Proof by contradiction is nothing complex. Fastest explanation: Let's say you know that A -> B, and you want to disprove A. Proof by contradiction is showing that B is false, which then proves that A is false. Because saying that A -> B is the same as saying that NOT(B) -> NOT(A).
That's actually proof by contrapositive.
http://en.wikipedia.org/wiki/Proof_by_contrapositiveProof by contradiction is done by supposing the proposition's negation is true and showing that this leads to a contradiction.
In this case, the negation would be "there exists a car that I can't see".
The problem with the original car example isn't really that it's false-- if we can agree that "all cars are visible" is reasonable, at this time-- rather, the problem is that not seeing a car at time t doesn't allow you to infer anything about the presence of a car at time t+a (for a=/= 0). You could obviously still be hit because of the time elapsed between looking and pulling out into traffic. But, you know, that isn't relevant to the overall point of the example.
Secondly, there IS a difference between "proof of" and "very good reason to think that". Again, not an opinion, but a fact.
That is essentially the difference between deductive and inductive reasoning. Yes, inductive reasoning doesn't assert that conclusions are absolutely true, rather that they are probably true; that there is good evidence to think that something's true. Inductive reasoning is what most of us use to function in everyday life. Inductively, it reasonable to assume that there is no car at time t+a (for small a) because there was no car at time t.
When people say "you can't prove that something doesn't exist", they usually mean you can't employ deduction in that situation because you can't possibly check every case (as Sissyl said). Sometimes you can deductively show that things don't exist because you don't have to check every case."No dogs are made of fire." This is easy because humans have made a definition for what it means to be a living thing. Living things are made of "stuff" (that's the scientific term), but fire is not "stuff". Stuff that is occupying the same space as flame is said to be "on fire" not "made of fire".
I never realized that I was so bigoted toward elementals.
Some definitions of god make proving nonexistence possible ('proof' in the deductive sense, as you were using it). Defining god as "the creator of the universe" is vague enough to make it impossible. Defining it as "unknowable" makes proof impossible by definition.
Science (and philosophy) is done using a combination of deduction and induction. Sometimes science-hostile types claim that there's no proof for scientific theories because "they're just theories". This is partly a misunderstanding of the word "theory", but it's also a claim that science is likely wrong because it often employs inductive reasoning (especially to formulate initial hypotheses) and scientists can't possibly check every case of something. Having to check cases is not usually a desirable way to prove something; alas, large primes are a pain.
Legal proof is just evidence, not proof. People use "proof" in conversation differently that it's used in logics.
It's true that oftentimes inductive arguments are specious, either because they don't make deductively valid arguments or they are bad at prediction. In the LGBT Gamer thread, they're arguing about the usefulness of hospitals asking for a patient's gender. One argument goes, "If I know a patient's gender and age, I can reasonably conclude some information about the patient that is necessary to patient care." For any claims made about people based on their gender, I can easily find lots of counterexamples.
Medical claim: "Women have ovaries."
"This woman does not have ovaries."
Medical claim: "Define 'women' as those individuals that have XX sex chromosomes. Women have ovaries."
"This person with XX sex chromosomes does not have ovaries."
[Opinion] It is obvious to me that any assumptions you try to make will be wrong for some patients (or many, depending on the assumption), so it doesn't seem very useful. Hence, if you need to know about ovaries-- or really any detail about a patient's body-- I think you should investigate that thing rather than rely on generalizations about humans. It would be embarrassing to recommend that a patient have their gall bladder removed only to be told that they've already had it removed.
This, however, doesn't mean that inductive reasoning never produces anything useful.