I don't understand your argument or its notation. "Believing that X could be true" is not the same as "Believing that X is true and not knowing how to prove it".
Which goes back to Graham's point - you seem to be wanting me to "pick a side" instead of treating it like any other statement that one can be uncertain about.
I am sorry about my mistakes. I completely forgot that sets must be represented by upper case letters and not numbers. My argument is based on the exactly meaning of the two words. I agree that there are connotations, but this do not change the meaning of the words.
The acceptance of a possible existence imply a belief in the subject existence. Of course you could argue that the existence could be true or is true. Whatever argument you choose, you have to accept the subject existence to determine if is true or could be true.
Let's imagine three hypothetical dialogues:
- Do you believe in the existence of Santa Claus?
- Yes.
- Why?
- Because is possible that a old man dressed in red clothes are living somewhere in the north pole.
- Do you believe in the existence of Santa Claus?
- No.
- Why?
- Because is impossible that a old man dressed in red clothes are surviving in the north pole under extreme low temperatures.
- Do you believe in the existence of Santa Claus?
- No...
- Why?
- I do not believe Santa Claus, but is possible that a old man dressed in red clothes are living somewhere in the north pole.
Notice the contradiction in last dialogue. The hypothetical character answering the questions first refuse the existence of the subject, then when required to justify his/her answer, he/she assumes the existence of the subject. This contradiction is not present in the first two dialogues because once the hypothetical character express his/her belief or disbelief, he/she straightly assumes the possible or the impossible existence of the subject.
My argument using mathematical notation is:
p ∈ B
The possible existence of the subject is element of the belief.
i ∈ D
The impossible existence of the subject is element of the disbelief.
B ≠ D
Belief and disbelief do not share any common element, therefore there is not any intersection between theism or atheism.
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