('Probably not what you think)
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The "a" as a prefix means "non-" or "not", ergo "atheism" means to not possess a belief in the existence of any 'god' or 'gods' as any given theist does, that is, all atheists do not possess any theistic beliefs.
All theists possess theistic beliefs, and all atheists do not possess any theistic beliefs.
The following assertion is slightly contentious ONLY in the sense that I assert that the above is the only reasonable definition of atheism, and the "hard atheism" or "explicit atheism" assertion (to assert a belief that there are no existing 'gods') should be (actually deserves to be) called something more. (More about this later). I contend that to hold the position of atheism proper is to make no actual assertions at all.
Consider the following argument:
Atheism from a logical perspective.
Consider, if 'x' is true, then '~x' ("not x") is false and if 'x' is false, then '~x' is true. Each assertion, 'x' & '~x' is the truth functional negation of the other, the "opposite" if you will. Both 'x' & '~x' are "contradictories" of each other. When one is true the other must be false, and they can never be both true or both false in the same instance. (That would be a contradiction).
The 'a' prefix in atheism (meaning 'not' or 'non-') works as a truth functional negation. If "Bob possesses theistic beliefs" is false, then "Bob does not possess theistic beliefs" is true, and if "Bob possesses theistic beliefs" is true, then "Bob does not possess theistic beliefs" is false. Both these positions cannot be both true or both false in the same instance. (See figure 1 above).
If we were to say that 'x' = "Bob possesses theistic beliefs", then '~x' would = "Bob does not possess theistic beliefs". As this correlates to the definition of "atheism" above, we can see that atheism as a contradictory of theism is logically consistent.
However, many would contend that atheism is 'the opposite' of theism in the sense that it asserts a belief that there are no existing 'gods', but this is not true, as this proposition (let's call it "proposition y") is a "contrary assertion" to 'x', not a contradictory position because both 'x' and 'y' can be false at the same time. In fact, logically manipulating 'x' cannot produce 'y' without first introducing 'y' as a proposition, as 'y' is not a contradictory of 'x', nor is it the converse, obverse or contrapositive of 'x'. It is in fact a contrary, which cannot be derived from 'x' at all using MODERN LOGIC. *
("*"= technical jargon you may not care about.)
Let's say that "Bob believes that the box contains a ham sandwich" is proposition 'A'. If A is true, then ~A is false, that would be that Bob does not possess a belief that the box contains a ham sandwich. We can see that "the opposite" of 'A' is not that Bob believes that the box does not contain a ham sandwich (lets call this proposition B), as it's possible for Bob to not possess any beliefs about what might be in the box. He might have some idea about some items in the box, but not ham sandwiches and therefore not assert any beliefs about alleged ham sandwiches at all. Since proposition B and proposition A can both be false in the same instance, then B does not equal ~A.
If we extrapolate the idea of a box allegedly containing ham sandwiches to that of the universe allegedly containing some existing 'god' or 'gods', then some would contend that Bob not knowing and therefore not possessing any belief on the manner is not atheism, but rather agnosticism. But this is to ignore that gnosis (Greek for "knowing") and agnosticism are about relative degrees of knowing (and/or theoretical know-ability), and theism-atheism are about relative degrees of believing. There IS a relationship between believing and knowing, but as a cursory examination of epistemology (the study of knowledge) will show, "B" (belief) is not the same thing as "K" (knowledge).
Let's say, as we suggested above, that '~x' = "not possessing any theistic beliefs" and 'y' = "to possess a belief that there are no existing 'gods'. Then members of class 'y' would necessarily be a subclass of class '~x' because everyone who possessed a belief that there are no existing 'gods' ("hard" or "explicit" atheists) would by default not possess any theistic beliefs. ALL atheists do not possess any theistic beliefs, yet SOME atheists assert a belief position that is contrary to theism in that they believe that there are no existing 'gods'. "Hard atheism" is a subclass of "implicit" or "weak" atheism (What I would term "universal atheism" and the only proper definition of atheism in my opinion. It is universal atheism because all atheists must be at least implicit atheists, that is, all atheists do not possess any theistic beliefs).
Just as everyone who holds the position of prince would be male and therefore "prince" is a subclass of the class "male", then just as "prince" means so much more than simply being defined as a male, then also "hard" or "explicit" atheism asserts so much more than simply not possessing any theistic beliefs and therefore deserves it's own particular nomenclature in my opinion, but shouldn't be called "atheism" as it doesn't represent all atheists.
"Atheists believe in no existing 'gods'" is wrong for the same reason "Christians believe in purgatory" is wrong. They are wrong because the subject terms are expressed as universal terms but the predicate descriptions are sub-classes of the universal terms used.
I say that "hard" or "explicit" atheism deserves to be called something more than atheism because 'atheism', with it's 'a' prefix, simply names the complement set of "theism". That's hardly befitting a "positive" assertion, as the category of "non-red cars" falls far short of being an adequate descriptoin of the assertion "blue cars". Calling the assertion that one believes that there are no existing 'gods' "atheism" is like calling ice 'non-warm H2O'. It's not entirely wrong, but it is certainly inadequate.
LOGOS
* ("Modern logic"; as opposed to Aristotlean logic, which did allow for contraries, sub-contraries and other forms of immediate inference that are now considered invalid. There used to be many different kinds of "opposites" of categorical statements that were considered valid in Aristotlean logic, but in modern logic, there is only one valid type of opposite, and that is the logical negation called a "contradictory".
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