Suppose the Brouwer axiom is false. Thus, there is some possible world w1 such that at w1 our world w0 is impossible. Here’s another world that’s impossible at w1: the world w⊥ at which every proposition is both true and false. Thus at w1, possibility does not distinguish between our lovely world and w⊥. But that seems to me to be a sign that the possibility in question isn’t making the kinds of distinctions we want metaphysical modality to make. So, if we are dealing with metaphysical modality, the Brouwer axiom is true.
You can't run this argument if you run this one.
If Brouwer is false, then necessity depends on the world you’re at, but the idea of necessity is that it should be independent from such things.
ReplyDeleteWell, I am thinking that the opponent of Brouwer opposed Brower precisely because she thinks there should be variation in necessity between worlds.
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