The Square of Opposition (for logic nerds only!)

The Square of Opposition is a useful tool for representing and understanding the relationships between categorical statements with the same subject and predicate:


The relationships are defined this way:

Contrariety: The statements cannot both be true, but can both be false.
Contradiction: The statements cannot both be true, and cannot both be false.
Subcontrariety: The statements can both be true, but cannot both be false.
Subimplication: If the universal is true, the particular must be true.
Superimplication: If the particular is false, the universal must be false.

Students might be interested to discover that the square of opposition can be created for non-categorical statements as well. Consider compound propositions (Such as those we learn about in Intermediate Logic). We know, for example, that “She is a princess and a queen” implies that “She is a princess” — that is, p • q implies p. Also, contradiction is equivalent to negation, so the contradiction of these statements would be ~(p • q) and ~p, resulting in this square of opposition:


You can work through the different relationships and see how they apply. For example, this says that propositions of the form p • q are contrary to statements of the form ~p. This means that the statements “She is a princess and a queen” and “She is not a princess” cannot both be true, but can both be false. You can look at all the other relationships this way.

Try to come up with your own square of opposition for non-categorical statements.

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