Take one hundred people. Throw an idea at them. Watch them divide into three camps. Three, not two. Group one believes the idea is true. Group two believes the idea is false. Group three reserves judgment.
Of course, for this simplistic model to apply, the idea itself should be simple, not one that allows for responses like 'true up to a point'. Simple examples might be the moon is made of cheese, or Jesus was born of a virgin, or, that most contentious of all, God exists.
This is the archetypal existential statement. Although there are three possible responses to it (yes, no, don't know), logically there are only two possible situations (God exists, God does not exist). Now let's talk about cheese.
If I say cheese exists, again there are three possible responses. Cheese lovers shout- yes, cheese exists! Cheese deniers- no, cheese is a lie! Cheese innocents- perhaps cheese exists, who knows? (but they won't shout; it's not important enough!)
If I then produce half a Camembert and a pound of Stilton and parade them round the room for all to see, the Innocents will say, you were right. Cheese exists. So will most of the Cheese deniers (but grudgingly, because they have lost face). Most, but maybe not all. One or two will probably refuse to see, smell or taste the cheese and will hold fast to their position that cheese does not exist. And everyone else will call them mad.
The point here is that all I had to do to end the argument was produce the cheese for all to see. To prove existence we need merely point to one example. But proving non-existence is logically impossible. Ninety-nine empty cheese-boards do not preclude the existence of one well-laden one. (I like cheese - can you tell?) Now, back to God.
The statement, God exists is logically identical to cheese exists and therefore it also divides the world into three camps:
Please note - Thus far, we've been dealing with absolutes, not probabilities. Later, we'll look at statements like God probably exists, but not yet. Also, we're saying nothing about second order effects like zeal. Some people believe in God harder than others, but that's psychological, not logical.
Having divided the population into three groups by their response to the proposition God exists, let's now look more closely at each group in turn.
The proposition was God exists. There are many names for God. So our group of believers will include Muslims, Christians, Jews, Hindus and many more. Some will believe that only the names are different and so all believe in the same God. Others believe that only their version of God is true, so most of the Believers group members are 'deceived'. Wars are fought over this. But back to the cheese. Let him who believes in the Camembert God simply reveal Him (not a book about Him, but Holy Camembert Himself) to the others, and all is settled. It hasn't happened yet. The fact is, believers have no proof, only faith.
You might think that this group is more homogeneous than the believers, but that's not necessarily true. A large number of Agnostics, perhaps the largest number, are apathetic; the proposal, God exists, simply doesn't interest them. Probably, they don't even know they're agnostic! Another Agnostic group is interested in the question, and would welcome an answer, but is not willing to make a leap of faith in the absence of evidence. This group is truly Agnostic (with a capital A), because they know they don't (can't) know. Within this group of true Agnostics there is a smaller group of Rationalists who actively try to espouse no belief (on any subject) without evidence.
The Atheist who says there is no God is not necessarily unaware of the logical fallacy of the statement. The thinking Atheist, by such a statement, means God's non-existence is far more likely than his existence. This might seem not an unreasonable position. However, at best it is intuitive. S/he really means My feeling is that God's non-existence is far more likely than his existence. You might think I'm being very pedantic here, but consider this: no-one has the ability even to work out the probability of finding a Ford Escort parked on the White House lawn on Christmas day. There are too many variables. Anyone who pretends to be able to calculate the probability of God's existence is a charlatan. The mathematics to do it doesn't exist. For this reason, I think that the strict Atheist position is impossible to justify. To summarise:
If someone says there is no God, not as a belief but as a falsifiable proposal, that is a perfectly sound position. A single verified manifestation of God falsifies the proposal, and so knowledge advances. Strictly, this is not an Atheist position, but a Rationalist one, because it is neither necessary nor desirable (in falsification theory) to believe your proposal true. It is simply a way of testing the water.
Similarly, a Rationalist could put forward the proposal God is omnipresent, without having to believe it. The proposal is falsified by the discovery of a single half brick that can be shown not to contain God. However, the problem here is that no-one yet knows how to test for the presence of God in a brick.