Ontological atheism is when one says that (what are generally considered) "gods" do not exist, whereas qualitative atheism is when one says that (what are generally considered) "gods" exist but are not gods. So a qualitative atheist can say that for example Yahweh, Allah, Satan, Prometheus, Lucifer, Aphrodite, Queltzalcoatl etc. exist but are not gods. (Some) Christians and Muslims may be considered qualitative atheists when they say that Satan exists (in addition to Yahweh/Allah) but claim to be monotheists rather than duotheists. In some interpretations, since the Sun, Earth and Moon may be considered gods, an atheist who believes that the Sun, Earth and Moon all exist may be considered a qualitative atheist.

Another distinction is between global and local atheism. Global atheism is atheism towards all deities, whereas local atheism is atheism towards a particular deity. This compounds into four categories:

Global ontological atheism: "no deity exists" i.e. "none of what is (regarded as) a deity exists"
Global qualitative atheism: "no deity exists" i.e. "none of what is (regarded as) a deity is a deity (though it may exist)"
Local ontological atheism: "this specific deity does not exist"
Local qualitative atheism: "this specific entity that is regarded as a deity is actually not a deity (though it may exist)"