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I found an exercise page 126 of the book 'Logic for Philosophy' by Sider. The exercise asks to prove in second-order logic the identity of indiscernibles. I tried to write a proof ex absurdo, but I cannot find the contradiction in the hypothesis that the principle is false. I attach the demonstration: which step is missing?

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  • Use the SOL definition of identity.
    – Mauro ALLEGRANZA
    Sep 16, 2022 at 18:20
  • Why can 'a = b' be defined as 'For all X (Xa iff Xb)'? Does this not require a further demonstration?
    – Frank
    Sep 16, 2022 at 18:39
  • The def of identity is "two objects are identical iff they share all ppropertjes". From this Identity of indiscernibles immediately follows.
    – Mauro ALLEGRANZA
    Sep 17, 2022 at 7:10
  • My question is: Why is identity defined in that way?
    – Frank
    Sep 17, 2022 at 17:22
  • Because if two thins are different, i.e. not identical, they differ in some aspect.
    – Mauro ALLEGRANZA
    Sep 17, 2022 at 17:45

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