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Ideas from 'The Identity of Indiscernibles' by Max Black [1952], by Theme Structure

[found in 'Metaphysics - An Anthology' (ed/tr Sosa,E. /Kim,J.) [Blackwell 1999,0-631-20279-x]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10196 The Axiom of Choice needs a criterion of choice
                        Full Idea: Some mathematicians seem to think that talk of an Axiom of Choice allows them to choose a single member of a collection when there is no criterion of choice.
                        From: Max Black (The Identity of Indiscernibles [1952], p.68)
9. Objects / A. Existence of Objects / 5. Individuation / b. Individuation by properties
10194 Two things can only be distinguished by a distinct property or a distinct relation
                        Full Idea: The only way we can discover that two things exist is by finding out that one has a quality not possessed by the other, or else that one has a relational characteristic that the other hasn't.
                        From: Max Black (The Identity of Indiscernibles [1952], p.67)
                        A reaction: At least this doesn't conflate relations with properties. Note that this idea is clearly epistemological, and in no way rules out the separateness of two objects which none of us can ever discern. Maybe the Earth has two Suns, which imperceptibly swap.
9. Objects / F. Identity among Objects / 5. Self-Identity
10193 The 'property' of self-identity is uselessly tautological
                        Full Idea: Saying that 'a has the property of being identical with a' is a roundabout way of saying nothing - a useless tautology - and means not more than 'a is a'
                        From: Max Black (The Identity of Indiscernibles [1952], p.66)
                        A reaction: This matter resembles the problem of the number zero, and the empty set, which seem to be crucial entities for logicians, but of no interest to a common sense view of the world. So much the worse for logic, I am inclined to say.
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
10195 If the universe just held two indiscernibles spheres, that refutes the Identity of Indiscernibles
                        Full Idea: Isn't it logically possible that the universe should have contained nothing but two exactly similar spheres? ...So two things would have all their properties in common, and this would refute the Principle of the Identity of Indiscernibles.
                        From: Max Black (The Identity of Indiscernibles [1952], p.67)
                        A reaction: [Black is the originator of this famous example] It also appears to be naturally possible. An observer at an instant of viewing will discern a relational difference relative to themselves. Most people take Black's objection to be decisive.