I'm an atheist...

[macskeptic]

Translating to mathematics (or logic)

P: There is evidence for fairies
Q: We can say that fairies exist for sure

P => Q (P is a sufficient condition for Q, as Q is a necessary condition for P)

We know for sure that P is false, right?
But even then, there is no logical, rational way to get from there to a point when we can say anything about Q.

Bad use of logic.

P => Q means that P is a sufficient condition for Q, but not vice-versa. Just because someone (or all of us) can say "fairies exist for sure" doesn't mean there is evidence for them.

Indeed, if we "know for sure" that P is false, then Q *is* false by definition.

Actually I will have to disagree.

To make things easier I will throw a much less complicated example.

P: I threw a lit torch on the forest
Q: The forest caught on fire

P => Q (let's assume that if I do throw the torch, there is NO WAY it won't caught on fire, just for the sake of the argument)

You may say that if Q (the forest caught on fire) that means that certainly P (I threw the lit torch in it) [this is wrong, I will get there in a second]
You may also say that if P is false (I didn't threw the lit torch), then Q is false (the forest didn't caught on fire) [this is also wrong]

Both are wrong because the forest caught of fire because of this:
R => Q

R: A lightning struck a tree on the forest

In other words... in this simplified scenario the jumped conclusion was incorrect due to lack of the total variable involved in the problem.

That was to demonstrate that my logic concept is right =P

Now, for the fairies problem, it is a little more complicated - because the way I constructed it (which might be wrong, but the logic is not) "P => Q" states that there can be fairies (Q) without necessarily being evidence for their existence (~P).

The possible outcomes of a P => Q expression are:

if P is true and Q is true then the expression is true
if P is false and Q is true then the expression can be true*
if P is false and Q is false then the expression can be true*
if P is true and Q is false then the expression is false

* With these two comes that the lack of evidence (~P, or false P) is not enough information for us to affirm that fairies exist or not (Q or ~Q).

Just to clarify...

Yes, I think it is pretty useless to adopt a completely neutral posture with things like fairies, or unicorns, or gods because of these logical deadlocks. After all, it would get in the way of our thought process.

My point is just that the logical construct "lack of evidence" doesn't contribute to proving or disproving anything logically.