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In denying World Haecceitism, Lewis forgoes a one-to-one correspondence between maximal possibilities and possible worlds.
This correspondence allows ersatzists to treat talk of maximal possibilities and possible worlds as largely interchangeable for most theoretical purposes.
Some formulations of quidditism invoke either maximal possibilities or possible worlds, while modalist formulations of quidditism do without either resource.
But, for most realists about possible worlds, quantification over maximal possibilities requires or is simply equivalent to quantification over possible worlds.
Some of these frameworks involve a commitment to possible worlds or maximal possibilities, while other frameworks aim to do without these commitments.
Although formulations of haecceitism tailored to various ersatzist views differ significantly, they typically share a commitment to a one-one correspondence between possible worlds and maximal possibilities.
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Typically, such realist views of possible worlds identify each maximal possibility with a unique possible world and non-maximal possibilities with sets of possible worlds.
So, for any possibility, a maximal possibility includes that possibility or its negation.
Similarly, if it is possible for twins to swap their birth orders while leaving all qualitative matters unchanged, that maximal possibility also differs haecceitistically from actuality.
Similarly, if there is a maximal possibility according to which all actual causal-nomic facts are the same except that some actually uninstantiated property, schmass, occupies the causal and nomic role of mass, this maximal possibility would also differ quidditistically from actuality.
If, for example, there is a maximal possibility, otherwise indiscernible from actuality, where mass and charge swap their respective causal-nomic roles (e.g., where objects resist acceleration in virtue of their charge), this maximal possibility would differ quidditistically from actuality.
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