2. If God exists, then he is an all-merciful judge.
3. An all-just judge treats every offender with exactly the severity that he/she deserves.
4. An all-merciful judge treats every offender with less severity than he/she deserves.
5. It is impossible to treat an offender both with exactly the severity that he/she deserves and also with less severity than he/she deserves.
6. Hence, it is impossible for an all-just judge to be an all-merciful judge (from 3-5).
7. Therefore, it is impossible for God to exist (from 1, 2, and 6).
I have heard it said by Christians that the way God judges offenders depends on whether or not they are true believers. If they are, then he is lenient with them, but if they are not, then he treats them with exactly the severity they deserve (which can be pretty bad). By this Christian way of speaking, God is said to be both an all-just and an all-merciful judge. He is all-just in giving everyone an equal opportunity to become a true believer and thereby come to receive leniency, but he is also all-merciful in that every true believer, without exception, receives mercy. This way of viewing matters would be an attack on both premise 3 and premise 4, above.
I would respond by maintaining that premises 3 and 4 come closer to capturing ordinary language than the given Christian way of speaking. According to the latter, God treats some offenders more leniently with regard to what they deserve than he does other offenders. It does not seem that such a judge would (or should) be called "all-just." And similarly, since he does not treat all offenders less severely than they deserve, he would not (and should not) be called "all-merciful" either. Instead of being both all-just and all-merciful, the Christian God, as described, would be neither.
As with many of the previous attacks on the incompatible-properties arguments, this one turns on semantical issues. In a sense, it is all a matter of semantics, for the issue of whether or not certain property ascriptions conflict with certain other property ascriptions depends very much on what exactly they mean. Theists could defend against the arguments by denying that the property terms in question mean what the proponents of the arguments claim they mean. Often such denials lead to still other difficulties for the theist. A full presentation and defense of incompatible-properties arguments should explore such implications and fully pursue the many issues, whether semantical or not. That project is beyond the scope of the present essay.
My aim was simply to survey several of the more common (and a few not so common) incompatible-properties arguments for the nonexistence of God. Just which of those arguments are sound and which of them are most effective in discussions and debates with theists are further issues that are certainly worth pursuing.