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)
label or self-identification “atheist”= ~Bp
label or self-identification “Theist” = ~B~p
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 arguably for the same reason of a “lack of evidence for the non-existence of Gods”.
This follows form ~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 (~Bp) and as such if can self-identity as “atheist” they are also EQUALLY justified to self-identity as “theist”.
This follows also from a previous argument:
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). (disjunction 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.
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.
“There is nothing that places agnosticism closer to atheism than to theism.” – Dr. Graham Opy