This should be fun.
Let the domain of x be the set of all things. If God exists, then some x is God. If God does not exist, then no x is God. Simple:
In words: there exists a thing (∃x), and that thing is God (Gx).
In words: for all things (∀x), no thing is God (~Gx).
Atheists often place the burden of proof on theists, because, atheists claim, "one cannot prove a negative." In this context, the term "prove" does not mean the same thing as it does in mathematics, but, instead, it means to verify empirically. Since it is impossible to empirically verify that every single thing is not God, then one cannot prove that no thing is God. However, in principle, only a single empirical verification is needed to prove that God exists. There is a logical asymmetry: in principle, atheism cannot be proven and theism can. The burden of proof, therefore, is on the theist.
In the meantime, however, the atheist claims that every new thing we discover that is not God decreases the probability that God exists. Atheists claim the existence of God is highly improbable, because we have searched and searched and yet no thing discovered so far has been God. The structure of the argument is inductive, e.g.
~Ga, ~Gb, ~Gc, ~Gd ⊢ ∀x[~Gx]
Atheists observe singular instances of things not being God (~Ga, ~Gb, etc.) and conclude that no thing is God (∀x[~Gx]). Like all inductions, the argument is strictly invalid, i.e. the truth of the conclusion is not necessary given true premises. Instead, the singular instances are said to make the conclusion more probable. In this case, the evidence appears to support or partially justify the claim that God does not exist. Each observation of something new that is not God, supports the conclusion even more.
Let y be the degree to which a set of premises supports a conclusion (where 0 > y < 1). Appending y to the previous argument, we get a partial entailment:
~Ga, ~Gb, ~Gc, ~Gd ⊢y ∀x[~Gx]
The more y increases, the more ∀x[~Gx] is supported. In the limiting case, every thing has been observed not to be God, y equals 1, and the argument is purely deductive. In other words, in the hypothetical scenario in which our evidence exhausts all possibilities, the support for the conclusion becomes comprehensive. That said, this is impossible to achieve in practice, and hence the burden of proof is on the theist. I mention this limiting case to illustrate the relationship between induction and deduction.
From here out, to keep things from getting too cluttered, I am going to refer to all observations of things that are not God as "the evidence" or just "e," and the claim that no thing is God will be "the hypothesis" or "h." The previous induction can now be restated:
e ⊢y h
To relate this idea of evidential suppor to probability, atheists rightly claim that the degree of support (y) which is given by the evidence (e) to the conclusion (h), is equal to the conditional probability of h given e.
(1) e ⊢y h ⇔ p(h|e) = y
Given the premises, the conclusion of a deductive argument must be true on pain of contradiction. That is, the deductive relation between the premises and conclusion is a necessary truth. It follows, that for any deduction, a corresponding material implication, with the premises as antecedent and the conclusion as consequent, will be a tautology.
A ⊢ B ⇔ ⊢ A → B
We can extend this principle to our partial entailment or induction.
(2) e ⊢y h ⇔ ⊢y e → h
And from (1) and (2) we get
(3) ⊢y e → h ⇔ p(e → h) = y
And from (1) and (3), we get
(4) y = p(e → h) = p(h|e)
That is, the degree of evidential support for the hypothesis is equal to the probability of e → h and the probability of h given e. So far, this agrees with the atheists intuition of how evidential support and probability behave.
When changing the probability of the hypothesis in light of the evidence, it is necessary to also change the probability of all logical consequences of the hypothesis. The set of all logical consequences of the hypothesis has the same probability as the hypothesis, because they are logically equivalent. Now consider the following a valid deduction from the hypothesis.
(5) h ⊢ e → h
That is, it is impossible to update the probability of h in light of e without also updating the probability of e → h in light of e, however, the probability of e → h given e actually declines rather than rising:
(6) p(e → h|e) < p(e → h)
This result is necessary supposing the neither p(h) or p(e) are equal to 0 or 1. Since y is equal to the probability of e → h, it follows that after updating the probability of h in light of e, y actually decreases.
This argument demonstrates something incredibly counter-intuitive. Supposing that the existence of God is initially assigned a probability greater than 0, it follows that as the probability that God does not exist increases, the evidential (or inductive) support that God does not exist actually decreases!
Thus, by atheists own standards, they have no right to say that the evidence does not support the existence of God. In fact, every time they observe something and it is not God, the evidential support for the conclusion that some thing is God keeps growing even while its probability is decreasing. It's counter-intuitive but true: higher probability reduces evidential (or inductive) support!
Alright, I'm done. Whew!