Evaluating PE and IPI

§1.2 is shorter but makes a very important point about PE and IPI, reductionism and dualism. Let physical reductionism be the claim that for the theory, formulated in a suitable language, that contains all the lawlike truths of science (including physical science), every scientific predicate is definable in physical terms.  That is to say, for every n-place predicate P, one can derive using only the laws of science, a formula of this form:

$(\forall x_{1})\dots (\forall x_{n}) (\textup{P}x_{1} \dots x_{n} \leftrightarrow \textup{A})$

Here, A is a finite sentence that contains only physical vocabulary as the nonlogical terms and n distinct variables, $x_{1} \dots x_{n}$. These equivalences are provable within scientific theory and are logical consequences of its laws. What may come as a shock to those who want to avoid mysterious dualism by adhering to such a strong form of reductionism is that even this reductionism is compatible with ontological dualism.

Hellman sets out the following basic theory to make this point. Let $\Sigma$ be the theory that contains only two one-place predicates, P and Q and these non-logical axioms:

$(\exists x)(\exists y) (x \not= y \wedge (\forall z) (z = x \vee z = y))$
$(\exists x) (\textup{P}x \wedge (\forall y) (\textup{P}y \rightarrow y = x))$
$(\exists x) (\textup{Q}x \wedge (\forall y) (\textup{Q}y \rightarrow y = x))$
$(\forall x) (\textup{P}x \vee \textup{Q}x)$

All that $\Sigma$ says is that there are just two objects and that just one of them is a P and just one of them is Q and everything else is either a P or a Q.  It follows from $\Sigma$ that Q is definable in terms of P, but $\Sigma$ gives no assurance that every object is exhausted by things that are P –as a matter of course, every interpretation where the axioms of $\Sigma$ are true partitions the domain into disjoint subsets of P and Q type things.  And dualism here is a minimal case, since the reasoning can be carried over to any finite number of predicates.

This is a quick and elegant way of showing that reductionism is no shelter from dualism.

What Hellman concludes here is that while PE is necessary for physicalism, neither it nor IPI are sufficient for these reasons: PE does is silent about the scope and power of physical laws, while in IPI, quantification is restricted to the actual world and consequently neither can be used to voice the physicalist thesis that all phenomena are determined by physical phenomena.

We’ll see in §2 what the precise connection is between determination, definability and reduction.