It Adds Up After All: Kant's Philosophy of Arithmetic in Light of the Traditional Logic¹

Officially, for Kant, judgments are analytic iff the predicate is “contained in” the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like ‘7 + 5=12.’ Kant is correct. Operation concepts (> 7+5 >) bear two relations to number concepts: >7< and >5< are inputs, >12> is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic.