### Contraries

If one looks in a general dictionary one will not a find a definition of contrary that is philosophically useful.i

I looked the word up in a philosophical dictionary but that was not satisfied with the provided definition:

A pair of categorical propositions which (provided that we assume existential import) cannot both be true, but can both be false. In the traditional square of opposition, an A proposition and its corresponding E are contraries. Thus, for example:

All cars are green and No cars are green are contraries.ii

The reason why I am not satisfied with the above is that it is too narrow, i.e. that some contraries are not captured by the definition. I’m looking for a more rigorous definition. It seems that I have to create my own.

The problem with the above is that it only works for categorical propositions but contraries are not restricted to categorical propositions. For instance, two scientific theories may be contraries but they are not categorical propositions, or can be composed as a set of categorical propositions.

A more rigorous definition is this:

Two propositions, p and q, are contraries iff they belong to a set of propositions in which at most one proposition is true.

Note that it may be that no propositions in the set is true.

More formally we may define contraries as this:

For all x and for all y and for all z, iff x belongs to z and y belongs to z and at most one proposition in z is true, then x is a contrary of y and y is a contrary of x.

If one looks at the definition of contradictory the situation is better, though I still want to clarify the definition.iii

A contradictory may be defined as this:

Two propositions p and q are contradictory iff p and q belong to a set of propositions where exactly one of the propositions is true.

Note that this is assuming the law of the excluded middle.iv

More formally we may define contradictories thus:

For all x and for all y and for all z, iff x belongs to z and y belongs to z and exactly one proposition is true in z, then x is a contradiction of y and y is a contradiction of x.

Note also that all contradictories given my two definitions are contraries and that some contraries are contradictories.