subcontrarias

Subcontrarias are a pair of categorical propositions in the Aristotelian square of opposition, specifically the particular affirmative and the particular negative: Some S are P (I) and Some S are not P (O). They are called subcontraries because they occupy the lower pair of the square, and their logical relation is that both statements can be true at the same time, but they cannot both be false (at least within the traditional, nonempty subject class).

In contrast, contraries (All S are P and No S are P) cannot both be true, and subalternation relates universal and particular propositions (A implies I, E implies O) within the traditional framework. This latter relation assumes existential import for universal statements, so that the truth of A or E can entail the corresponding I or O only if the subject class S is nonempty.

Formally, I is expressed as ∃x(Sx ∧ Px) and O as ∃x(Sx ∧ ¬Px). The defining feature of subcontraries is that there can be a situation where both I and O are true—some S are P and some S are not P—while it is also possible for only one of them to be true. They cannot be simultaneously false only under interpretations that preserve existential import for S.

Historically, subcontrarias were central to the medieval and early modern treatment of syllogisms and the square of opposition. In contemporary logic, the precise relationships among categorical propositions are often treated via first-order logic, where the classical mutual exclusivity of the pair depends on assumptions about existence and the interpretation of universals. Subcontrarias remain a useful concept for understanding classical syllogistic structure and historical debates about existential import.