Hellman then introduces:
Identity of Physical Indiscernibles (IPI): (∀u)(∀v)((∀φ)(φu ↔ φv) → (u = v), where φ ranges over physical predicates and u and v range over arbitrary n-tuples of physical objects.
IPI says that if u falls under a φ-predicate if and only if v does, then u and v are identical. Letting ψ range over predicates not in Γ (i.e., non-physical predicates used to describe phenomena in any branch of science), and by contraposition of Leibniz’s laws, IPI is equivalent to:
IPI’: (∀ψ)(∀u)(∀v)(∃φ)(ψu & ¬ψv → φu & ¬φv).
IPI’ says that for any non-physical predicate and any distinction that it makes, there is a physical predicate that makes that distinction as well.
Now, in a footnote Hellman explains that IPI’ is different from IPI”, where the existential quantifier (∃φ) comes before (∀u)(∀v). IPI” says that for any ψ there is a φ that distinguishes just what ψ does. IPI” implies (∀ψ)(∃φ)(∀u)(ψu ↔ φu), in cases where ψ is neither universal nor empty, which says that every nonphysical predicate has the same extension as a physical predicate and is a limited form of reductionism. IPI’ is a lot weaker than IPI”, and doesn’t imply even limited reductionism.
The upshot will be that none of the principles, PE, IPI and IPI’ imply reductionism or even accidental extensional equivalence between the φ and the ψ.
