Tuesday, October 4, 2011

Does belief distribute over conjunction?

The distribution thesis is that:

  • Necessarily, if x believes that p and q, then x believes that p and x believes that q.
I'm going to give three arguments against the distribution thesis.

Argument 1:

  1. It is not possible to believe that p while believing that not-p.
  2. It is possible to believe that p&q while believing that not-p.
  3. So, it is possible to believe that p&q while not believing that p.
I am inclined to think (1) is false, but some people do accept it. As for (2), imagine a case where you believe not-p, and it's one of your slow days, and someone you trust tells you that p&q. So, you believe that p&q. But since it's one of your slow days, it takes an extra moment or two before you see that this conflicts with p&q, and until then you keep on believing both p&q as well as p.

Argument 2:

  1. If it is possible to assign a higher probability to the proposition that p&q than to the proposition that p, then it is possible to believe that p&q while not believing that p.
  2. It is possible to assign a higher probability to p&q than to p.
  3. So, it is possible to believe that p&q without believing that p.
Claim (4) is pretty plausible. If it's possible to assign a higher probability to the conjunction, it should be possible to have a case where the conjunction's probability is just above the cut-off line for belief and the first conjunct's probability is just below it. And if that's possible, it should be possible to believe the conjunction without believing the first conjunct. Claim (5) follows from Tversky and Kahneman's work on the conjunction fallacy.

Argument 3: My mathematics dissertation director gave me this advice on how to write mathematics papers: you can skip an obvious step, but you cannot skip two obvious steps in a row. The point is that two obvious steps in a row may be quite unobvious. Say that a belief "transfers over an inference" if and only if it is not possible to believe the premises of the inference without believing the conclusion of the inference. For instance, the distribution thesis says preicsely that belief transfers over conjunction elimination.

  1. Belief does not transfer over inferences that can be unobvious.
  2. The inference from the claim that p&(q&r) to the claim that q can be unobvious, since it takes two applications of conjunction elimination.
  3. If the distribution thesis is correct, then belief transfers over the double conjunction elimination inference from p&(q&r) to q.
  4. So, the distribution thesis is not correct.

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2 comments:

  1. Dan JohnsonOctober 4, 2011 at 8:51 AM

    Arguments 2 and 3 are a lot better than 1. Your argument for premise 2 of argument 1 just sounds to me exactly like an argument against premise 1 of argument 1 -- so it doesn't support the argument overall. Argument 2 is pretty convincing.

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  2. Alexander R PrussOctober 4, 2011 at 10:50 AM

    Here's another, perhaps less compelling but still interesting, line of thought.

    The logical form of "Sam is a short basketball player" is not "Sam is short and Sam is a basketball player", and one can't infer Sam is short from Sam being a short basketball player. Say that "short" in "Sam is a short basketball player" is "non-detachable" (I expect there is a term for this phenomenon but I can't think what it is--I am thinking here of Geach on
    "good"). But maybe the logical form of "x is a green ball" is "x is green and x is a ball". However, suppose I am not sure that "green" is detachable in "x is a green ball". (After all, it surely can be controversial whether an adjective is detachable. You might agree with Geach on the good or you might disagree.) Then if I am cautious, I will refuse to infer "x is green" given "x is a green ball". But in believing that x is a green ball, I really am believing the conjunction of the proposition that x is green with the proposition that x is a ball. It seems plausible, then, that I might end up believing the conjunction without believe one of the conjuncts. (This makes me think of the Kripke puzzle.)

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