(This is inevitably when someone says “But if you don’t believe in God you’re an atheist! “. Well, sorry…that is just NOT true. That is how many atheist think of atheism as they are told that by other atheists…but logically, as I have shown, you do not have to be neither theist nor atheist. Not believing in God is a necessary, but not sufficient condition and does not by entailment nor by necessity mean that if you do not believe in Gods you are an atheist. Logic > misinformation you have been told. What they are merely doing is taking A V ~A and saying “Theist or “NOT theist” and relabeling “not theist” as “atheist”. This is like making a disjunction (p V q) and saying if not p then q. I can do that with anything. p = theist, q=car. p V q, theist or car, If you are not a theist you are a car. See?)
What atheism and theism actually are called contradictories. Meaning that if there is a God and theism is true, then atheism must be false…and vice versa, such that if there isn’t a God then theism is false and atheism is true. Neither can both true, and neither can be both false. One must be the the case, and the other must not be the case. This is slightly different than contraries such as atheistic religions or theistic religions where if one is true the other must be false, both can not be true, but both can be false.
Proof: If p=”at least one God exist”
p1) A lack of belief for p logically is ~Bp
P2) A lack of belief for ~p logically is ~B~p
p3) A lack of belief atheist holds to ~Bp
p4) Holding to ~Bp without holding to B~p must entail holding to ~B~p.
p5) A lack of belief atheist who holds to ~Bp (p3) but does not hold to B~p must then hold to ~Bp & ~B~p (p3-p4). (constructive elimination)
p6) Agnosticism holds to ~Bp & ~B~p
C) Agnosticism logically is the same as a lack of belief atheist as both hold to ~Bp & ~B~p.
(But they are epistemically different, much like “Christian” and “Jew” are logically the same, but epistemically different.)