In my intro logic class I’m using the WebCT software to help manage the class. WebCT has lots of nice features, one of which is a discussion board for the students to use. Right now on that board there is a little debate going about whether A entails B ® A. I am, naturally, rather pleased to see debates like this get going.

One thing that made the debate particularly interesting is that I didn’t suggest that A entails B ® A by just suggesting that ® , here still sort of meant to be representing English if, has the truth table of the material conditional. Rather, I just introduced the intro and elim rules for ® , and noted that this surprising result falls out of it. Some of the students are still, wisely, noting that the argument still looks fairly dubious. Others are, also wisely, noting that there are various tricks we can do to get into contexts where this argument sounds more natural. This is something that I don’t think would happen if I just taught the truth tables, or the semantic tableaux, for ®.

I’ve never been too concerned by this particular paradox of material implication, in the sense of instances of it do not seem as objectionable to me as they seem to some. The main reason I end up backing a theory that doesn’t make this valid is that I think entailment contraposes, and I think that some instances of ~(B ® A) therefore ~A are very counterintuitive. In a sense, this is how things should be since I support an epistemic analysis of indicative conditionals and a contextually sensitive epistemic theory of assertability. From those two it follows that most of the time that one is in a position to assert A, one is in a position to assert B ® A, modulo concerns about scalar implicature. The only exception is when raising B to salience changes to context to make live a possibility in which ~A. The argument from A to B ® A has many of the same virtues that Robert Stalnaker notes the argument from A Ú  B to ~A ® B possesses. Some people may take this to be a cost of the epistemic analysis of conditionals, or of Stalnaker’s explanation of what is good about the argument from A Ú  B to ~A ® B, but I think it is a nice virtue of each.