EXAMPLE 1: THE BASIC LOGICAL STRUCTURE
1.1. I want to go to the park
1.2. It's not the case that I want to go to the park
1.3. I want not to go to the park
In this analogy, a proposition is given in (1.1), "I want to go to the park." When we negate that proposition, we get (1.2). What does (1.2) mean? It means that I have no preference to go to the park. But the question is do I have a preference not to go to the park? We do not know in this case. We only know that I have no preference to go. This could mean that I have a preference to stay home. It would be a much more strong statement to say that I have a particular preference not to go to the park as in (1.3).
EXAMPLE 2: THE BASIC LOGICAL STRUCTURE USING BELIEF
2.1. I believe Elvis Presley is alive
2.2. It's not the case that I believe Elvis Presley is alive
2.3. I believe Elvis Presley is not alive
In this case we have a proposition in (2.1), this time a statement of belief, that the King is alive. In (2.2), I have no belief that Elvis is alive. This could mean that I don't know anything about Elvis, and therefore don't know one way or another whether Elvis is alive or not. All I know in (2.2) is that there is no belief of Elvis being alive. This does not mean there is a belief that Elvis is not alive, as (2.3) requires.
EXAMPLE 3: THE BASIC LOGICAL STRUCTURE USING THEISTIC BELIEF
3.1. I believe a God exists
3.2. It's not the case that I believe a God exists
3.3. I believe a God does not exist
So from the other instances, we find our central theistic proposition in (3.1). The negation of that (3.2) says that I have no belief that a God exists. As above, this could mean I don't know anything about God, and therefore don't know one way or another whether a God exists or not. As above, all I know in (3.2) is that there is no belief of the existence of a God. This does not mean there is a belief that a God does not exist, as in (3.3).
What this means is that theistic belief is well, either theistic or atheistic; it comes down to either (3.1) or (3.2). A person either believes or does not believe in the existence of a God. However just because a person does not believe does not mean they believe in non-existence. The strong atheist requirement (3.3) goes a step further than (3.2) and requires that a person conclude the non-existence of deities.
For additional discussion on the Strong vs. Weak distinction, see Austin Cline's Strong Atheism vs. Weak Atheism: What's the Difference?
