Ploucquet’s “Refutation” of the Traditional Square of Opposition

Abstract

In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”.