but neither C1 nor C2 wholly contains the other. Here is a brief argument from
Westerhoff’s account to Sommer’s Law.^13 Ontological categories are base sets. Base
sets are form sets. Some things belong to the same form set if and only if they are
intersubstitutable in all states of affairs. Assume that there are two base sets B1 and
B2 that merely overlap; let B- be the intersection of B1 and B2. Everything in B- is
intersubstitutable with everything in B1, and everything in B- is intersubstitutable
with everything in B2. So everything in B1 is intersubstitutable with everything in B2.
But then neither B1 nor B2 is a form set (rather the union of B1 and B2 is a form set)
and so neither B1 nor B2 is a base set. So our assumption was false; base sets cannot
merely overlap.
Why is this a worry? Westerhoff thinks that, far from being an objection, satisfying
Sommer’s Law is actually a good feature of an account. He is not alone: Amie
Thomasson (2004, section 1.5) also claims that a minimal condition of adequacy is
that the categories be mutually exclusive, which implies Sommer’s Law.^14
Westerhoff (2005: 57–9) discusses three putative reasons for endorsing Sommer’s
Law. First, he notes that Sommer’s Law imposes a tree structure on the set of
ontological categories, and this tree structure is recognized by most systems of
ontological categories. Second, Westerhoff (2005: 58) discusses the work of Frank
Keil (1979), which on Westerhoff’s view supports the claim that“categorizations
with non-overlapping categories (which are hierarchies and not lattices) are particu-
larly psychologically natural.”Third, Westerhoff claims that Sommer’s Law allows us
to prove that every system of ontological categories has a topmost category. (The
details of the proof are not provided by Westerhoff; I cannot speak of whether it
is successful.)
None of these arguments is particularly compelling. That many ontological
systems have a tree structure does not show that all plausible systems must. (We
will see shortly that some do not.) Second, the fact that we are attracted to systems of
overlapping categories shows only that we will be psychologically hesitant to accept
systems of non-overlapping categories. But does it provide a reason to think that
such systems could not be ontological systems or that we are unlikely to have good
reasons for endorsing them as such?^15 Third, the technical result that every system of
ontological categories must have a maximal category, is not an advantage, but rather
is possibly a problem, since some systems of ontological categories, such as the
Aristotelian system discussed earlier, lack a unique maximal element. If neutrality
(^13) Compare with Westerhoff (2005: 138).
(^14) Nolan (2011: 78) by contrast refrains from insisting on exclusivity and exhaustivity. Paul (2013) also
does not appear to demand the conceptual necessity of exclusivity or exhaustivity, though on her preferred
ontological system, there is just one ontological category. Van Inwagen’s (2014) account of ontological
categories also is neutral on exclusivity; it is less clear to me whether he wishes to take a stand on
exhaustivity. 15
Rosenkrantz (2012: 86) tells us that we should prefer systems of ontological categories in which none
of them merely overlap, but he offers zero justification for this.