We say that “C counterfactually implies D (in our world)” if D is true in w, where w is the closest world to our own where C is true.
This has a somewhat subtle error in it: C []-> D is true in w iff there is no world in which C is true and D is false that is closer to w than every world in which C is true and D is true. In other words, C []-> D is true in w iff for every world in which C is true and D is false, there is a closer world in which both C and D are true. (At least, I think those two statements are equivalent.)
Your definition requires that there be a closest world to w in which C is true. But there is no guarantee that there is such a world. There might be a continuum of worlds approaching w but no closest one.
My post is an “intuitive introduction to the subject”, which is “no substitute for rigorously going through the formal definitions” :-) But yeah, it can get complicated, especially as the “distance measure” need not be a metric at all.
This has a somewhat subtle error in it: C []-> D is true in w iff there is no world in which C is true and D is false that is closer to w than every world in which C is true and D is true. In other words, C []-> D is true in w iff for every world in which C is true and D is false, there is a closer world in which both C and D are true. (At least, I think those two statements are equivalent.)
Your definition requires that there be a closest world to w in which C is true. But there is no guarantee that there is such a world. There might be a continuum of worlds approaching w but no closest one.
My post is an “intuitive introduction to the subject”, which is “no substitute for rigorously going through the formal definitions” :-) But yeah, it can get complicated, especially as the “distance measure” need not be a metric at all.