Our way of mathematical thinking is totally controlled by a simple two-valued logic $(\mathbf{false}, \mathbf{true})$. All deductions are due to this logic and we are unable to think otherwise. But there are signs seemingly suggesting that this logic is oversimplified. Even if $$(A\iff\neg A)\iff\mathbf{false}$$ there is a difference between the paradoxical $(A\iff\neg A)$ and $\mathbf{false}$ since we easily can accept that a statement $A$ is false but have problems with the paradoxes, as is exemplified in the paradox of Russell.
There is a simple way to correct this defect and extend logic to a three-valued logic $(\mathbf{false}, \mathbf{true},\mathbf{paradoxical})$ with a total correspondence regarding the deduction rules, by introducing a finer equivalence relation '$=$' than logical equivalence. There is a commutative semiring $$\big(\{0,1,2\},+,\cdot\big)$$
$$ \begin{array}{cc} \begin{array}{c|ccc} + & 0 & 1 & 2 \\ \hline 0 & \mathbf{0} & \mathbf{1} & \mathbf{2} \\ 1 & \mathbf{1} & \mathbf{2} & \mathbf{1} \\ 2 & \mathbf{2} & \mathbf{1} & \mathbf{2} \end{array} \qquad \qquad & \qquad \qquad \begin{array}{c|ccc} \cdot & 0 & 1 & 2 \\ \hline 0 & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ 1 & \mathbf{0} & \mathbf{1} & \mathbf{2} \\ 2 & \mathbf{0} & \mathbf{2} & \mathbf{2} \end{array} \end{array} $$ Interpreting $0$ as $\mathbf{false}$, $1$ as $\mathbf{true}$, $2$ as $\mathbf{paradoxical}$, $+$ as $\mathbf{exclusive\;or}$ and $\cdot$ as $\mathbf{and}$, all ordinary logical expressions can be developed: $$ \begin{array}{|l|l|l||l|l|l|} \hline 0000 & \textbf{false}\quad & \textbf 0 \quad & 1000 &\textbf{neither A nor B} & \textbf{1+A+B+AB}\\ 0001 & \textbf{A and B}\quad & \textbf{AB} \quad & 1001 &\textbf{A if and only if B} & \textbf{1+A+B}\\ 0010 & \textbf{A but not B}\quad & \textbf{A+AB} \quad & 1010 & \textbf{not B} & \textbf{1+B}\\ 0011 & \textbf{A}\quad & \textbf A \quad & 1011 & \textbf{if B then A} & \textbf{1+B+AB}\\ 0100 & \textbf{not A but B}\quad & \textbf{B+AB} \quad & 1100 & \textbf{not A} & \textbf{1+A}\\ 0101 & \textbf{B}\quad & \textbf B \quad & 1101 & \textbf{if A then B} & \textbf{1+A+AB}\\ 0110 & \textbf{either A or B}\quad & \textbf{A+B} \quad & 1110 & \textbf{not both A and B} & \textbf{1+AB}\\ 0111 & \textbf{A or B}\quad & \textbf{A+B+AB} \quad & 1111 & \textbf{true} & \textbf 1 \\ \hline \end{array} $$ It can be proved that any law of deduction and any tautology that is valid in ordinary logic also is valid in this extended logic. All calculations and deductions can be performed as usual and the only thing to keep in mind is that $0\neq 2$, but $0\iff 2$.
Now consider the disreputable construction $$ X=\{x|x\notin x\} $$ Obviously $X\in X\iff X\notin X$ and therefore $|X\in X|=2$, which e.g. could be denoted $X\overset{p}{\in}X$. And everything is OK due to the extended logic. Elementary set theory with this logic correspond to ordinary elementary set theory and also point set topology adapt to the concept. One could ask, can this be used somehow? E.g. in quantum physics?
A qubit in superposition state, does not have a value in between $\textbf 0$ and $\textbf 1$. There is no way to tell which of the two possible states forming the superposition state actually pertains.
That lead me to ask if the mathematics for qubits cooperate with this logical semiring? Unfortunately I have very little understanding of quantum phenomena, but it seems nice to have a logic that works also with quantum statement and not only with human everyday statements. I'm aware of that the question is imprecise due to my ignorance, but maybe there are some known references that may still my curiosity?
First, I'm almost certain some mathematicians do not think in a binary way. They probably don't make up the majority of mathematicians, but they do exist. See http://girard.perso.math.cnrs.fr/linear.pdf for instance.
I haven't studied linear logic, but some people told me it might work. You can try having a look at https://en.wikipedia.org/wiki/Linear_logic if the paper above gives you too many headaches.
PS : You might want to learn a bit of proof theory first, if you don't already know about it. The Wikipedia page https://en.wikipedia.org/wiki/Proof_theory should give you enough sources.