What is a one-level criterion of identity?
Harold W. Noonan · Analysis · 2009
Standardly, a one-level criterion of identity1 is given in the form: (1) ∀x∀y(Kx & Ky → (x = y ↔ Rxy)) where ‘K’ denotes the kind of thing for which the criterion is being given and ‘R’ denotes the criterial relation. Thus, we have, for example, the criterion of identity for sets: (2) ∀x∀y(Setx & Sety → (x = y ↔ ∀z(z is a member of x ↔ z is a member of y))) and for composites: (3) ∀x∀y(Compositex & Compositey → (x = y ↔ ∀z(x has z as a proper part ↔ y has z as a proper part))) and for events: (4) ∀x∀y(Eventx & Eventy → (x = y ↔ ∀z(x is a cause or effect of z ↔ y is a cause or effect of z))). (1) is equivalent to the conjunction of: (1a) ∀x(Kx → Rxx) and (1b) ∀x(Kx → ∀y(Ky & Rxy → (x = y))),