An example of logical incoherence: an atheist who makes objective moral judgments.
This is incoherent (even amusing, in my better moments), since if God does not exist, then objective moral values and duties do not exist.
The atheist is left with utilitarianism (there's no real "right" and "wrong"; it's only "pleasure" and "pain"), emotivism (moral judgments are equivalent to "Hurray!" and "Boo!", or (worse yet) some evolutionary argument for objective moral values (worse, because such arguments commit the genetic fallacy).
My interest in the foundations of morality (metaethics) began in 1972, when I was a philosophy major at Northern Illinois University. I was given a copy of C.S. Lewis's Mere Christianity. Lewis spoke of morality (as existent, in humanity) as the key to the meaning of the universe. Surely objective morality, on atheism, is nonexistent.
On atheism, morality is subjective and non-binding. See, e.g., J.L. Mackie, Nietzsche, and Sartre. These three "saw the nonexistence of God as morally relevant... For these thinkers, atheism didn't mean business as usual when it came to ethics. It meant fundamental rethinking of what ethics is all about, because they recognized the long history of a perceived connection between God and morality. Thus they stand in contrast to those who think that eliminating God from the moral equation changes little or that including God adds nothing of consequence." (In Baggett and Walls, Good God: The Theistic Foundations of Morality)
Way back in the 70s I read Sartre's famous essay "Existentialism Is a Humanism." Sartre wrote:
"Towards 1880, when the French professors endeavoured to formulate a secular morality, they said something like this: God is a useless and costly hypothesis, so we will do without it. However, if we are to have morality, a society and a law-abiding world, it is essential that certain values should be taken seriously; they must have an a priori existence ascribed to them. It must be considered obligatory a priori to be honest, not to lie, not to beat one’s wife, to bring up children and so forth; so we are going to do a little work on this subject, which will enable us to show that these values exist all the same, inscribed in an intelligible heaven although, of course, there is no God. In other words... nothing will be changed if God does not exist; we shall rediscover the same norms of honesty, progress and humanity, and we shall have disposed of God as an out-of-date hypothesis which will die away quietly of itself. The existentialist, on the contrary, finds it extremely embarrassing that God does not exist, for there disappears with Him all possibility of finding values in an intelligible heaven. There can no longer be any good a priori, since there is no infinite and perfect consciousness to think it. It is nowhere written that “the good” exists, that one must be honest or must not lie, since we are now upon the plane where there are only men." (Emphasis mine.)
So: If God does not exist, then objective moral values and duties do not exist.
(For reading suggestions on metaethical matters see my post "Morality Needs God."