Square of Opposition in Aristotelian Logic

Aristotelian logic, or the traditional study of deduction, deals with four so-called categorical (or subject-predicate) propositions, which can be defined by:
⇔ all is (universal affirmative or A proposition),
⇔ some is (particular affirmative or I proposition),
⇔ no is (universal negative or E proposition),
⇔ some is not (particular negative or O proposition).
is called a subject (or minor) term and is called a predicate (or major) term of the proposition. We could think of and as one-place nonempty predicates or sets.


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A diagram showing the relations among categorical propositions is known as the traditional square of opposition [2, p. 217]. Two propositions are contradictories if one is the negation of the other. Propositions A and O, and propositions I and E are contradictories. Two propositions are contraries if they cannot both be true. Two propositions are subcontraries if they cannot both be false. Proposition A is called superaltern, I is called subaltern, and the corresponding relation is called subalternation. The same definitions are applied to E and O [2, pp. 214–217]. Under the assumption that class is not empty, propositions A and E are contraries, propositions I and O are subcontraries, and superaltern implies subaltern.
[1] L. Carroll, Symbolic Logic and The Game of Logic, New York: Dover Publications, 1958.
[2] I. M. Copi and C. Cohen, Introduction to Logic, 9th ed., New York: Macmillan Publications, 1994 pp. 214–218.
[3] J. M. Bocheński, A History of Formal Logic, 2nd ed. (I. Thoma, trans. and ed.), New York: Chelsea Publishing Company, 1970 p. 235.
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