Aristotle represented the relations of the four logical propositions by what is termed the _square of opposition_. Viewed from the standpoint of the square, the relations may be summed up as follows:
As related to each other, A and E are said to be contrary because they seem to express _contrariety_ to the greatest degree.
If one is true, the other must be false, but both may be false.
(1) If one is true, the other must be false; e. g., if A is true, as “All metals are elements,” then E is false, as “No metals are elements.” Or, if E is true, as “No birds are quadrupeds,” then A is false, as “All birds are quadrupeds.”
(2) Both may be false. If A is false, as “All men are wise,” then E may be false, as “No men are wise.”
Propositions I and O are said to be related to each other in a subcontrary manner because they are contrary as to each other and “_under_” their universals A and E.
If one is false, the other must be true, or, both may be true.
(1) If one is false, the other must be true.
If I is false, as “Some metals are compounds,” then, O is true, as “Some metals (at least) are not compounds.” Or, if O is false, as “Some metals are not elements,” then I is true, as “Some metals are elements.”
If I is true, as “Some men are wise,” then O also may be true, as “Some men are not wise.”
Etymologically considered subaltern means _under the one_, thus proposition I is under A, and O is under E.
Subalterns are related to each other as are the universals and particulars; hence,
(1) If the universal is true, the particular under it is also true; while if the particular is true, the corresponding universal may, or, may not, be true.
(a) If the universal is true, the particular under it is true.
If A is true, as “All metals are elements,” then I is true, as “Some metals are elements.” Or, if E is true, as “No metals are compounds,” then, O is also true, as “Some metals (at least) are not compounds.”
(b) If the particular is true, the corresponding universal may, or, may not, be true.
If I is true, as “Some men are wise,” or, “Some men are mortal,” then A may be false, as “All men are wise,” or, A may be true, as “All men are mortal.” Or, if O is true, as “Some men are not wise,” or, “Some men are not immortal,” then E may be false, as “No men are wise”; or, true, as “No men are immortal.”