Two propositions are contrary if they cannot both be simultaneously true but it is possible for both to be simultaneously false. For example, the proposition that “every man is just” is contrary to the proposition that “no man is just,” since both propositions may be false if some men are just.
Two propositions are contradictory if both cannot be simultaneously true and both cannot be simultaneously false. The proposition that “not every man is just” is contradictory to the proposition that “every man is just,” because both cannot be simultaneously true, nor can they be simultaneously false.
Note that contraries are two universal propositions and contradictories must have one universal and one existential proposition. And note that one proposition is the negation of the other — but there are two kinds of negation: contrary and contradictory.
Fregean logic handles these two kinds of negation by segregating them: contradictory (outer) negation goes before the quantifier and contrary (inner) negation goes after it. So these expressions are equal:
All x aren’t y as −∀x: x ⊂ y = ∃x: x ⊂ −y
Some x aren’t y as −∃x: x ⊂ y = ∀x: x ⊂ −y.
The other purpose of quantifiers is to bind a variable as universal or existential.
George Spencer Brown’s Laws of Form does something similar in two dimensions with his “cross” symbol ( ⏋). Contradiction is represented in the horizontal dimension via the Law of Calling. Contraries are represented in the vertical dimension via the Law of Crossing.
The intersection of horizontal and vertical crosses is a single cross, which in the interpretation for logic represents negation. With a variable under or ‘inside’ it, the cross represents “non” or “no” as in “non-A” or “no A”.