Should we accept a symmetric accessibility relation for possible worlds semantics within metaphysics?

Possible worlds semantics is a common tool for doing metaphysics. An interesting question is the question of which properties should be ascribed to the accessibility relation between the possible worlds. Now, surely, we need reflexivity: each possible world can access itself. On standard Kripkean semantics, this guarantees that necessarily true propositions in each world are also true in that world, which is a principle without which possible worlds metaphysics becomes senseless. Moreover, we need the accessibility relation to be transitive. After all, if possible world w2 is possible from the perspective of possible world w1, that is, if w2 is accessible from w1, and if possible world w3 is possible from the perspective of possible world w2, that is, if w3 is accessible from w2, then, plausibly and reasonably, possible world w3 is also possible from the perspective of possible world w1, which means that w3 is also accessible from w1. Indeed, transitivity guarantees that necessity is stable or robust: if a proposition p is necessarily true in some possible world, then p is also necessarily necessarily true in that world. Now, what about symmetry? Should we also require that the accessibility relation be symmetric, meaning that if possible world w2 is accessible from w1, then w1 is also accessible from w2? This doesn't seem to be the case. Here is a proposed counterexample. Consider a possible world w1 and consider the conjunction p of natural laws in w1. Suppose that in w1 the natural laws are necessarily true, so that p is necessarily true in w1. Thus, p is true in each possible world accessible from w1, which includes w1 itself due to reflexivity. Now, consider a possible world w2 in which p is false and whose conjunction of natural laws, say q, thus differs from p, and in which the natural laws are merely contingently true. So, there is at least one possible world accessible from w2 in which q is false and whose conjunction of natural laws thus differs from q. In this example, since p is false in w2, w2 is, given Kripkean semantics, not accessible from w1. But plausibly, w1 is still accessible from w2. For while the natural laws in w2 are contingently true and have q as conjunction, it's from the perspective of w2 reasonably still metaphysically possible that they could have been necessarily true and have p as conjunction. That is, w1 is possible from the perspective of w2. In this specific plausible case symmetry thus does not hold. It follows that we should not accept that the accessibility relation is symmetric. Therefore, the most adequate modal logic for doing possible worlds semantics within metaphysics is S4 (reflexivity and transitivity) rather than the stronger S5 (reflexivity, transitivity and symmetry) or the weaker KT (only reflexivity).

Is God necessarily necessarily good?

On mainstream theism God has his properties necessarily. Take one of these properties, say, goodness. Theists normally hold that God is necessarily good. It is simply impossible for God not to be good. But is God also necessarily necessarily good? That is to say, although God is in fact necessarily good, could it have been otherwise? Could reality have been such that God is not necessarily good? And if not, is that necessarily so?

A Scotistic Cosmological Argument

The following modal cosmological argument goes back to John Duns Scotus.

(Premise) Possibly, there is an object that cannot have a cause,
(Premise) Every contingent object, possibly, has a cause,
(Conclusion) There is a necessary object.

The logical derivation is quite straightforward. According to the first premise there is a possible world that contains an object, let us say A, that is uncaused in every possible world in which it obtains. Now, due to the second premise, object A cannot be a contingent object. Therefore, object A is a necessary object.

Both premises seem to be plausible. Yet, together they entail an very interesting metaphysical conclusion, that is, the conclusion that there is something in the actual world that exists necessarily.