For this discussion the precise meaning of God doesn’t matter as long as the same meaning is used throughout. There are two different atheisms:
- Logical Atheism. Logical atheism (LA) is the proposition that God does not exist. As a proposition, LA enters into logical relations with other propositions. If P is some proposition, then LA either entails that P; or is consistent with P; or contradicts P; or etc. An argument against the existence of God concludes with logical atheism. For instance: (1) If God exists, then P; (2) however, it is not the case that P; (3) therefore, LA. The conjunction of (LA & not LA) is a contradiction.
- Psychological Atheism. Psychological atheism is the belief that God does not exist. As a belief, it is the belief of some mind. Thus psychological atheism is a proposition of the form Believes(x, Logical Atheism), or, for short, B(x, LA). The proposition B(x, LA) is not equivalent to LA. Nor does LA entail B(x, LA) nor does B(x, LA) entail LA. The logic of belief is very complicated, very controversial, and very different from logics that merely involve propositions without intentional operators.
The difference between logical and psychological atheism is significant. For any believer x, (B(x, LA) & (not B(x, LA))) is a contradiction. Thus
((x believes that God does not exist) & (x does not believe that God exists)) is a contradiction.
However, for any believer x, (B(x, LA) & B(x, not LA)) is not a contradiction. Thus
((x believes that God does not exist) & (x believes that God exists)) is not a contradiction.
It is not a contradiction because B(x, P) does not entail P.
This careful distinction is relevant to the definition of atheist. An atheist is a person who holds a certain belief. Thus one may say that
(1) x is an atheist if and only if B(x, LA).
Now, it is rational to assume that there are no self-inconsistent objects. That is, there are no objects that satisfy contradictions. Thus for any property F, there is no object y such that (F(y) & not F(y)). On this assumption, there is no person x such that
(2) ((x is an atheist) & (not B(x, LA))).
However, unless you add other assumptions, B(x, not LA) does not contradict B(x, LA), that is, (B(x, LA) & B(x, not LA)) is perfectly consistent. Hence the following is also perfectly consistent:
(3) ((x is an atheist) & B(x, not LA)),
which is equivalent to
(4) ((x believes God does not exist) & (x believes God exists)).
Thus, unless you add other assumptions, it is entirely possible that
(5) x is an atheist and x believes in God.
The other assumption that has to be added to make (5) contradictory is a principle which we may call No Self-Deception (NSD):
(6) if x believes P then x does not believe not P.
which is formally
(7) if B(x, P) then (not B(x, not P)).
NSD is a very strong assumption, and it’s not at all clear why it should be added it to the logic of belief. Thus consider self-deception.
(7) x is self-deceived if and only if there is some P such that (B(x, P) & B(x, not P)).
Adding NSD obviously rules out self-deception. So here is a very diffferent definition of atheism than the one given in (1). This new and distinct definition is:
(8) x is an atheist if and only if (B(x, LA) and NSD).
Clearly, NSD is a very strong proposition which is entirely independent of LA and even B(x, LA). The first definition of atheism in (1) makes no psychological demands of the atheist beyond B(x, LA). The second definition of atheism in (8) makes an extremely strong psychological demand of the atheist. It says that an atheist cannot be intellectually dishonest, cannot be self-deceived, cannot be in bad faith, etc. etc.
And all of this lands us right into issues dealing with the ethics of belief. Since NSD is hardly a logical principle, you might try to argue that it is an ethical principle: for any mind x, for any proposition P, if B(x, P) then it ought to be the case that (not B(x, not P)). If you believe P, then you are obligated to not believe not P. But once more, this is a very strong demand. Anyone who adds it to the definition of atheism has expanded the definition far, far beyond merely B(x, LA). For now an atheist is a person who believes that God does not exist and who also believes that self-deception is ethically forbidden.
Thus (1) and (8) are very different definitions of atheist.