That would be one place to start, though you appear to be spoilt for choice.
Logic is a tool like mathematics. It depends on the assumptions you base your calculations on but it quickly becomes apparent when they're wrong. I'm no logician but when you start out with proven facts any conclusions logic allows you to draw will be equally true, so long as your method is sound.
Beliefs often feel like knowledge but are in danger of falling over when tested. You can sincerely believe that you're able to walk on water and you'll be safe in that belief so long as you don't embark on a journey over deep water on foot. Your belief may turn out to be right but that has no relationship with knowledge arising from sound reasoning based on solid facts.
You believe you're right and that makes you happy. Clearly you are not but, as I say, so long as you don't embark on any serious endeavour which relies on your belief being true, it really doesn't matter. In fact it doesn't much matter to me either way, so good luck.
First let me state for the record that I am no scientist or philosopher (I do dig science people though). I only have my education to go on but it seems to me that some people are arguing that you CAN in fact prove a negative. I agree with clivephoto etal that science does not set out to disprove anything. Science in fact sets out to test what you've said and see if it holds up. People make claims and then others try to duplicate the results. The more times you can duplicate the result the stronger the theory becomes. In the case of gravity, it is the best we have so far. Certainly though it is incomplete and there are things yet to be discovered that will make that theory better.
"Christian Science" (oxymoron I know) does at times set out to specifically disprove scientific claims that have mountains of evidence. One such claim is irreducible complexity. Scientifically though it fails under the most rudimentary scrutiny. Science (IMHO) has never been about disproving god but only explaining the unexplainable. As the centuries have progressed more and more things have been explained and taken from the realm of mysticism and squarely placed in the land of science - things like floods, earthquakes, electrical storms etc - are no longer angry gods but acts of nature. And all this thanks to science.
So I don't quite understand the argument for proving a negative. It's just not how science works (at least not at the university where I studied).
Besides that I guess "There are no extraterrestrials" is not really a scientific question. The statement is not testable and therefore is not considered science. If it is considered philosophy though that is different - no one can ever be wrong when in a philosophical discussion.
Science doesn't deal with the supernatural, ie imaginary, world of god, pixies etc., only the natural. If science can be applied then it's no longer supernatural. If it were possible to travel back in time with an Uzi and a projector or any number of modern wonders, you'd be probably be taken as a powerful magician. The Uzi would be especially useful in this respect.
Extraterrestials belong firmly in the natural world because, since we know we exist, it's perfectly reasonable to assume there could be other life forms in the universe.
Gods, being by their very nature, supernatural, are not the province of science and never will be. Any god which can affect the natural world is no longer supernatural and hence not a god, just something more advanced than we might currently understand. Extraterrestials may well appear to be god-like to us should any ever turn up.
Apart from the rather annoying claims that god exists because without a god to create us we couldn't, there's not a shred of evidence for any of the many thousands of gods men have invented to explain that which they don't yet understand.
I can't remember who said it, but when asked by a christian why he didn't believe in god the response was along the lines of- "I contend that we are both atheists. I just believe in one fewer god than you do. When you understand why you dismiss all the other possible gods, you will understand why I dismiss yours"
Logic, on the other hand, demonstrates the impossibility of omnipresence, omnipotence and all the other omni's claimed by the fathful. You can argue the finer points of logic up hill and down dale for as long as you like but I think Epicurus has it nailed down when he says-
Is God willing to prevent evil, but not able? Then he is not omnipotent. Is he able, but not willing? Then he is malevolent. Is he both able and willing? Then whence cometh evil? Is he neither able nor willing? Then why call him God?
Basically, if god can't be relied on to do something useful when needed, then why bother?
To put it in the simplest form, I say that knowledge requires evidence, but believing requires no evidence. Evidence is in the province of science. Belief and language is in the province of philosophy.
The philosophers say that they can prove a negative just by thinking logically about it. They require no evidence. By my definition, not requiring evidence is believing. I do not say their belief is valid or not, just that it is only belief, not knowledge. They contend it is in fact knowledge.
I noticed a lot of activity on this thread lately so I thought I'd step in and see what all the fuss was about... Philosophy, of course.
Now, if this were a scientific topic it would be worded something along the lines of "Can a universe come into existence on its own and organize itself in a manner that forms intelligent life and how can we test it?" Then at some point if the answer to that question was yes, then the next question is, "Can the forces that created such a universe be reproduced or manipulated?". We haven't even reach a point that this topic can reasonably be asked.
So, to all the philosophers out there... enjoy a small bit of the playful criticism from one of the most eloquent critics of philosophy.
--Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong.
--We can't define anything precisely. If we attempt to, we get into that paralysis of thought that comes to philosophers… one saying to the other: "you don't know what you are talking about!". The second one says: "what do you mean by talking? What do you mean by you? What do you mean by know?"
I feel comfortable in saying that I know there is no god because I feel there is sufficient evidence to support that hypothesis and an utter lack of any to the contrary.
For me, sufficient evidence is enough. The lack of concrete proof in this circumstance is irrelevant, unnecessary, and leads to such time-wasting hypothetical pursuits as pondering alternate universes rather than living in the here and now. Life is too damn short to waste daydreaming about "what if."
I realize that evolution does not disprove the existence of god, but it utterly destroys creationism (see Q3 here to understand what I'm getting at). And the notion that a god could steer the process of evolution (as some believers want to believe) is absurd: it's natural (not supernatural) selection, stupid!
But evolution alone doesn't disprove the existence of god (Q5 from the FAQ). Religion takes the final steps toward doing that. Religious texts are so inwardly contradictory (in and of themselves) and so outwardly contradictory (to what science has shown us to be true, for example) that they are preposterous to anyone, of any age, capable of rational thought.
What we're left with if not atheism, then, is deism or spiritualism. The problem I have with these beliefs is explained in my second paragraph: I regard them as glorified daydreaming, no more, no less. My opinion is that those that take the position that "I don't believe in religion but I do believe in god" are clinging to that belief rather than examining the evidence with any degree of real effort. If you want to waste your life doing that, then go right ahead. But for me, reality is too damn interesting to pass by. I'd rather think about what is rather than what I'd like there to be.
Finally, I for one do not "trust scientific fact and reasoning" as OP puts it, and I don't recommend anybody does. I doubt it all the way. Science and scientists have been wrong in the past. They will be wrong again. But there is built-in self-correcting machinery: knowledge evolves. Perhaps the most beautiful thing about science is that it does not require belief in order to work. In fact, I have found that doubting it (this is different than denying it, I should add) makes it work best.
Hypothesizing is a human thing to do; it's part of the thinking process. But remember that theories evolve when there is lack of any contradictory evidence to an hypothesis: they fail to be dis-proven. Neither "god does not exist" nor "god exists" have been disproven concretely, but only one of those statements has any supporting evidence that has withstood intense scrutiny.
I think that belief in science is venomous. I feel that belief is antithetical to the Scientific Method. Beliefs opinions are like assholes. Just the facts, ma'am.
A quick clarification and I'm done. In the last paragraph I was equating belief and trust.
I also wanted to add that it's dangerous to believe in science. That's how frauds (homeopathy anyone?) continue to survive.
... but my edit timer ran out.