From Pascals Wager:
Look at it this way: suppose Pascal is racing a horse, and he's betting his entire fortune on it winning the race. Is that a wise bet? Surely this depends on what other horses are running. Pascal's bet has assumed there's only two. This is the classical fallacy of bifurcation (www.infidels.org 
).
The fallacy is this: Some people take a true statement "you must choose X or ~X" and pervert it into "you must choose X or Y" where Y is only a subset of ~X.
You then prove that Y is not a good option and claim X, ignoring the whole range of options in ~X that are not Y.
That is one kind of a False Dichotomy. The other is where Y is a proper superset of ~X. Saying "Either X or Y must be true" when Y is a superset of ~X is not fallacious, but saying "X and Y can't both be true" is.
However, it is NOT a Bifurcation Fallacy to offer a choice between X and ~X. Just be careful about dealing with the scope of ~X. The advantage of using this sort of binary choice algorithm is that it allows proving X is not a good option, and then breaking down ~X into Y and ~Y and dealing with Y, etc. etc. until one can arrive at two choices where both the first choice and its complement are easily dealt with.
The subtlest bifurcation fallacies occur when Y looks very much like ~X. For instance: "Either such-and-such is good or it's bad" (no; it might be neutral).
If Y were a superset of ~X, then it would partially intersect with X, which seems unlikely to happen in a real argument. But maybe somebody can think of an example?
How about the phrase in a song: "Don't Mess with Mr Inbetween" or the verse "I would that you were cold or hot, but because you are lukewarm, I will spew you out of my mouth".
How do the above examples demonstrate the fallacy of using X or Y where Y is a proper superset of X?
How about this for an example: a number is either 5 or not 5. That is true. Then, you pervert it into something along the lines of: a number is either 5 or 13. Assuming that a number is either 5 or 13 puts you in a situation such as this: 3+67 does not equal 5, therefore 3+67=13
This is a bad example. What if the number in question is sqrt(25), which is both 5 and -5? It is both 5 and not 5. Your logic assumed there were two exclusive possibilities, with no provision for both, neither, or third options. Classic Bifurcation Fallacy. OK, I take it back. Good example. -- Mental Nomad
That's not a good counterexample. When the output of sqrt is in the context of a number, usually the principal square root is chosen, which would be the number 5. When it's in the context of having multiple roots available, then the output is no longer a number; it's a set of numbers, so the original statement's condition doesn't apply. For that matter, the set {5, -5} isn't both 5 and non-5; it's just non-5, since {5, -5} (the set) is not mathematically equivalent to 5 (the number); it just contains it without being identical to it, and is therefore a distinctly non-5 entity.
The Hot, cold lukewarm example is from Revelation 3:16 ("So then: because thou art lukewarm, and neither cold nor hot, I will spue thee out of my mouth."). ''[It may also be said about tea. Isn't this a little more understandable than X this and Y that? Especially if you like tea? -- Donald Noyes
[Unsupported reference to Socrates deleted.]
Pascal's wager also does not take into account the time one would waste in life studying and believing in something. One could argue that so much time might be wasted, that being agnostic would be a better bet - this way one takes a "shrugged shoulders" innocent stance and focuses his life not worrying so much about religious "truths" (or non truths).
See original on c2.com