In traditional logic the square of opposition summarizes the logical relationships between the four forms of subject-predicate proposition known as A, E, I, O: All X are Y; no X are Y; some X are Y; some X are not Y.
In the diagram, contradictories are propositions that cannot both be true and cannot both be false. Contraries cannot both be true, but can both be false. A proposition is subaltern to another if it is implied by it, but does not imply it. Subcontraries can both be true, but cannot both be false. The relations as indicated depend upon reading existential import into, for example, ‘All X are Y’. In modern quantification theory the generalization carries no implication that there are any things that are X, and indeed, is bound to be true when there are none (see vacuous).