square of opposition a graphic representation of various logical relations among categorical propositions. (Relations among modal and even among hypothetical propositions have also been represented on the square.) Two propositions are said to be each other’s (1) contradictories if exactly one of them must be true and exactly one false; (2) contraries if they could not both be true although they could both be false; and (3) subcontraries if at least one of them must be true although both of them may be true. There is a relation of (4) subalternation of one proposition, called subaltern, to another called superaltern, if the truth of the latter implies the truth of the former, but not conversely.
Applying these definitions to the four types of categorical propositions, we find that SaP and SoP are contradictories, and so are SeP and SiP. SaP and SeP are contraries. SiP and SoP are subcontraries. SiP is subaltern to SaP, and SoP is subaltern to SeP. These relations can be represented graphically in a square of opposition: The four relations on the traditional square are expressed in the following theses:
Contradictories: SaP S -SoP, SeP S -SiP
Contraries: -(SaP & SeP) or SaP P -SeP Subcontraries: SiP 7 SoP Subalterns: SaP P SiP, SeP P SoP For these relations to hold, an underlying existential assumption must be satisfied: the terms serving as subjects of propositions must be satisfied, not empty (e.g., ‘man’ is satisfied and ‘elf’ empty). Only the contradictory opposition remains without that assumption. Modern interpretations of categorical propositions exclude the existential assumption; thus, only the contradictory opposition remains in the square. See also SYLLOGIS. I.Bo.