In the comments below Kai von Fintel pointed me to this very interesting paper by Sabine Iatridou, On the Contribution of Conditional Then. It’s a very nice paper, and I recommend you read it. Roughly, her conclusion is that then is only acceptable in p -> q if ~(~p -> q) is true. There’s a fair bit to be said about this, but let me make three quick points.

This theory predicts, as far as I can tell, that then is bad in sideboard conditionals. As noted in the comments on the earlier thread, opinion about this is mixed.

Iatridou notices something I kicked myself (hard) for not having seen before. Lots of phrases that we usually interpret as meaning something very close to conditionals do not allow then. Here’s one consequence of that. William Lycan has argued that If P, Q means In the event that P, Q, and it seems clear that Lycan means this to apply to if P then Q as well. (At least I couldn’t find anything from a quick skim through his book to indicate otherwise. On page 39 he suggests the ‘then’ carries temporal connotations, but the analysis in terms of events is still meant to hold, just with different restrictions on the tacit quantifiers.) Lycan doesn’t think this is just a paraphrase. His ‘event theory’ is meant to be syntactically plausible. But note the differences between (1) and (2).

(1) If Jack doesn’t show up to the party, (then) Jill will be furious.
(2) In the event that Jack doesn’t show up to the party, (*then) Jill will be furious.

Having said that, a point I think I learned from Lycan tells against one of Iatridou’s other arguments. She says that her theory explains why (3b) is bad.

(3) a. Even if John is drunk, Bill will vote for him
     b. *Even if John is drunk, then Bill will vote for him.

The idea is that (3b) implicates that Bill will vote for John no matter what, and in particular that he will vote for John if John is not drunk. In general Even if p, q entails q and that entails If not p, q which rules out the use of then. Well, that’s the story. But it’s not in general true. You can’t ever, as far as I can tell, have Even if p, then q, even when the If not p, q is not entailed in the circumstances. Consider the following warning given to new employees at a well known hotel chain.

(4) The one thing you cannot do here is flirt with the boss’s family. If you flirt with his sister, then you’ll be fired. If you fllrt with his daughter, then you’ll be fired. If you flirt with his wife, then you’ll really be fired. Even if you flirt with his second cousin twice removed, (*then) you’ll be fired.

The last sentence doesn’t entail that you’ll be fired, or that if you don’t flirt with the boss’s second cousin twice removed, you’ll be fired. But then is still bad. This isn’t I think a problem for Iatridou, because there could still be a syntactic explanation for this. But it means even if conditionals are not evidence for her theory.