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Ideas for 'Structures and Structuralism in Phil of Maths', 'Scientific Objectivity' and 'Intro: Theories of Vagueness'

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2 ideas

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5

9065	S5 collapses iterated modalities (◊□P→□P, and ◊◊P→◊P) [Keefe/Smith]
	Full Idea: S5 collapses iterated modalities (so ◊□P → □P, and ◊◊P → ◊P).
	From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §5)
	A reaction: It is obvious why this might be controversial, and there seems to be a general preference for S4. There may be confusions of epistemic and ontic (and even semantic?) possibilities within a single string of modalities.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

10166	ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price]
	Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements.
	From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
	A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do?