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.” _Second Relation._ (2) If the universal is false, the particular under it may or may not be true, but, if the particular is false, the universal above it must be false. _Illustrations._ (a) If the universal is false, the particular under it may or may not be true. If A is false, as “All metals are compounds,” or “All men are wise,” then I may be false, as “Some metals are compounds,” or, I may be true, as “Some men are wise.” Or, if E is false, as “No men are mortal,” or, “No men are wise,” then O may be false, as “Some men are not mortal,” or, O may be true, as “Some men are not wise.” (b) If the particular is false, the universal above it must be false. If I is false, as “Some men are trees,” then A is false, as “All men are trees.” Or, if O is false, as “Some men are not bipeds,” then E is also false, as “No men are bipeds.” 4. _Contradictory Propositions._ _Why so named._ The propositions A and O, likewise E and I, are called contradictory propositions because they oppose each other in both quantity and quality. They are mutually opposed to each other or _absolutely_ contradictory. _Relation stated._ If one is true the other must be false. _Illustrations._ (1) A and O compared. If A is true, as “All metals are elements,” then, O is false, as “Some metals are not elements.” Or, if O is true, as “Some metals are not compounds,” then A is false, as “All metals are compounds.” (2) E and I compared. If E is true, as “No birds are quadrupeds,” then I is false, as “Some birds are quadrupeds.” Or, if I is true, as “Some birds are bipeds,” then E is false, as “No birds are bipeds.” The chief value of the square of opposition springs from the contradictory propositions. The square shows conclusively that any universal affirmative assertion (an A) may best be contradicted by proving a particular negative (an O). For example: To satisfactorily refute the statement that, in this section, all birds migrate to the south in winter, it would be sufficient to prove that the English sparrow and starling do _not_ migrate to the south. The square likewise makes evident that any universal negative (an E) may be conclusively denied by establishing the truth of a particular affirmative (an I). To illustrate: The easiest way to prove the falsity of “No trusts are honest” is to present facts showing that at least trusts A and B _are_ honest.