Now, before moving on to §1.2, we follow Hellman in noting that IPI an IPI’ don’t imply PE, but they do imply that there can be no more than one entity apart from the sum of all basic physicl entities. IPI and IPI’ are stronger than PE, since everything may be exhausted by mathematic0-physical entities, but physical language may not be powerful enough to tell among nonidenticals. In any case, like I said in the previous post, the upshot is that while IPI and IPI’ make use of the expressive power of the physical language, none of the principles, PE, IPI and IPI’ imply reductionism or even accidental extensional equivalence between the φ and the ψ. And, regardless of its complexity, no physical predicate covers the the extension of any biological or psychological predicate.
To summarize, in §1.1, Hellman has accomplished the following:
1. He has set out the physicalist thesis:”Everything is accounted for by mathematico-physical things satisfying predicates in Γ”.
2. He shows us how to build up a physicalist ontology using Γ.
3. Given the physicalist ontology, he defines the principle of Physical Exhaustion (PE): (∀x)(∃α) (x ∈ R(α)), where R(α) is a rank in the hierarchy.
4. Then he introduces the principle of the 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.
5. And follows up by introducing IPI’: (∀ψ)(∀u)(∀v)(∃φ)(ψu & ¬ψv → φu & ¬φv), which says that for any non-physical predicate and any distinction that it makes, there is a physical predicate that makes that distinction as well.
6. He points out that PE, IPI and IPI’ do not imply reductionism or even accidental coextension between the physical and non-physical predicates.