## Friday, March 19, 2010

### Indicative conditionals

Here is a proposal. It can't be right—it's too messy and ad hoc—but it matches my intuitions about how conditionals are used. (Some people will say that the latter fact is evidence against the theory!) The proposal is this:

1. (if p, then q) iff [P(q|p) is high and (q or not p)].
I don't know if P should be subjective or objective probability, what the background for P is, and what level of "high" counts. It may be that all these things are determined by context.

The proposal solves the main problem with simply identifying the indicative conditional with a high conditional probability: a high conditional probability does not support modus ponens. But (1) supports modus ponens, since it makes the indicative conditional be stronger than the material conditional.

The proposal handles the problem with the material conditional generated by false antecedents. It isn't true that if the Queen will invite me for dinner tonight I will go in my pajamas, because although the material conditional conjunct on the right hand side of (1) is true, P(I go in my pajamas | the Queen invites me) is low.

The problem of true consequents is handled in at least a lot of cases. It's true that I won't accept the Fields Medal this year, because I won't be offered it. But the following indicative conditional is false on the proposal: If I am justly offered the Fields Medal, I won't accept it. It is false because although the consequent is true, it has low probability conditionally on the antecedent.

The proposal is problematic in the case of impossible antecedents, at least if P is objective probability, but everybody knows per impossibile conditionals are tricky. It is also problematic in the case of zero-probability antecedents, but there one can at least hope that in the important cases one can give a limiting case interpretation of the conditional probability.

Another problem is with contrapositives. If it's true that if p, then q, we'd like it to be true that if ~q, then ~p. That works fine for the material conditional. The problem is with the high-probability conjunct in (1). We're going to have this problem when P(q|p) is high and the material conditional holds, but P(~p|~q) is not high. These cases are also going to be a problem for the no-truth-value conditional-probability view of indicatives.

#### 1 comment:

Alexander R Pruss said...

Here's a problem. George doesn't have syphilis. "If George has syphilis, he does not have paresis" is true on (1). For most people who have syphilis don't have paresis, and so the conditional probability conjunct is true. But "If George has syphilis, he does not have paresis" doesn't seem right.