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: 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.
A: There is already an accepted answer that describes the issue pretty well.  To add to it, however, contrary and contradictory are used with distinct meanings in Aristotelian logic.  The square of opposition shows four types of relations between sentences.  These are:


*

*contrary;

*subcontrary;

*contradictory; and

*subalternation.


Therein, the contrary of the sentence "All S are P" is "No S are P", while its contradictory is "Some S is not P."  The contradictory is logically equivalent to the negation, but the contrary is not.  The linked Wikipedia article covers all the cases in much more detail.  You're probably not likely to encounter these terms in casual use, or even in formal logic today unless someone is making a point of speaking about Aristotelian logic.
