Metaphysics and metaphysical presuppositions are a necessary feature of any logical philosophy, at least under our current conceptions of logic. Although they have some similar features, metaphysical presuppositions are very different from scientific hypotheses.
Both metaphysical presuppositions and hypotheses have the same logical function: They are axioms (a.k.a. premises) from which statements are derived using logical deduction. The difference is that if a statement contradicts a metaphysical presupposition, the statement is considered false. If a statement known to be true contradicts an hypothesis, the hypothesis is considered false. The italicized qualifier is critical: Hypotheses are offered to explain and account for statements known prior to logical deduction to be true.
Note that the definition of "hypothesis" offered above is a metaphysical presupposition. A statement that contradicts that definition is false by virtue of contradicting that definition. Implicit in this metaphysical presupposition is that we cannot know the the truth of any statement independently of deduction from the metaphysical presupposition.
One feature that distinguishes different metaphysical systems is whether the metaphysical system is open or closed. A closed metaphysical system enumerates all the statements that are considered true by definition; only those statements which can be rigorously derived from the metaphysical presuppositions are true. An open metaphysical system gives us some system for adding additional statements which are true by definition, not by derivation.
All empirical metaphysical systems are open by definition because they include the metaphysical presupposition: All statements of subjective experience are known directly to be true or false. (True statements about experience are properly basic).
Technically, from logic and mathematics, an axiom is simply a statement held true by definition; the set of axioms defines a formal system. But because of the traditional use of logic in mathematics, axioms are usually chosen so that interesting statements can be formally derived from those axioms. This procedure works well in mathematics, but, as the Logical Positivists showed and as I discuss in more detail on my series on The Scientific Method, it fails miserably in trying to deal with statements of experience: Even when known directly to be true, statements of experience are too complex to draw any logical deductions from. We thus label statements about experience as evidence to distinguish them from mathematical axioms suitable as a basis for deduction.
Since we cannot straightforwardly draw deductions from experiential statements, we need a different meta-system to do more than simply enumerate such statements. We thus define "ontology": An ontological system is an axiomatic formal system in which true statements about experience are specific theorems of that formal system. In other words, an ontological system is a series of axioms—in the sense that they are suitable for derivation—which logically account for the evidence. Since these ontological axioms are not held by definition to be true in the same sense as metaphysical presuppositions or mathematical axioms, we label them as hypotheses.
There's one more problem: We can derive any finite set of theorems from an infinite number of axiomatic formal systems. At the very least, we can always add an irrelevant axiom—an axiom which does not entail any new theorems or nontheorems—to any axiom set. Therefore we add one additional metaphysical presupposition: If two ontological systems account for the same evidence, then the simpler system, the one with the fewer hypotheses, is better than the more complex system.
To recap, we have three metaphysical presuppositions:
- All statements of subjective experience are known directly to be true or false.
- An ontological system is an axiomatic formal system in which true statements about experience are specific theorems of that formal system.
- If two ontological systems account for the same evidence, then the simpler system, the one with the fewer hypotheses, is better than the more complex system.
Given this definition, we are in a position to discuss the existence of God according to Metaphysical naturalism.
Before denying the existence of God, the metaphysical naturalist must assign meaning to the statement, "God exists". This task is rather difficult, since Metaphysical Naturalism does not directly assign any meaning at all to "God" or "exists". The statement, "God exists" is not by itself meaningful according to Metaphysical Naturalism. Because Metaphysical Naturalism only talks about axiomatic formal systems, not specific statements (other than statements of experience, and "God exists" is not a statement of experience), we must embed the statement in some hypothetical formal system, and evaluate the formal system according to Metaphysical Naturalism. This task is non-trivial: Every different sect of every religion defines a different formal system, and there is considerable individual variation even within the same sect. Even so, an examination of the most popular formal systems in which "God exists" is a valid theorem reveals three broad classes which can be evaluated in general:
- Trivial definitions (e.g. Einstein's God)
- Hypothetical systems
- Metaphysical definitions
Sometimes "God exists" is embedded in a prosaic formal system: "God exists" is an hypothesis in a collection of hypotheses, and the hypothetical system entails statements about experience. (I'll discuss truly metaphysical conceptions of God in a later essay.) What we invariably find, though, is that either the entailment is "degenerate" (the system entails statements like "the sun will rise tomorrow or the sun will not rise tomorrow"), false-to-fact (the system entails that if you pray for something it will be or become true), or are simply irrelevant elaborations of simpler ontological systems.
It is typically on the evaluation of hypothetical ontological formal systems which include "God exists" either as an hypothesis or a theorem which follows from other hypotheses that the atheist denies that "God exists" is true. Furthermore there are a sufficient number of hypothetical ontologies which are specifically false-to-fact that the active denial, "God does not exist," is meaningful and true.
 The derivation method also defines the formal system, but almost all formal systems use propositional calculus as a derivation method, so this criterion does not draw many distinctions in ordinary practice.