When the ontological status of a property or attribute is in question, the central issue is the ontological status of an object’s possessing or exemplifying a property. When the ideological status of a property or attribute is in question, the central issue are the kinds of predicates that express the property. The following example brings this out starkly: A property can be mental, but not physical (ideologically), but can be ontologically physical, but not mental.
If a mental predicate is not lawlike coextensive with a physical predicate , where and are appropriate vocabularies, the attribute expressed by is mental, but not physical. Nevertheless, by the principle of physical exhaustion (PE), the attribute is physical, but not mental (ontologically), as it is possessed solely by physical objects.
Entity-wise, every attribute is mathematical-physical, but only those attributes are mathematical-physical that are expressible by mathematical-physical predicates. So, if physical reductionism is false (which is likely the case), there are attributes which are mathematical-physical entities, but are not mathematical physical attributes.
The confusion arises when terms like “mental” or “material” are used to call out properties or attributes without clearly setting out whether one is talking about the ontological or ideological status of the attribute. This is an important point by Hellman, and we will see how this distinction and physicalist materialism in general are useful in clarifying questions of reductionism and materialism in the philosophy of mind when we turn to an analysis of Chalmers’ anti-materialist claims.