What kind of system are we in if we explicitly take its negation as an axiom?

Here's the identity of indiscernibles.

(1) ( ∀P.P(x)↔P(y) ) → x = y

Here it is written in prenex normal form

(2) ∀x∀y.∃P.(x=y) ∨ (P(x)↮P(y))

The negation of (2) can be though of as giving us two indistinguishable but otherwise propertyless constants c and d in our domain.

(3) ∃c∃d.∀P.(c≠d)∧(P(c)↔P(d))

Are there any go-to paradoxes / counterintuitive results if we take (3) as an axiom in a second-order logic system?

  • Maybe useful Max Black, The Identity of Indiscernibles, (Mind, 1952) and Identity. – Mauro ALLEGRANZA Dec 24 '18 at 19:54
  • The thought experiment in the first source seems to mostly be about whether a notion of identity that captures an intuitive notion of equivalence would have identity of indiscernibles or not. Personally, I'm okay if = describes an equvialence relation that is so fine-grained as to be useless in practice. In the first part though, there's a bomshell: Q(x) := (x = c) is now ruled out as a predicate. I am also okay with saying that "pseudopredicates" that contain = aren't predicates and aren't quantified over by second-order variables, but is that fix enough to make a coherent system? – Gregory Nisbet Dec 24 '18 at 20:58
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    The position that denies the identity of indiscernibles is called haecceitism, the haecceity (literally, thisness) is that which distinguishes things that share all of their predicates. There are some formalizations of haecceitism in modal logic that are referenced in the SEP article. See also conceptualizations of the non-idenity of indescernible quantum objects. – Conifold Dec 24 '18 at 21:47
  • @Conifold re that qt-idind stuff, the sep article seems correct to me, as far as it goes, but these so-called particles are typically really field quanta, i.e., they don't actually exist as particles, per se, until/unless "pulled out of" the field by some observational/experimental act. So calling them "identical" can be a bit premature, so to speak. Maybe very loosely analogous to calling two babies(twins) identical -- before they're even conceived. – John Forkosh Dec 25 '18 at 8:50
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    Indiscernability of identicals is not the principle you mention in logical form, but its (much less discussed) converse. This is identity of indiscernibles. – Quentin Ruyant Dec 27 '18 at 16:45

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