inconsistency

noun

1. The state of being inconsistent.

   “It was a strange inconsistency, but never till then had I been so much struck with the worthless and frivolous life of society. Never till then did I feel the deep and eternal debt of gratitude that human nature owes to those who assert its higher influence; who feel their generous activity stirred by a thrice noble emulation; who appeal from the present to the future, and redeem their kind, by shewing of how much that is good and great ambition and genius are capable.”

2. An incompatibility between two propositions that cannot both be true.