To me it seems very mysterious (and unexpected) that (classical propositional) logic can be "lifted" to powersets and Boolean algebras, in the sense that elements of a powerset behave like propositions (and logical connectives like or and and are "lifted" to set-theoretic union and intersection) and they satisfy the axioms of a Boolean algebra.
And even more mysterious is the observation that intuitionistic propositional logic is the logic of open sets of a topological space and of Heyting algebras.
And then quantum propositional logic seems to be the logic of orthocomplemented lattices of closed subspaces of a Hilbert space (and what algebra?).
And then the modal logic S4 seems to be the logic of interior algebras (and what space?).
And, along this line of thought, David Ellerman has some really interesting stuff about the logic of partitions (of a set), which is dual to the logic of subsets (of a set) and which seems to be related to the fact that subsets are dual to quotient sets (and then he relates partition logic to information theory in the same way that subset logic is related to probability theory).
So logics seem to be related (or, equivalent, even) to various objects in various branches of math. Eg.
\begin{array}{c|c|c} \text{classical propositional logic} & \text{subsets of a set} & \text{Boolean algebra} \\ \hline \text{intuitionistic propositional logic } & \text{open sets of a topological space} & \text{Heyting algebra} \\ \hline \text{quantum (propositional?) logic} & \text{closed subspaces of a Hilbert space} & \text{?} \\ \hline \text{S4} & \text{?} & \text{interior algebra} \\ \hline \text{partition (propositional?) logic} & \text{partitions of a set} & \text{?} \\ \hline \text{classical first-order logic} & \text{?} & \text{?} \\ \hline \text{classical $n$th-order logic, for all $n$} & \text{?} & \text{?} \\ \hline \end{array}
- Why are logics related to lattices and algebras? What can be said about this relationship? Are there any books that talk about this relationship (and the table above) in detail?
- How can the $2$nd-to-last row in the table above be filled (ie. for classical first-order logic)?
- Is there a mathematical object that contains all classical logics of all degrees, all at once? Kinda like the exterior algebra of a vector space contains all exterior powers of all degrees, all at once, together with a meaningful way in which the elements of different degrees interact (like graded-commutativity and the exterior product)?