Alternatively, you could just leave the definitions as is. That wasn't the point. The point is that if you define an atheist as "not a theist", then this there is something that must define a theist for an atheist to not be one. Defining atheism as not theism and theism as not atheism is logically consistent but of no use for my purposes.
I don't agree with the definition of atheism I used, and the first time I presented my pseudo-proof (actually, each time) was to counter the argument that we can just define an atheist as "not a theist" and be done. However, the reason I had to use a pseudo-proof (apart from the fact that most people don't know formal logic), is because of the way predicates are represented in classical logic. "not being capable of having" is even worse than "believe" for a formal representation, although the added difficulties don't make it any less a pseudo-proof than the one I gave. The definition itself isn't a problem, as we could just define them as I think you suggest above: in terms of one another. Formally, that's fine. Practically, it's useless. I have to define an atheist as "not a theist" simply because that's the definition I'm addressing, and in order to say anything useful that can be represented in a real formal derivation I need to be able to negate things without ambiguity. So, for example, if I represent Gx as "x is capable of having a belief in god", I cannot I use it to apply the necessary inference rules even if I restrict the domain to people.
Tx = "is a theist"
Gx = "x is capable of having a belief in god"
Ax = "x is an atheist"
(∀x) (Ax→ ~Gx) |A
(∀x) (Ax →~Tx) |A
(∀x) (Ax→~Gx) | Reiterated
(∀x) (Ax→ ~Tx) |Reiterated
Au→ ~Gu |universal quantifier elimination
Au→ ~Tu | ditto
Tu → ~ Au | negation elimination and MT
Gu → ~Au |ditto
Tu ⇔ ~Au | biconditional intro
(∀x) (Tx ⇔~Ax) | universal quantifier intro
I could have shown the same formalizing my own proof using Bxy to be "x believes in y", having g = god, Tx and Ax as is, and starting with the necessary definition of atheist. The problem is that in predicate logic one can only negate the entire predicate. Mental state predicates trigger indirect speech, which means that slots aren't like normal. I can't avoid negating the predicate and at the same time negate the belief. Not in classical logic.
But "not having a belief" isn't the definition. It's a byproduct. The definition is that atheists are not theists. Also, my goal wasn't to present a logical formulation of the definitions, but to link the definition demanded with what it entailed. However, as I said, there isn't a way to do that well in classical logic because it isn't suited for propositional attitudes.
True. Also, although I could be wrong, I don't think it would satisfy the ones who argue that atheism is a lack of belief, because at least most still define atheist solely as "not a theist" and I doubt I could convince someone of what this entailed without using that as the definition. Perhaps most importantly, my goal that prompted not the pseudo-proof but the one you responded to was not relating atheists to beliefs. I already did that in a way I could represent in classical logic, and it was already denied because my use of inference rules led to the statement that atheists believe gods don't exist (or they don't believe gods exist), and as that isn't the definition then it wasn't accurate. Once simple inference rules are denied, the project became more an interesting exercise than anything else.
However, in trying to look at logics that are designed to deal with things like mental state predicates/propositional attitudes/etc., I realized that arguably all of formal logic is about the truth value of propositions, and that as belief is an epistemic expression of the likelihood of the truth of some proposition, it's not just that the logics designed to deal with beliefs can't express what certain atheists have stated here. It's that the reason they can't is because absence of belief isn't possible (the possible exception being utter ignorance and thus incapable of comprehending a proposition). I had said that earlier, but then I was thinking more about cognitive science, neuroscience, and the philosophy of language. It didn't occur to me when, even before my pseudo-proof, I said classical logic wasn't equipped for this that actually even the logics designed for this don't lend incorporate ways to deal with absence of belief. And for similar reasons. The epistemic scale ranges from certainty of untruth/false to certainty of truth, but any epistemic commitment to a proposition means locating one's
belief on that scale, and logics just try to formalize this.
You have always "not believed" in a god you've never heard of but to disbelieve in this god you must have heard of him. You really don't understand the difference?
