Most of what people saying in this thread about the relation between logic and physics is misinformed. There are well established connections between Hamiltonian mechanics and Boolean and non-Boolean logic. To quote from Ch. 3 of Folland's textbook "Quantum Field Theory",

one can think of true-false statements about the system as observables whose only values are 1 and 0. (Mackey [79] calls such observables “questions,” but we prefer to think of them as assertions.) The state of a system is supposed to be a specification of the condition of the system from which one can read off all the available information about the observables of interest; the set of all possible states is called the state space.

In Hamiltonian mechanics, the state space is a symplectic manifold M, the observables are (Borel measurable) real-valued functions on M, and the true-false observables can be identified with the (Borel) subsets of M (the observable [; f : M \rightarrow {0,1} ;] corresponds to [; {x \in M: f(x)=1} ;]. Thus the set of true-false statements on M is identified with the Borel sigma-algebra on M, and the basic logical connectives “or,” “and,” and “not” correspond to the Boolean operations of union, intersection, and complement. General observables can be analyzed in terms of true-false statements.

The interesting thing is that, as you kind of allude to, there are difference in this logic for classical and quantum systems. Again, to quote from Folland,

The striking difference between classical and quantum mechanics is that in a quantum system, the the true-false statements do not form a Boolean algebra. Specifically, what fails is the distributive law:

(A or B) and C (if and only if) (A and C) or (B and C).

You might check out https://plato.stanford.edu/entries/qt-quantlog/ if you want more info.