One way to answer this question, I think, would be to consider the history of science. Ptolemy, for example, believed that the earth was at the center of the universe, and that the sun and other planets revolved around it in roughly circular orbits, except for "eccentricities" accounting for which was much of what astronomers did in those days. Copernicus corrected part of that, holding instead that the sun was at the center of the universe and that the Earth and other planets revolved around it, with only the Moon (now not considered a planet) revolving around the Earth. But Copernicus too thought that the orbits of the planets were roughly circular, except for eccentricities By the time of Descartes, it was realized that the sun is not at the center of the universe, but it one star among many, though Descartes did think the sun was at the center of (what we would now call) the solar system. Kepler would later replace the view that the orbits of the planets are circular with the much more nearly...

I know several people who believe such things, or at least say they do. One group thinks that there are true contradictions that involve very special cases. The usual example is the so-called liar sentence, "This very sentence is not true". There is a simple argument that the liar sentence is both true and not true, and some people believe just that. Other people, though, think there are contradictions involving much less special cases. An example would be what are called "borderline cases" of vaguepredicates, like "bald". People often want to say that there are somepeople who aren't bald and aren't not bald either. But the so-called DeMorgan equivalences entail that this is equivalent to saying that theperson is both bald and not-bald (or, strictly, both not-bald andnot-not-bald). People who hold such views are known as "dialetheists". See this article for more.

To ask a question our illustrious leader, Alexander George, has several times asked here: What's meant by "prove"? If what's meant is what's ordinarily meant by "prove", then it's not clear that a single person on this planet can prove human beings evolved from apes. Nor can anyone prove that the Loch Ness monster does not exist. But that simply doesn't mean that there can't be good reasons to believe that human beings evolved from apes or that the Loch Ness monster does not exist. There can be, and there are. Now what exactly that has to do with the rest of the question is not yet clear. But have a look here http://plato.stanford.edu/entries/pascal-wager/ for some thoughts (not mine).

I don't know the context of this claim, nor why Dawkins thinks---I take it he does think this---that no-one has any "evidence" for religious belief. Most theistically inclined epistemologists of religion, in the analytic tradition, anyway, think we do have certain kinds of evidence for belief in God. Dawkins might not find the evidence impressive, or he might disagree as to the evidential facts themselves, but it would be a parody of religious faith to think people believe on absolutely no basis. Just for example, suppose one is some kind of coherentist. Then you might think belief in God forms part of an overal "theory" of the world, and the evidence one has for it is that this theory is coherent, more successful than alternative theories, etc. You've got the same kind of evidence for your belief in God, ultimately, as for anything else you might believe, though belief in God, in such a system, will be deeply embedded, like very high-level theoretical claims, rather than towards the periphery, where...

It's hard to be sure what "rationality is normative" means, but I think I know what someone who would write that would mean. (That is: what they meant, whatever it is their words meant.) The word "rationality" is an abstract noun, formed from the adjective, "rational". So we ought really look at that adjective. So we say things like, "It is rational to do X", or "Doing X is (or would be) rational", and the like. To say that these are "normative" statements is to contrast them with merely "descriptive" statements: They are normative in the sense that they say, in some sense, what one ought to do. Now exactly how claims about rationality are related to claims about what one ought to do---that's the really hard question.

There could be several different sorts of irrational action. Some actions might be irrational in the strong sense that the actor did not have much approaching a reason for the act. The person might "just be reacting". Another possibility would be actions for which the actor has reasons but for which s'he does not have anything approaching a good reason. In this case, one might act contrary to reason, and such an action might reasonably be called "irrational". A really good answer to this question would need more context.