The most elementary formulation of Quantum Mechanics (the one usually formulated in Hilbert spaces) can be constructed starting form a lattice of all the elementary propositions which can be tested on a given quantum system obtaining, as the outcome, YES or NOT.
This can be similarly done for classical mechanics and, in that case the elementary propositions are described by a class of sets in the space of phases of the physical system, an elementary proposition $P$ is true (YES) if the state of the system belongs to $P$ at the considered time. That class of sets/propositions has to be closed under the action of logic operators/set operations. For instance $P$ AND $Q$ corresponds to $P \cap Q$, and so on...
A state of the system is, in fact, a map associating each elementary proposition to a number in $\{0,1\}$, where $0$ means NO and $1$ means YES.
If instead the whole set of the outcomes $[0,1]$ is allowable, the state is probabilistic, it is the probability that a elementary proposition is true
(this is the standard, for instance, dealing with statistical mechanics).
The fact that a state is a probability measure (a Dirac measure for sharp states) forces the class of subsets describing propositions to be a $\sigma$-algebra, which is a notion a bit more complicated than a Boolean algebra.
In quantum mechanics one could try to adopt a similar point of view, singling out the elementary YES-NO propositions from scratch when describing a quantum system. The point is that there are pairs of elementary propositions that cannot be joined by connectives: They are the famous incompatible propositions. Here is a typical example. $P=$ "the particle has momentum $p$" and $Q=$ "the particle has position $q$". There is nothing like P AND Q in Nature. No physical experiment can associate a value YES/NO to $P$ AND $Q$.
Physicists say that $P$ and $Q$ are incompatible. A Boolean structure cannot be used here!
However the situation is not so desperate as it may seem at first glance, because there are mathematical structures (non Boolean lattices) able to grasp and describe the physics of elementary propositions of a quantum mechanical system. As a matter of fact these structures are isomorphic to the lattices of orthogonal projectors of Hilbert spaces. The states can be defined similarly to the classical case, (generalized) probability measures, and they turn out to be associated with the normalized vectors of the mentioned Hilbert spaces (I refer here to the so called pure states only).
The point is that these mathematical structures, non Boolean lattices, embody operations that just resemble the classical connectives AND, OR, NOT, $\Rightarrow$ and so on. They are generalizations of the classical corresponding and reduce to them when dealing with sets of pairwise compatible propositions.
Here a twice possibility arises. One can ignore this fact and exploit it as a merely technical opportunity. Alternatively one can assume that these pseudo connectives are the connectives of the quantum world which, in that precise sense (connectives with different properties), satisfy a logic different from the classical one. This was the idea of von Neumann and Birkhoff, who started the first investigation of that mathematical world. Nowadays, quantum logic is a (wide) research area more proper of logicians rather than physicists. I mean, for several reasons even of practical nature, von Neumann and Birkhoff were not able to convince physicists to abandon classical logic in favour of (some) quantum version.
In any cases, there are no problems in handling quantum logic, because it is nothing but a formal procedure to construct statements when initial statements are given. To this regard it is not different from any other mathematically formalized theory. We can handle quantum logic using intuitive logic exactly as we can handle some formalization of classical logic using intuitive logic.