# What if what is possible follows from logic alone

Suppose that every imaginative variation [I don't mean this in any technical sense] of what is physically possible is necessarily possible: because what I can imagine follows from logic alone.

already: □ (◊A → ◊A) what i think i am asking is: ◊A → □◊A and so: ◊A → □◊A → A

or have I misinterpreted that something follows from logic alone?

If not, and if you equate imaginability with metaphysical possibility, then anything about the physical world that is metaphysically possible is metaphysically actual. Which is, I confess, somewhat absurd.

• You seem to be committing an error here: □◊A → A. The necessarily possible is not necessarily the actual. Merely because the converse holds (A → □◊A → ◊A) does not mean the opposite holds. – virmaior Aug 19 '14 at 10:43
• i'm not sure that in my haste i'm not misreading it but: "□◊A → A is provable from A → □◊A" – user6917 Aug 19 '14 at 11:31
• really, you main it is possible to prove that the necessarily possible is actual because the actual is necessarily possible? – virmaior Aug 19 '14 at 11:33
• To put it another way, at a fork in the road, I go right. This makes it necessarily possible that I go right. Not sure why the converse follows: if it is necessarily possible to go right, I go right. – virmaior Aug 19 '14 at 11:34
• "Dual to the theorem B, S5 has the theorem BO. which means that whatever is possibly necessary is simply so." This would mean (1) <>[]A -> A; not(1*) []<>A -> A. The reason (1) is valid in S5 is that every world is accessible from every other world, so if it is true in world w that there is some world w* such that A is true in every world accessible from w* A is the case, then A must be the case in w also--as w is accessible from w*. That wouldn't be the case in sentence (1*) though. The order of the modal operators matters. – shane Aug 19 '14 at 12:40