What are some examples of things that cannot be explained by logic?

Actually, logic by itself explains very little. You need a set of premises to draw any conclusion, and logic will never tell you if these premises are correct or not. At most it will tell you they are contradictory. Rationalising is a different story. It's mostly an act of arrogance, but it's not logic's fault. The word itself has different meanings (think rationalising resources), but in the sense you use it seems to refer to: "attempt to explain or justify (one's own or another's behavior or attitude) with logical, plausible reasons, even if these are not true or appropriate" (From the Mac OS X dictionary). The arrogance here comes from the belief that one can know the "premises" that drive oneself or other's behavior. There is an expression from people who do computer models that is "garbage in, garbage out". If one's assumptions about other people/situations are inaccurate, the conclusions will be inaccurate, no matter how good your logic is. So logic almost (with permission from Gödel's incompleteness theorem) never fails. What fails are our hypotheses about the world, and in most real life situations the amount of parameters is huge, and some of them are impossible to know. But thinking that the we can explain anything that happens, that everything follows an order that is within our grasp, make us feel safe, so no wonder many people feel compelled to do it.

Everything yields to logic through http://en.wikipedia.org/wiki/Reductionism. The problem with this is, the resulting models end up explaining exactly what they set out to explain and so risk being examples of http://en.wikipedia.org/wiki/Begging_the_question. One part of making peace with the idea that everything must have an answer: realizing that our definitions of nature, and of science, are bounded by perception. If we can't perceive it - if it is impossible ever to perceive it, whether directly or indirectly - we can't apply science to it. But correspondingly (we reason, fallaciously), if we can't perceive it - if it is impossible ever to perceive it, whether directly or indirectly - it is literally of no consequence. The gray area that we wring our wrists over is populated by phenomena to which we ascribe indirect causes that are beyond our current (and perhaps ultimate) perceptual limits. I suspect consciousness is still in the realm of things that defy logic. The best (ish) that we can do is call it an emergent property of a complex system, and then play at writing computer programs to ape what we observe about it.

The continuum hypothesis. Cantor went nuts trying to rationalize it. Then Godel came along and proved that it was impossible.

Even fully mathematical problems can be undecidable. Here is a list of proven undecidable problems in formal language theory. http://www.cs.wcupa.edu/rkline/fcs/grammar-undecidable.html. One of my favorites is the complement of a context free language is not even recursive (decidable by Turing Machines). For more undecidable problems in other fields of mathematics http://arxiv.org/abs/1204.0299

Following two things (arguably) cannot be explained by logic. Randomness cannot be explained by logic. (by definition). Illogic cannot be explained by logic: In general as soon as one chooses a "logic" i.e. a system of propagating truth and falsehoods, one ends up choosing "illogic" as well. e.g. "Socrates is a man, all men are mortal therefore Socrates is immortal" is illogical in the sense of classical logic. (In turn, Classical logics can be said to have roots in our perception of the world and can be said to be an extension of set theory (labeling sets as True/False etc)). The trouble is, humans (arguably) cannot rationalize about the world much beyond set-theory (why ? because thinking about the world in that way appears to solve most our problems. Why? See * below) ) That's why one can argue -- logics that we create are worthy of being called "logics" only if they satisfy certain basic set-theoretic principles. One can try to be creative about the meaning of logic, but one cannot go far beyond set theoretic ideas for that to be called "logic". If so, illogical statements exist, and those cannot be explained by "logic". Note that if we could construct "illogical" statements using the rules of "set-theory-based-logic", then we would see at least 1 statement that is both logical AND illogical, which is not allowed in our systems of set-theory-based-logics. Thus, we cannot construct illogical statements from rules of logic (See arguments against http://en.wikipedia.org/wiki/Dialetheism) The real question is : can these above two concepts be applied to things we see in the real world ? to which i don't think there is a consensus yet. Randomness: I do not know if randomness truly exists in the real world. I have seen arguments that it does (http://arxiv.org/pdf/0911.3427v3.pdf), but i am not really convinced because maybe i do not understand the field deeply enough. Illogic: As the to existence of illogical things in the real world, we do not know about that either except only in this sense : We all can definitely imagine illogical statements in our heads. The fact that we imagine something means that things in the real world joined forces for our brain to experience that feeling. Thus, there should be something ingrained in natural laws that made us imagine/think illogical things. We already know that logical things cannot construct illogical things. Thus, based on Penrose's argument for "Godelism" (http://math.stanford.edu/~feferman/papers/penrose.pdf), i reason that illogic (in general imagination) must have a physical basis, something inherent in physical nature that allows our brain to explore that dimension. Thus at least a bit of illogic must exist in the real world. Further, in the real world, if we see something defying belief, e.g. wave-particle duality, we just accept that as a premise, but we don't alter our system of logic . We  try to create "extensions" though e.g. Fuzzy logics, temporal logics etc. *One potential reason reality can be explained using the tools of mathematics and set-theory-based-logic is this http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis where the argument is that our universe is so easily analyzable with the tools of math and logic, becuase, (quote): "Only Godel-complete (fully decidable) mathematical structures have physical existence".  My extension to this is:  reality may be more complex, e.g. it may have elements of randomness and illogic sprinkled in it - but mostly fully decidable mathematical structures which has been the reason for our success story in understanding the universe using the tools of set-theory-based logic.

Out of series of events when the probability of occurrence of an event is really low then we call it a random event. We don't have the tool or the technology to measure the occurrence of that event. For example. uncertainty principle foundation relies on the way we see things that is "reflection of light". Now if we have a way to measure the position of the electron other than by using "light photon" then there will be no uncertainty hence no randomness. I think it's totally logical.