Question on the logic behind quantum computing

According to Wikipedia,

"Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.[1] Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits), which can be in superpositions of states."

Which implies that in a Quantum computer a qubit can have the values 0 and 1 at the same time. With the help from Quantum mechanical phenomena such as entanglement and superposition.

But according to fundamental logic a statement can have only one truth value.

"The truth or falsity of a statement is known as it's truth value. For an expression to be a statement, it is not necessary that we actually know whether it is true or false, but it must clearly be the case that it is one or the other" -Analysis (Steven R.Lay)

With the facts being so, how can quantum computing be possible? Doesn't it violate basic logic? Or am I getting the whole thing wrong?

Also I thought this is the best SE to post my question as SEs regarding computing are not familiar with QM.

• propositional logic is not the only logic around. See for instance en.wikipedia.org/wiki/Fuzzy_logic – user83548 Sep 11 '15 at 17:40
• Part of the confusion may arise from the fact that classical computers are overwhelmingly not being used to express the truth value of logical propositions but they are being used to approximate real numbers with IEEE floating points. While you can use a classical computer to prove logical statements, that's not their main application. Neither will it be the main application of quantum computers. We are still going to be mainly interested in calculating numerical approximations of problems. – CuriousOne Sep 11 '15 at 17:55
• This link may go a long way toward answering your question: cds.cern.ch/record/383367/files/… – Ernie Sep 12 '15 at 0:36
• @Ernie that seems to answer my question. I'll read the whole thing leisurely. Tnx :) – slhulk Sep 12 '15 at 2:42

Before the measurement, however, it gets tricky. In this realm, quantum mechanics breaks logic--in particular, Bell's inequality. Let's say we have three two-valued properties: A, B, and C. We assume their properties are predetermined (even if we can't observe them) and the properties are local, i.e. they do not effect each other. Then the probability $P(X,Y)$ of finding the properties of $X$ identical to the properties of $Y$ is
$$P(A,B)+P(A,C)+P(B,C)\ge 1$$