If you don’t mind, to simplify things for purposes of this dicussion, let’s leave out the word “system”. I’m not even sure what a belief system is. I mean, I’m sure it can be defined as a compound term, but the question at its root really is a question simply of belief. If there is a system underpinning it all, I think it can be examined separately.
And so, belief and its object — there can be no belief without a predicate: it is necessary to believe or not believe something or not something — is what we should examine. If the object were this board, for example, we might have the following statements:
- I believe this board exists.
- I do not believe this board exists.
- I believe this board does not exist.
- I do not believe this board does not exist.
Note that (1) is a positive affirmation of both belief and existence. (2) is a negation of belief. (3) is a negation of existence, but a positive affirmation of belief. And (4) is a negation of both belief and existence. So we can characterize the four statements this way:
- Having a belief that affirms the predicate.
- Having no belief that affirms the predicate.
- Having a belief that denies the predicate.
- Having no belief that denies the predicate.
Because these are modal statements, they are statements about belief, just as “I have no bananas” is a statement not about bananas, but about possession. The “no bananas” merely constitute what you “do have”. And so if we make the predicate about God rather than the board, we have:
- I believe that God exists.
- I do not believe that God exists.
- I believe that God does not exist.
- I do not believe that God does not exist.
There are four statements about belief concerning two different predicates.
The problem with saying that denial of belief is an “absence” of belief is that it is not — a negation is not an absence. -2 is not the absence of 2. Not A is not the absence of A; it is the contradiction of A, just as A is the contradiction of Not A. Every positive affirmation has a negation and vice-versa.
Having no belief is not the absence of a belief, but a negation of having a belief. “No belief” is the kind of belief you have, just as “no preference” is the kind of preference you have if asked what you’d like to eat. For an absence of preference, you have to negate the having, not the preference.
So, “I do not prefer bacon” is different from “I do not have a preference for bacon”. The former is a statement about what you prefer, while the latter is a statement about what you have. Likewise, you can make a statement about belief in God such that belief is absent by not modalizing the belief itself:
“I do not have a belief that God exists” expresses the absence of belief. Here, the predicate is about belief of a certain kind. And the statement is about a negation of inclusion. This may sound nitpicky to people with little knowledge of grammar or logic, but it is in fact very helpful in communicating exactly what it is you believe or don’t believe or have or don’t have.
I think that eliminating the equivocations helps us to communicate better.