in isolation and assuming the premises are true, we can say that the person in question is more likely to be B than not.
Finally, let's assume that we have both arguments and their second premises refer to the same person. Is it likely that that person is B? If I use arguments' conclusions as premises for my third argument then the inference is trivial.
Argument #3
P1) This person is likely to be B. (From argument #1.)
P2) This person is likely to be B. (From argument #2.)
C) This person is likely to be B.
I expect to come to the same conclusion if I combine premises from arguments #1 and #2.
Argument #4
P1) If a person is A, then it's likely that that person is also B.
P2) This person is A.
P3) If a person is C, then it's likely that that person is also B.
P4) This person is C.
C) This person is likely to be B.
However, if I try to prove this using conditional probabilities I can't get a definitive answer. What we need to prove now is that Pr(B|A,C) > 0.5. However, as was shown in Information paradox: the more we know, the less confident we are, premises
P1) Pr(B|A) > 0.5 and
P3) Pr(B|C) > 0.5
do not guarantee that
Pr(B|A,C) > 0.5
There must be a mistake in the way argument #3 or argument #4 is constructed but I don't see it. Or my assumption that the two ways of argument construction are equivalent may be wrong. So why the conclusions are different?