they can not say they believe in niether without claiming that they believe both to be equally likely as true or false
Doesn't "equally likely" imply a certain degree of consideration of the probability?
For example, If I flip a coin that can come up heads or tails and I ask if believe the outcomes is heads and you say, "No" because you have no reason to believe the outcome is heads. And then I ask you if believe the outcome is tails and you say, "No" because you have no reason to believe the outcome is tails. Can I infer that you believe the outcomes to be equally likely? (The answer is no, I can't, there is no way for you to know if the coin is fair or reason for me to think that you've considered the chances carefully to arrive at a conclusion)
A better example: What if I roll a six-sided die and ask you if the outcome was a "one" or "not a one"? You would still lack evidence to say one way or the other, but you couldn't say you believed the outcome to be "one" or "not a one" without some degree of investigation. For example, if you assumed the die was fairly weighted and also assumed that only one of the six sides had a "one" , then you could make a calculation that the chances of a "one" is 1 in 6. In this case, he would expect the outcome to be "not a one", but he wouldn't necessarily believe it wasn't "one". He wouldn't be manifesting a belief until he actually acted upon an expectation.
So when a person says he doesn't believe in either the proposition that "God does exist" or that "God does not exist", he is not necessarily committing to an investigation of the odds.
That's not grammatically logical.
If the statement is that "god exists" (i.e. "p") is true, then its negation is that the statement is not true, rather than that god does not exist.
As an aside, you shouldn't mistake a statement (i.e. "p") for reality.
Let p be statement, "This statement is false."
In this case, the negation of the statement "god exists" is the negation of the statement "god does exist". To say that "god does exist" is false is to say "god does not exist". I was very careful to include those statements in my example, so that it would be clear.
In the case of the statement, "This statement is false." The same logic applies.
Let q be the statement "This statement is false."
q is false implies ¬q is true.
¬q is false implies q is true.
¬q means "not (this statement is false)" which means "this statement is not false"
I could say that I don't believe q is true and that I don't believe ¬q is true.
The problem is "This statement is false" is nonsensical.
"This statement" is not a statement to which a truth value may be ascribed. It's similar to saying something like "The number three is continuous." "continuous" is a property associated with functions and "The number three" is not a function. Therefore, the statement "The number three is continuous" is nonsensical.
If you claim that "This statement" simply refers to the statement in which the phrase "This statement" is embedded, then you are self-defining. For example, if I were to say Let t be equal to t+1, when I solve for t, I get that 0=1. The problem is defining t in terms of itself.
In the case of "god does exist", god is something to which the property of existence can be ascribed.