Towards a Modal Syllogistic

How many valid syllogisms do we obtain when the four classical categorical forms (‘All A are B,’ ‘No A are B,’ ‘Some A are B,’ and ‘Some A are not B’) are extended to twenty modal forms by introducing a possibility operator P and a necessity operator N—allowing forms such as N(All A are B), P(Some A are B), ‘Some A are not P(B)’, and ‘All A are N(B)’—and given a semantics that rigidly fixes the actual world, where N(All A are B) means that in all possible worlds all A are B, ‘All A are N(B)’ means that everything that is A in the actual world is B in every possible world in which it exists, and ‘Some A are P(B)’ means that there is something in the actual world that is B in at least one possible world?