Is classical logic inconsistent with some facts in QM? 
Quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper1 by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum.

The emphasis in mine. Is classical logic actually inconsistent with some facts in QM?
Is it surprising that a theory could be developed which was not logically consistent in this way?
 A: I think @kpv is correct: wikipedia's choice of the word "inconsistent" isn't entirely appropriate; "inapplicable" might be a better choice.
Classical logic provides rules of inference that may or may not correctly characterize systems whose behavior we want to describe formally. One classical rule of inference is that if $A\Rightarrow B$ and $A\Rightarrow C$ then $A\Rightarrow B\& C$. Sounds pretty reasonable, doesn't it? But suppose you're standing at a vending machine that dispenses candy for a dollar and also dispenses coke for a dollar. You might write that as $1\$\Rightarrow\mbox{candy}$ and $1\$\Rightarrow\mbox{coke}$. But then inferring that $1\$\Rightarrow\mbox{candy&coke}$ is clearly wrong. The dollar gets "used up".
That's called a resource-aware logic, with http://plato.stanford.edu/entries/logic-linear/ the paradigmatic example. Here, premises (our dollars) are discharged (our "used up") during proofs. So the preceding classical rule doesn't exist. And this kind of "resource awareness" may be applicable to quantum mechanics, e.g., http://boole.stanford.edu/pub/ql.pdf since if you prepare an initial state and subsequently measure it, that prepared initial state gets "used up"/changed during the measurement (unless, of course, it's already an eigenstate of the measured observable). So classical rules of inference are simply inapplicable to our universe of discourse when we're discussing quantum phenomena.
A: QM and classical physics both are logical. The difference is that QM defies our common sense. Reason is our common sense developed per our classical experiences. We have no real life experiences with quantum physics and so have not developed that common sense. However we are building that common sense slowly based upon the experimental results and an attempt to describe the behavior in terms of classical sense.
In classical physics, we first gain common sense and then we formulate it mathematically. This process has reversed in case of quantum physics - mathematical formulation was developed way ahead of theoretical sense. 
Once we will grasp workings of QM by extending our common sense, it will not appear as weird as it does today.
So, classical logic is not inconsistent with QM, it is just behind it by nearly a century in context of being able to describe QM workings.
A: There is one inconsistency which is very important: in classical logic, it is assumed that there is no limit to what you can measure other than the quality of your equipment.  In quantum mechanics, there are limits to what can be measured (the uncertainty principle).
The truth is, classical logic came first.  QM came later.  At first, classical logic was consistent with reality, because the quality of our measurement tools was far worse than the limits provided by the uncertainty principle.  However, as our tools got better, we started noticing that there were effects that were not well modeled in classical logic.  QM was formulated to achieve predictions which mirrored reality in these extreme scenarios.
If one thinks of classical logic as a "good approximation" for reality, then any inconsistencies really just turn into limitations of that approximation.  Indeed, classical logic was always just an approximation, just as quantum mechanics is just an approximation... we just got used to it being such a good approximation that we mistake it for reality.
If you look at the realm where classical logic is still a good approximation, you will find that it's predictions and those of QM are very much in line.  It's only the fringes where the two differ, and in those cases, QM is currently a better model of reality.
