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Related to intrinsicality: Extrinsicality
See: character
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See, for example, Rorty, "Daniel Dennett on Intrinsicality," 98-121.
Given Best's deference to Wittgenstein, he might more consistently conclude that the notion of intrinsicality is essentialist baggage, and more trouble than it is worth.
The standard counterexamples to David Lewis's account of intrinsicality involve two sorts of properties: identity properties and necessary properties.
So far as I can see, at this point there are two possible avenues to take as regards contingent intrinsic predication, once both simplicity and truthmaker necessitarianism are countenanced: (a) maintain that apparently intrinsic, contingently true predications are actually extrinsic, or (b) maintain the intrinsicality of contingently true predications while denying that this poses a problem for the contingency of certain predications.
In response it proposes an account of intrinsicality that is compatible with both priority monism and priority pluralism.