Is this statement about quantum mechanics valid? In Philosophy of Language by William G. Lycan, there are the lines:

Even apparent truths of logic, such as truths of the form "Either P or not P", might be abandoned in light of suitably weird phenomena in quantum mechanics.

I really don't know much about quantum mechanics, is this statement valid?
 A: Let me preface this by saying that I don't think it's semantically appropriate to ask

Is this argument valid?

because I don't think any argument has been made here.  Instead, the author is making a statement, and we might be inclined to judge whether it is true or false.
In my opinion, his statement is (a) vague (at least when out of context), so it would be very difficult to assign it a truth value and (b) misleading.  Quantum mechanics is a model for the physical world based on mathematics.  In particular, every mathematically precise statement that is ever made in quantum mechanics can be written, just as any purely mathematical statement can, using standard rules of mathematical logic.
A: Yes, the statement in the book is sensible.
Consider a physical system with two possible "states", like: Electron spin = pointing up or pointing down, or light polarization = clockwise/anti-clockwise (with respect to it's direction of propagation). 
Examples like door = open/closed and schrodinger's cat = dead/alive make it sound totally weird since we never experience such in our daily experience. So I would prefer to avoid them. There are reasons why such quantum effects are observed only on tiny scales, and not in day to day classical phenomena.
Intuition from day-to-day life suggests that the electron spin can be either up, or down, but not both. But that's an incorrect picture to describe the system. It is a better description to say that the electron is in a "superposition" of the two states... like say $0.8|up\rangle + 0.6|down\rangle$. Note that the coefficients don't sum to one.  Instead, the squares of coefficients sum to 1. So these coefficients are a little different from usual probabilities -- they are called probability amplitudes. So not that here, the electron is in a state which has both P=up (partly) and (~P = NOT up) partly. Genralizing from this example, states that are completely P (or ~P) are edge cases. A generic state will be a combination of the two possibilities. Now you can understand what the author means by that statement. Classically you would have said that the electron spin must have been up or down, but not both based on the probability rule prob(up) + 
prob(not up) = 1. But that's not true any more and the system simultaneously is both up and down, in a specific way.
Important note : This does NOT mean that quantum mechanics is illogical. In fact, this means that the usual rules of probability are too crude and not subtle enough to describe the behaviour of physical systems on small scales. We have developed/found other mathematical structures to precisely and accurately describe quantum phenomena.
An electron can remain in this state till it's "measured". The best description we have so far is that as soon as you measure whether such a system is spin up or spin down, the electron immediately falls into one of the two states! (It's like the electron doesn't want to be caught with it's pants down and one foot in each state, but it also can't make up it's mind till it's forced to do so by someone measuring it. And don't take this explanation too seriously.)
A: Such statements emerge when trying to capture quantum physics in classical physics terms. In other words, statements like "P or not P is not a valid proposition in a quantum world" are typically made by philosophers who don't really understand quantum physics and stubbornly attempt to map the quantum onto their classical intuition. 
A much more insightful remark would be to state that quantum physics forces us to abandon classical probability theory and to replace it with what one could refer to as a Pythagorean probability theory. In other words, quantum physics gives a simple and straightforward linear description of our world in terms of squareroots of probabilities (referred to as probability amplitudes). And yes, this yields counterintuitive effects, but there is absolutely no need to say goodbye to logic.
A: Superposition isn't the only case where seemingly exclusive event both seem to happen.  For example in the double slit experiment "P" could be "photon went through right slit" and "not P" could be "photon went through left slit" and the interference pattern seems to imply that it actually went through both.
From a philosophical perspective I think a more reasonable approach would not be to abandon logic but rather recognize that logic may not be an accurate description of the quantum world.
