(2017-10-25, 11:13 AM)Laird Wrote: OK, so, I can't resist. I'll bang on the drum a little more. I think what you've written above is a perfect example of "everyday" logic with respect to implication. I also think that you'd concede that formal logic goes a little further: it stipulates that also, "p implies q" means that "in all cases where 'not p' is true, the implication is true". This is where the beat becomes more intense... but I'm still waiting for somebody to start dancing to my drumming!
I don't think I'd describe that as going further (because I think it's implicit in what I said), but I agree that the hypothesis "p implies q" is true (though not very useful) if the condition p is never true. So I think you could also say it's true for the subset of cases where p is not true. Though there may be a danger of confusion if we start talking about hypotheses being true for some cases and not others.