Repeat
1. For all practical purposes, "I do not believe in the existence of God" does mean "I believe that God does not exist". Raising shifts the negative from the subordinate clause where it logically belongs to the main clause, when the main clause’s verb is suppose, think, believe, seem, etc..
For example:
"I don't believe the egg was eaten" does mean "I believe the egg was not eaten". It does not mean that I don't have any beliefs about the egg being eaten/not eaten. Both these sentences are valid english usage and are understood as meaning same. Here the subject is eating of egg.
2. Absence of hair can never mean presence of hair. 'Absence of X negates presence of X' logic applies to objects and not to cognition itself.
"X" is an object. It can either 'exist' or 'not exist'. Whereas, 'belief', is a symptom of cognition that is everpresent in any assertion. Assertions such as 'I have lack of belief of God' or 'I don't believe in existence of God' do not happen without cognition.
These assertions do not mean that "I do not have any belief about existence of God".
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