A central Aristotelian insight is that substances exist in a primary way and other things—say, accidents—in a derivative way. This insight implies that use of a single existential quantifier ∃x for both substances and forms does not cut nature at the joints as well as it can be cut.
Here are two pieces of terminology that together not only capture the above insight about existence, but do a lot of other (but closely related) ontological work:
a fundamental quantifier ∃u over substances.
for any y, a quantifier ∃yx over all the (immediate) modes (tropes) of y.
We can now define:
a is a substance iff ∃u(u = a)
b is a (immediate) mode of a iff ∃ax(x = b)
f is a substantial form of a substance a iff a is a substance and ∃ax(x = f): substantial forms are immediate modes of substances
b is a (first-level) accident of a substance a iff u is a substance ∃ax∃xy(y = b & y ≠ x): first-level accidents are immediate modes of substantial forms, distinct from these forms (this qualifier is needed so that God wouldn’t coount as having any accidents
f is a substantial form iff ∃u∃ux(x = f)
b is a (first-level) accident iff ∃u∃ux∃xy(y = b).
This is a close variant on the suggestion here.
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