Some Instances of the Hexagon of Opposition in Mathematics, Aesthetics, and Politics

One of the things that logicians study is how opposition is expressed in formal or informal languages. The most obvious type of opposition appears when we negate a simple sentence: (1) “the cat is black” becomes (2) “the cat is not black”. In this case, (1) and (2) are strongly opposed to each other: they are contradictories. This sense of binarity is even stronger in mathematics, where natural numbers can be either even or uneven, with no other options. But often, we find more sophisticated ways to express opposition. A good example is that if we have eight people in front of us, we can refer to all, some, or none of them. We could, for example, say that all the people at the party have grey hair. What oppositions exist when we have not two, but three basic terms? It turns out that there are three types.

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