Atheism isn't an ideology, it's just a single idea - that there are no Gods.
You're describing strong atheism, but weak atheists are atheists too.
Theists: Those who believe that gods do exist.
Strong atheists: Those who believe that gods do not exist.
Weak atheists: Everybody else.
So .. are weak atheists a subset of agnostics, or is it the other way around?
First, lets set out the two most common systems of nomenclature. I don't know which is older, but I call them
newsys and
oldsys.
Newsys and Oldsys are two ways of naming the following five categories:
A. Those who believe that gods do exist.
B. Those who believe that gods do not exist.
C. Everybody else (those who don't believe either way).
X. Those who know (or think they know) whether gods exist.
Y. Everybody else (those who don't know whether gods exist).
Here are the two most common systems of nomenclature (I don't know which is older, but I call them newsys and oldsys).
Oldsys:
A. Theists
B. Atheists
C. Agnostics
X. Gnostics (not capitalized except mostly as the first word of a sentence)
Y. Agnostics.
Newsys:
A. Theists
B. Strong atheists
C. Weak atheists
X. Gnostics
Y. Agnostics
Oldsys uses the word
agnostic for both category C and category Y, which leads to endless confusion.
Newsys uses agnostic only for category Y. This facilitates communication.
People who identify as atheists overwhelmingly prefer newsys, so much so that someone referring to herself as an atheist presumptively using newsys.
Now to your question:
So .. are weak atheists a subset of agnostics, or is it the other way around?
Oldsys doesn't use the term
weak atheist. So, it (oldsys) doesn't see weak atheism as a subset or superset of anything.
In newsys, theism/atheism describes belief and gnosticism/agnosticism describes knowledge. Agnosticism and weak atheism
Neither system sees weak atheism as a subset of agnosticism, or vice versa.
We often hear that theism/atheism and gnosticism/agnosticism are
orthogonal. I hadda look it up.
Dictionary.com: "
... orthogonal is also sometimes used in a
figurative way meaning unrelated, separate, in opposition, or irrelevant. In this sense, it means about the opposite of
parallel when parallel means corresponding or similar.Example:
Not everything happens according to a grand scheme—some events are simply orthogonal to each other.
There are numberless other nomenclatures, other systems of naming the five categories. But newsys and oldsys are the common ones. Whatever's in third place is so rare that we don't have to worry about it. If you elect to use a third system, you will have to tediusly explain it to each person you engage in conversation on the topic.