Not necessarily possible
1. Equivalence: For any worlds A and B, if A is accessible from B, then B is accessible from A.
2. Reflexivity: Every world is accessible from itself.
This fevered adding of axioms renders accessibility redundant and meaningless, since all possible worlds become equally accessible from all others. Will’s argument comes from two premises:
1. (Roughly, there was no hand out): Truths about what is possible are grounded in the actual world.
2. It is possible that there could be nothing at all.
The argument then is, call the actual world WA: it seems possible grounded in the actual world that there could have been nothing. Let us call this possibility WN. WN is accessible from WA. But WA is not accessible from WN, since if WN was the actual world, then there would be nothing in WN that would ground the truth of the possibility of facts in WA. Further more, it might also be claimed that WN is not accessible from itself. So WN provides a counterexample to equivalence and reflexivity.
A lot of the discussion focussed on the nature of N. People found reflexivity and equivalence to be so “intuitively plausible” that the argument just showed that N was impossible. But this kind of thing I found weak for this reason: the intuitive nature of the two dodgy axioms rests on a few examples. But all this shows is that reflexivity holds in some central cases, as does equivalence. To jump from this to the claim that it necessarily holds in all cases requires more than considering a few tailor made examples. It involves a proof, or at very least a confrontation with ordinary usage which in fact throws up many counterexamples. I’ve never seen anything like a proof, and when ever I have offered counterexamples I usually meet with hostile verbal evasion of this kind “You are confusing metaphysical modality with epistemic modality”.
Will began with the intuitive case for the two axioms by giving an example of something possible but not actual. He was actually standing, but he could have been sitting. Reflexivity and equivalence clearly hold in this case. Had he been sitting then it would have been possible that he was standing, and given that he was standing, it is clearly possible that he was standing. So far all we have got is a generalisation based on a single case. 1 is prime, 3 is prime, 5 is prime, 7 is prime, therefore all odd numbers without exception are prime.
Later, for contrast something impossible was considered. Tim is Portugal was the example. This is supposed to be clearly impossible. Why? Because it is nonsense, because it is impossible to imagine. But I thought we were talking about metaphysical possibility, not conceivability or semantics. A state can be defined in terms of its citizens and its territory and perhaps it constitution and economy. It is possible for a state to lose all its territory and all its citizens but one and that one could be Tim. The remaining citizen would then embody the whole of the state. If the state was democratic, and Tim was the last citizen, then the constitution of Portugal would be identical with Tim’s will, its territory identical with Tim’s property and body. Tim would be Portugal in this case. “I am the state of Portugal.” He could say with uncharacteristic grandeur. Is this scenario possible? What relevance does it have to the accessibility relationship?
Perhaps this is not possible because “Tim” and “Portugal” are rigid designators, and this scenario just changes Portugal too much for us to have a hold on what the possibility is supposed to consist in. With this in mind lets go back to the central example. It is possible that William Bynoe (rigidly designated) is sitting down at time t, when actually he was standing at that time. This is clearly not epistemic, since we know he was actually standing. So the accessibility relationship holds symmetrically between WA ( the actual world) and W1 (where William Bynoe was sitting not standing at time t). Now let us consider a second possibility: It is possible that William Bynoe (rigidly designated) does not exist. This is clearly not epistemic since in order to rigidly designate William Bynoe he must exist. The reference fixing must be grounded in the actual world. So the accessibility relationship holds between:
WA (the actual World) and W2 (a world in which William Bynoe does not exist). But is it symmetric? Well no, because in W2, William Bynoe doesn’t exist, so the possibility that William Bynoe is standing is not grounded in W2 and therefore can’t be a possibility. Complicated? Baroque? A misunderstanding of some kind? I don’t think so. Here are two propositions that, realist as you like about possibility, are straightforwardly true:
If William Bynoe did not exist then it would be impossible that he is now standing.
It is possible that William Bynoe did not exist.
There for it is possible that it is impossible that he is now standing.
This is a counterexample to the equivalence relation. It is also a counterexample to the axiom that whatever is actual is necessarily possible, since if William Bynoe is in fact now standing, then it is still possible that it is impossible that he is now standing.