The same basic language as was used for the basic finite probability logic need not change, but the semantics is slightly different: for every state w ∈ W, the component Pw of a modal probabilistic model is replaced by an entire probability space (Ωw, Aw, μw), such that Ωw ⊆ W and Aw is a σ-algebra over Ωw.
For B to be closed under resplicing is for it to be the case that for any property P such that its extension Pw in a world w is the extension of some property in B, P is a member of B. We have already noted that it is controversial whether such property-forming operations are legitimate, and, even if they are, we are often interested in property sets that are not closed under them.
Hale's dilemma is directed primarily against fictionalists who take the literal content of their fiction to be false, and those fictionalists prepared to ascribe some other status to their fictional claims avoid the dilemma as initially stated (though this route may encounter difficulties of its own, especially if it retains some sort of conditional analysis of the "according to PW" prefix).
On the other hand, if the fiction of possible worlds (PW) is only contingently false, Hale claims this also lands the Rosen's fictionalism in trouble, since if its falsehood is only contingent, then the fiction might have been literally true (or it is possible that the fiction be true).
However, it is precisely PW, or the Lewis story, that the fictionalist wants to argue is false in the actual world.
Maimon claims that the representations of space and time as intuitions arise as the result of the faculty of imagination, which is, as he describes it, the faculty of fictions (GW III, 61 | PW 37).
(p 65) Hale claims this is so because the claim "According to PW, there is a possible world at which PW is true" is equivalent for Rosen's fictionalist to "If PW were true, there would be a world at which PW is true": and this conditional is one which would be true whether or not PW was true.
Now, it is not true of the actual world that there is a king of France, and it is not true of any of the (supposedly fictional) possible worlds that PW is false there (after all, in the Lewis story, the Lewis story (or PW) holds at every world).
On the second horn, even fictionalists who accepted that they were committed to analysing their claim that PW could have been true as "If PW were true, there would be a world at which PW is true" could dispute Hale's claim that this is inadequate (see for example Divers 1999b, pp. 325 326).
We now know that for all the claims made by supporters and apologists, who say she hassled the apartheid regime for the release of Nelson Mandela, she actually made little or no effort to raise the issue during talks with the hardline South African president PW Botha at Chequers on 2 June 1984.
A formula P ≥ q is true at a pair (M,w), written (M,w) ⊧ P ≥ q, if and only if Pw({w′ | (M,w′) ⊧ ϕ}) ≥ q.
