However, by far the most common form of illogical reasoning is the type where someone equates two totally unrelated things as being the same, or where someone claims that two things which are similar in some superficial way as being the same thing. For example, take this gem:
"Imagine that you could listen to or talk face-to-face with satan about any subject that has to do with God.
Then bring to remembrance anything you've heard atheists say against God. (Or if you are an atheist, you know your own sentiments and thoughts).
Now, if you try, you will find that it is impossible to imagine satan expressing sentiments about God different than what atheists express. Satan certainly is not going to speak positively about God, and neither are atheists. So, what both think and verbalize is in complete harmony with one another.
Atheism is therefore a doctrine of demons, and in many, if not all cases, atheists are demon-possessed."
The reasoning behind this kind of claim goes something like this:
X is A
Y is also A
Therefore X is Y.
The problem with this kind of thinking should be pretty obvious. For example, swans are white (yes I know there are also black swans). A Boeing 747 is also white. Therefore swans are simply miniture Boeing 747s. Swans fly. Boeing 747s fly. Swans really must be Boeing 747s.
Obviously, this is not true.
This kind of reasoning only works if A, that characteristic which is shared by both subjects, X and Y is a defining characteristic of either X or Y. Let me explain.
If, for example, you said that birds were defined as "animals which have two feathery wings", then anything that has the characteristic of "having two feathery wings" would be classified as a bird. Now:
Birds have two feathery wings.
Swans have two feathery wings.
Swans are birds.
This works because of the use of a defining characteristic. Defining characteristics are such that anything with that characteristic falls into that classification; and anything that is NOT classifed as such CANNOT have that characteristic.
"Two feathery wings" is a defining characteristic of birds. Anything with that characteristic (anything with two feathery wings) falls into that classification (is a bird). Anything that is not classified as such (anything that is not a bird, such as a gazelle, or a whale) CANNOT have that characteristic (they CANNOT have two feathery wings). This is what is meant by a defining characteristic.
Being white and flying is not a defining characteristic of Boeing 747s or swans, so this kind of reasoning fails in such a case.
Also note that this kind of reasoning only works when describing categories, such as "birds". This reasoning allows one to classify subjects but not to make assertions that one thing IS another thing. For example:
Swans have two feathery wings.
Geese have two feathery wings.
Therefore swans are geese.
This does not work; "geese" applies to a specific bird (or a handful of species) rather than to a broad "class" of birds.
When we look back at the reasoning given in the original fundie argument, we see it violates both of these rules. It does not pretain to defining characterists (anti-God sentiments surely are part of being an atheist, but are not a defining characteristic; it is entirely plausible that one can be "anti-God" but not be an atheist), and it does not apply to categories (neither "Satan" nor "atheist" are categories).
Alas, this kind of reasoning is rampant amongst fundies. A vast number of arguments can be countered based on the fallacy outlined above alone. Its a shame (and kinda funny) that so many people keep coming up with this kind of argument and expect it to stand up to scrutiny.