I’m dissatisfied with the notion that a conditional with a false antecedent is true. The way I think about conditionals is much closer to probabilistic conditionalization, in which, under an event space interpretation, P(A|B) is a statement about the subspace for which B is true, and has nothing at all to do with the cases in which not B. I tried using that idea to reduce the notion of “if a is true of c, then b is true of c” to “for all c such that a, also b”, but then I found out that “such that” phrases in logic are just dressed up version of the old material implication. So that sucked. I thought I had away around the truth functional interpretation, and then I ended up where I started.
But then I read “Conditional Assertion and Restricted Quantification”, {Belnap 1970}. And he seems to think that you can write a conditional A->B which only says things “in the case that” A is true, and in that case it says “B”. I put “in the case that” in quotes because it seems to hide the same mystery as “such that”, so I’m not sure he’s actually reduced the conditional. However, that’s the natural language description of what he’s doing. Maybe the math part actually is interesting? I don’t know enough to evaluate it. The anecdote-joke at the end of page 8 seems to offer some evidence that his restricted domain quantifiers and conditionals match up with intuition. But then after the paper, Quine comments briefly, and he seems to be saying Belnap’s idea is trivial or confused.
So, anyone who’s dissatisfied with the material conditional, maybe in regard to spurious conditionals in ADT, does Belnap say anything interesting? Does it require a more precise sense of “possible world”?
I’m dissatisfied with the notion that a conditional with a false antecedent is true. The way I think about conditionals is much closer to probabilistic conditionalization, in which, under an event space interpretation, P(A|B) is a statement about the subspace for which B is true, and has nothing at all to do with the cases in which not B. I tried using that idea to reduce the notion of “if a is true of c, then b is true of c” to “for all c such that a, also b”, but then I found out that “such that” phrases in logic are just dressed up version of the old material implication. So that sucked. I thought I had away around the truth functional interpretation, and then I ended up where I started.
But then I read “Conditional Assertion and Restricted Quantification”, {Belnap 1970}. And he seems to think that you can write a conditional A->B which only says things “in the case that” A is true, and in that case it says “B”. I put “in the case that” in quotes because it seems to hide the same mystery as “such that”, so I’m not sure he’s actually reduced the conditional. However, that’s the natural language description of what he’s doing. Maybe the math part actually is interesting? I don’t know enough to evaluate it. The anecdote-joke at the end of page 8 seems to offer some evidence that his restricted domain quantifiers and conditionals match up with intuition. But then after the paper, Quine comments briefly, and he seems to be saying Belnap’s idea is trivial or confused.
So, anyone who’s dissatisfied with the material conditional, maybe in regard to spurious conditionals in ADT, does Belnap say anything interesting? Does it require a more precise sense of “possible world”?