Agnosticism, atheism, theism and LOGIC 101:
~p is the negation of the proposition
This is called a tautology. It means it is the case p is true or it is not the case p is true. Which is the same as saying p is either TRUE or FALSE. (This is know as the Law of Excluded Middle…that a proposition can only be T or F.)
Given the proposition of theism:
p=”at least one God exist”
It is either the case that p is T or p is F. Or in other words it is the case that “at least one God exist” or it is NOT the case that “at least one God exist”. This is an ontological dichotomy (God(s) exist or God(s) do not exist).
The theist affirms p to be true. That they believe it is the case that “at least one God exists”. The atheist (philosophical atheism) affirms the negation that p is false that it is not the case that “at least one God exists”.
This is logically shown as:
Bp = I believe p is true (Theist)
B~p= I believe p is false (Atheist)
Bp V B~p however is not an ontological (or even epistemic) dichotomy. One does NOT have to believe either that p is true or p is false.
The agnostic does not hold to either theism nor atheism and does not believe p to be true, nor believes p is false. They logically hold to ~Bp (I do not believe p) and ~B~p (I do not believe not p).
The agnostic suspends judgement on the proposition…but because they do not affirm theism, they do not believe in Gods, however they are not asserting that they do not exist. This is often the stance of “weak” atheist (~Bp) who while not believing God(s) exist, do not affirm the negation of theism or that God(s) do not exist (doesn’t explicitly state they also hold to ~B~p, however a logical proof can show that all who hold to ~Bp that do not hold to B~p must then hold to ~Bp & ~B~p (proof at bottom)).
Agnostics are not theist, nor are they atheist.
(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?)
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). (elimination disjunction)
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.)