I see your point, but it still sounds like a fallacy. Probably, there is some missing denominator probability in the equation.
P(FF) ... the probability of the great filter being an event in the future
P(ET) ... the probability of extra-terrestrial life existing or having existed
Apply your best guess as you see fit.
[MATH]P(FF|ET)=\frac{P(ET|FF)\times{}P(FF)}{P(ET|FF)\times{}P(FF) + P(ET|\neg{}FF)\times{}P(\neg{}FF)}[/MATH]
---------- Post added at 06:07 PM ---------- Previous post was at 05:55 PM ----------
"we haven't yet seen aliens doing a bunch of stuff some people think they would do, therefore aliens don't exist".
The main alternative explanations of the non-visibility of ET life everywhere are:
1.)That we're the first intelligent species to arise in a galaxy of a 100 billion stars that's 10 billion years old.
2.)That every other intelligent species has decided not to settle the galaxy and stood by that decision for billions of years.
The first option might be valid if we discover our system to be somehow unique, or that the entire galaxy was unsuitable to life until about the time it started on Earth.
However, if this galaxy is anywhere near as planet-dense as the Kepler sample and one in a million planets holds life, the second hypothesis is entirely ridiculous. If it's that common, than the odds are ridiculously low for us having been the first.