A possible world is a way in which it is logically possible (does not result in contradiction) for reality to exist. There is a possible world in which everything is exactly the same, but it is 1 degree warmer than it is now; there is a possible world in which COVID-19 never existed; there is a possible world in which everything is exactly the same, but there is +1 more person existing right now. What is the nature of these possible worlds, and specifically, what is the nature of my relationship to myself in these possible worlds?
If I have a square table, there is a logically possible world in which everything else is exactly the same as it is in our actual world, except that the table is circular. But it cannot be the case that table Ca in the actual world (where a is the referent table and C is the predicate “circular”) is equal to Sb in this possible world (where S is the predicate “square”); it is false that Ca = Sb, because if a = b then we would have to say that:
Where x and y are quantified over everything in the actual and possible universe respectively. In this conditional, we know that we cannot have Ca∧Sa∧Cb∧Sb because this is saying “a is circular and a is square and b is circular and b is square” which is a contradiction and so by modus tollens we know that (a=b→Ca∧Sa∧Cb∧Sb) is false and so, once again by modus tollens, we know that ((a=x∧b=y)∧(Ca∧Sb)∧(x=y)) is false. Thus, x and y are quantified over different things: various possible worlds are ontologically distinct from one another.
This has consequences for free will. Free will means that, given a decision between P and Q, if we say that Pa is my decision in the actual world, then Qb is my decision in the possible world. But, if a = b is false, since a = x and b = y and x and y quantify over different, distinct things, then the actual version of myself a and the possible version of myself b are distinct people (it is not the case that the referents of “Tom in the actual world” and “Tom in the possible world” are coextensive). This means that a did not have a choice but to choose P because the possible version of me that chose Q is not the same person that chose P.