The newer more atheist-accepting nomenclature holds:
Theist: B(G)
Atheist: ~B(G)
With that as the foundation, there is thus:
Weak atheist: ~B(G) and ~B(~G)
Strong atheist: ~B(G) and B(~G)
Notice the similarity: ~B(G)
The old school system is substantively different:
Theist: B(G)
Atheist: B(~G)
Agnostic: ~B(G) and ~B(~G)
The old school atheist definition is equivalent to the new school strong atheist.
The old school agnostic definition is equivalent to the new school weak atheist
The new school agnostic has been elevated to ~K(G) and presumably ~K(~G) as well.
The description that I think captures the transition is the older view distinction as being opposites whereas the newer view distinction is negation.
The "lack of belief" phrasing comes across as a learned adhoc response.
The idealized atheist progression of logic purportedly flows from because we have no empiracically reliable evidence, we therefore remain without a belief that there is a God. The truth, however, more likely flows for many in the other direction. We are told there is a God and tend to believe what we're told but after questioning the belief tend to have a diminished belief. The reason for the lack of belief is not exactly the lack of evidence first but rather lack of evidence second. These atheists (true atheists) are ones that have actually come to hold a belief opposite of theists. Their justification for their belief is something they think is lacking--lack of empiracle evidence.
When the theist turned atheist holds the position and then later comes to grips with why they hold the position, the reasoning is posthoc and conflates the what with the why. Imagine making a decision to offer a product without knowing until later the full scope of the benefits to the customer and later claim that the reasoning for the decision came from wanting customers to reap those benefits.
The link between lack of evidence and lack of belief is a big deal to atheists, and I'm saying the order of recognition is not to be taken lightly. There are those that develop their beliefs and later give reason, and I'm not talking about theists or atheists specifically at the moment. The tell tale, if ya will, is in examining the presentation and how it accords with belief formation. In fact, it's quite common that people look for reasoning to justify their positions. When their positions precede their justifications, I'm not saying it's a bad thing--just telling.
An atheist does lack belief in God, but I question the proper interpretation of that. I don't think that truth should be regarded as foundational. It's the belief in opposition (not negation) that marks the distinction--as I temporarily argue for the old nomenclature. The fact that an atheist lacks belief in God's existence, therefore, is a contingent truth, not a necessary truth. The "atheist (thus a strong atheist in this view) lacks belief "is true but not true by definition ... since the marked difference stems from opposing (not negating) beliefs to theists--it's the belief that's important--not the lack of.
At any rate, what captures my attention most is the mathematical influences of people's interpretation to language. An atheist is not a theist, but who's not a theist need not necessarily be an atheist: the terms under the old nomenclature are not collectively exhaustive, but thats just what's happening with the newer atheist-identifying definition.
There never was an 'old school' - just a misunderstanding.
Theist/Atheist is about belief.
Gnostic/Agnostic is about knowledge.
This has always been the case; The two comparisons are, and always were, orthogonal. Both axes include weak and strong versions of their respective stances. A weak atheist says "I believe that there probably isn't a god'. That weak atheist is most likely to also say 'I do not know that there isn't a god', and so is also an agnostic. But it is perfectly possible (if rare) to say 'I know that there isn't a god, and I am unsure whether I believe in Him' - a weak atheist and gnostic stance. Another weakly atheist and gnostic position would be 'I know that there is a God, but I don't think I believe in Him'.
These are not common stances, of course. But they are possible.
And of course, there's no reason at all why someone cannot be strongly theist, and strongly agnostic: "I have complete faith in the existence of a God who is completely unknowable and for whose existence no evidence shall ever be found".
The idea that 'agnostic' sits somewhere in the middle ground between 'Atheist' and 'Theist' is simply an error. It was a common error in the past, and is less common now, but it was never not an error.
I don't think so.
We had the one side that believes. The side that believes yes.
And we had the other side that (also) believes. But, that's the side that believes no.
The theist, the person of agency that has either a forming or fully formed belief (the yes's of the world)
The atheist, who with at least as much contemplation has come to hold the position of no.
The theist believes that there is a God: B(G)
The atheist believes that there is not a God: B(~G)
Then comes the in-between, the one's on the fence that is asked of their belief. With words, the term, "knowledge" is used, but listen closely and feel the message as intended as my fictitious representative speaks: Sir, I have thought long and hard about what I believe. I don't hold a belief that there is, and I don't hold a belief there isn't. Maybe there is; maybe there isn't ... a God; I just don't know.
The position isn't about knowledge. It's about belief. The intent is to convey the position not of knowledge but of not knowing which group to place his belief ... of which he has none. If one doesn't hold a belief either way, then of course one doesn't know, and that's what is said.
The agnostic stance is that of neither believing yes nor no.
Oh, and the distinction between weak and strong under the new nomenclature has nothing to do with strength solidity of belief. One can be a strong atheist and have an underlying weak belief.
Switching to expound upon the new nomenclature, the foundation that ties the strong with the weak has to do with the shared fact that there is no belief whatsoever THAT THERE IS a God. Atheists do not all share the belief THAT THERE IS NOT a God.
Again:
Atheist: ~B(G)
Weak atheist: ~B(G) and ~B(~G)
Hence, not only is it so that he has no belief THAT THERE IS a God, he also has no belief THAT THERE IS NOT a God.
Strong atheist: ~B(G) and B(~G)
Hence, like his weak atheist friends, they share the fact that neither hold a belief THAT THERE IS a God, but unlike the weak atheists, the strong atheist has formed a belief about God's existence and that is that God does not exist.
Within this group, not everyone has a fully solidified strongly held belief. Again, the difference between weak and strong atheists have absolutely nothing to do with conviction--just the mere presence of belief. A strong atheist believes that there is no God (regardless of how strongly or weakly held that position is) while the weak atheist simply has no belief either way. In fact, a weak atheist can furiously oppose the strong atheist position with great conviction.