I'm an atheist...

[macskeptic]

Q isn't a condition for P, it is the result of P. That is the difference.

If P => Q and Q => P you have a circular argument. Either both are true from the outset, or both are false from the outset. You can't have P (false) => Q (true) as you could before, because if Q is true, P is true (since you claim Q is a neccessary condition of P).

Going back to your example, you are confusing things yet again. You said "Q is a neccessary condition to affirm P", yet your example to show this is and example of P => Q, not Q => P. You said you can't say that if you throw the ball, it didn't fall (an example of P => Q).

However, you can easily have a ball falling (Q) without P. For instance, if R is dropping a ball. R => Q.

So, we can demonstrate a logical contradiction:

P = Throwing a ball.
R = Dropping a ball.
Q = The ball falls.

P => Q
Q => P
R => Q

P is false (we're not throwing the ball). If R then Q (this is true since R => Q). If Q then P (taken from Q => P). Yet P is false and Q is true, and true => false isn't a case of Q => P.

Q.E.D

Ok, rephrasing your example:

The fact that the ball fell, isn't enough (sufficient) to say that I threw the ball. (so... no Q => P), I think we agreed until here, right?

Ok, so let's take Q => P totally out of the example, since we both agree it is wrong.

But, what I am saying is that, if you say that either P or R happened (be it letting go of the ball or throwing it), you will necessarily have to see/observe that Q happens, otherwise what you are saying with P => Q / R => Q is a lie (or, in other words, an invalid argument).

The mathematical terms are exactly these actually.
P => Q can be read: P is sufficient for Q or Q is necessary for P. (Reference: http://en.wikipedia.org/wiki/Necessary_a..._condition)
It applies in a strict logical sense, this should not be confused with cause and effect - we agree that Q is an effect that can be caused by either P or R.

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I believe our disagreement has much more to do with how it is versed in language (english) than the logics itself.