Is there a difference between 'inconsistent', 'contrary', and 'contradictory' Is there a difference between 'inconsistent' 'contrary' and 'contradictory'? As far as I understand, two statements are inconsistent when they can not both be true; two statements are contradictory when they can not both bear the same truth value. Does 'contrary' have a unique meaning?

A contradiction is an pair of statements of the form $p$ and $\lnot p$.

A set of statements is contradictory if there is a pair of statements in it that form a contradiction.

A set of statements is inconsistent if you can _derive_ a contradiction from them.

We will sometimes use the terms "contradiction" and "contradictory" more loosely to mean "obviously inconsistent."

"Contrary" is used informally to mean the negation of a statement. I don't think I've ever seen a set of statements called "contrary," for example, just because there is a contradiction. Perhaps there is a formal use of the term that I am missing.