Seeking to show that if a weak atheist justifies self-identifying as an atheist, the same reasoning can be used to show that a weak atheist can be justified to self-identity as a theist as well.
Weak atheist= Someone who merely holds to a lack of belief on the existence of Gods. (Denoted by ~Bp)
Weak theist = Someone who merely holds to a lack of belief on the non-existence of Gods (Denoted by ~B~p)
Atheist= Someone who holds theism as false (Denoted by B~p)
Theist= Someone who holds that the proposition of at least one God existing is true. (Denoted by Bp)
Agnostic = Someone who both lacks a belief in existence of Gods and in the non -existence of Gods (Denoted by ~Bp & ~B~p)
Weak atheist with the label or self-identification “atheist”= ~Bp
Weak theism with the label or self-identification “theist” = ~B~p
“Hold” = to exclusively only claim or adhere to. (Ex: “hold to ~Bp would mean to explicitly only claim ~Bp).
1) Given S holds to ~Bp because of “lack of evidence for the existence of Gods”, and does not hold to B~p it follows that S must also hold to ~B~p on pains of irrationality and arguably for the same reason of a “lack of evidence for the non-existence of Gods”.
This follows from ~Bp is either because S holds to B~p which entails ~Bp or holds to ~Bp & ~B~p and is not convinced either way.
2) If S does holds to ~Bp but does not hold to B~p, it follows that S holds to ~Bp & ~B~p. If S adopts the label “atheist”, it follows S is just as justified then to hold to ~B~p and adopt the label “Theist”.
Conclusion: A weak atheist (~Bp) is also a weak theist (~B~p), and as such, if they can self-identify as “atheist” they are also EQUALLY justified to self-identity as “theist”.
This follows also from a previous argument:
If p=”God exists”
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 and a lack of belief theist holds to ~B~p
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). (Conjunction introduction)
p6) Holding to ~B~p without holding to Bp must entail holding to ~Bp.
p7) A lack of belief theist who holds to ~B~p (p3) but does not hold to Bp must then hold to ~Bp ^ ~B~p (p3-p6). (Conjunction introduction)
p8) Agnosticism holds to ~Bp ^ ~B~p
c) Agnosticism logically is the same as a lack of belief atheist (~Bp) and lack of belief theist (~B~p) as both actually hold to ~Bp & ~B~p.
Conclusion: Agnosticism entails weak atheism AND weak theism (granting they were not be “merely” here but that agnosticism holds to both ~Bp and ~B~p).
It then follows since agnosticism holds to both weak atheism and weak theism that it is no more acceptable to assign the label “atheist” to an agnostic than it is to assign the label “theist” to one.
Summary: If someone calls themselves atheist but does not hold a belief Gods do not exist, it is akin to someone calling themselves theist and not holding a belief Gods do exist. This seems to lead to a possible veridical paradox or “seemingly” odd contradiction of “theist atheist” if one merely defines these positions in the weak case. Since if once doesn’t accept the proposition of theism, lack of belief atheist adopt a ‘weak atheist’ position and call that atheism, to which a theist can call lack of belief of the proposition of atheism ‘weak theist’ and just call that theist. An agnostic then would be a “theist atheist” or “atheist theist”?
“There is nothing that places agnosticism closer to atheism than to theism.” – Dr. Graham Oppy
“For, if we accept that there is this distinction between strong atheism and weak atheism, we should surely accept that there is a similar distinction between strong theism and weak theism: strong theists reject the claim that there are no gods, while weak theists merely refrain from accepting the claim that there are no gods. And then we shall have it that agnostics are both weak atheists and weak theists.” -Dr. Graham Oppy
MORE CONCISE PROOF:
1. If ~Bp and not B~p, then ~B~p
2. If ~B~p and not Bp, then ~Bp
3. ~Bp and not B~p
4. ~B~p (MP 2,3)
5.~Bp and not B~p
6. ~Bp (MP 1,5)
7. ~Bp ^ ~B~p (Add 4, 6)
Venn diagram in image is the McRae-Noll Venn Diagram. (Computer scientist: Landon Curt Noll)