Theism is about belief in a god or gods. For any claim about the existence of a god or gods, theists accept that this claim is true. One way of stating this is that theists have the positive belief that the god claim is true.
Atheism literally means "without theism". An atheist is someone who does not accept, as true, claims that assert the existence of gods.
Considering the claims regarding the existence of a god, there are two possible claims:
1) God exists
2) God does not exist
For either claim, there are two positions one can take with regard to belief:
1) belief or acceptance of the claim
2) disbelief or rejection of the claim
For claim number 1 (God exists), the theist's position is one of belief, while the atheist's position is one of disbelief.
For claim number 2 (God does not exist), the theist's position is one of disbelief, while atheists can hold either position.
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Some atheists actively believe that no god exists while others hold that neither claim is sufficiently supported to justify acceptance. These positions are often labeled strong atheism and weak atheism, respectively.
Gnosticism (in the sense used here) addresses the issue of what one knows or claims to know. For any claim about the existence of god, gnostics are individuals who claim to know that the claim is true. An agnostic is, literally, someone who does not claim to know that such claims are true.
Therefore, as the terms aren't mutually exclusive, it's possible to combine them into four descriptions:
1. Agnostic Atheist (does not believe any god exists, but doesn't claim to know whether this is actually true)
2. Agnostic Theist (believes a god exists, but doesn't claim to know that this belief is true)
3. Gnostic Atheist (believes that no god exists and claims to know that this belief is true)
4. Gnostic Theist (believes a god exists and claims to know that this belief is true)
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It obviously can and will be argued that we can never truly "know" anything, yet we constantly make claims of knowledge