Let us pretend that the only fundamental predicates of individuals are shape
predicates, e.g.,“is a triangle,”“is a square,”and the only fundamental predicate of
sets is the subset predicate. The set of shape predicates is an intra-categorial set
of predicates, and the singleton set of the subset predicate is an intra-categorial set as
well. The shape predicates are all monadic, and their logical form permits construc-
tion of open sentences of the same kind as“Fx 1 .”There is one inter-categorial
predicate, namely, the two-place predicate“is a member of.”It is a dyadic predicate,
and its logical form permits the construction of the following kinds of open sen-
tences:“x 1 is an element ofs 1 ”and“s 1 is an element ofs 2 .”
The syntactic rules of this language prohibit each of the following:
(i) (^9) indx 1 , (^9) indx 2 :x 1 is a member ofx 2.
(ii) (^9) sets 1 :s 1 is a square.
(iii) (^9) indx 1 , (^9) sets 1 :s 1 is a member ofx 1.
Each of (i)–(iii) is a category mistake. In a metaphysically perfect language, they
are utterly without meaning, since the syntactical rules of the language disallow any
of (i)–(iii) from being well-formed sentences. These pseudo-sentences fail to respect
the logical forms of the predicates that appear within them.
Taking on board the full generality of the notion of a natural expression allows us
to provide a justifying rationale for the feeling that something is terribly wrong with
sentences like“my table has three subsets.”There are many ways for a language to fall
short of metaphysical perfection. One way is by having the wrong primitive predi-
cates. A language that has, for example, “is grue”and“is bleen”as primitive
predicates is less than ideal. Another way in which a language can fall short is by
having the wrong primitive quantifiers. A third and related way a language can fall
short is by having, in a sense, the right predicates, but having predicates whose
surface logical form does not match the logical form of the underlying properties
they stand for.
We have a predicate in English,“is simultaneous with,”and the apparent logical
form of this predicate is shown by open sentences such as“xis simultaneous withy.”
Let us suppose thatsimultaneity relative to a reference frameis a perfectly natural
three-place relation. There is still a strong temptation to think that our predicate“is
simultaneous with”refers to this relation. We learned that simultaneity does not have
the logical form we thought it had. We did not learn that“simultaneity”is like“filled
with phlogiston”in that neither corresponds with something in the world. Accord-
ingly, we can’t complain that“is simultaneous with”is defective in virtue of failing to
stand for something natural. Buthowit stands for something natural is defective: it
would be metaphysically better to have a predicate that matches the logical form of
the relation that is its content.
Similarly, perhaps,“is a member of”stands for a perfectly natural relation,is a
member of. It would be better to have a predicate that stands for this relation whilst
simultaneously not allowing sentences in which, as it were, material objects have