Connection between Boolean and Heyting algebras as models and as Lindenbaum-Tarski algebras I'm a bit confused about the role of Boolean/Heyting algebras in logic. It seems to me that there are two different facets of their usage that I'm trying to reconcile. The first one is the usual construction of a Boolean/Heyting algebra from equivalence classes of sentences of a theory, in classical/intuitionistic propositional logic - the Lindenbaum-Tarski algebra of the theory.
The other one is using Boolean/Heyting algebras as models or a domain of generalised truth values. By assigning to my propositional symbols values from an arbitrary 2-element Boolean/Heyting algebra, the connectives like and, not, or, and implication follow from the structure of the algebra. Can this be generalised beyond a 2-value Boolean algebra? Enderton, when introducing truth assignments, says that "[While we commit ourselves to 2-valued logic] a particularly interesting case is that for which the truth values form something called a complete Boolean algebra." Does this mean I can have "two-valued" models, "three-valued" models and so on, of a classical propositional theory?
The only connection between these two views I think is that we can view truth assignments/valuations/models of classical propositional logic as a Boolean homomorphism from the corresponding Lindenbaum–Tarski algebra to the 2-element Boolean algebra. Do I also get homomorphisms from a Lindenbaum-Tarski algebra to a n-element Boolean algebra that correspond to "n-valued" models of my theory? Does this work the same for intuitionistic propositional logic/Heyting algebras correspondingly?
What is the generalised connection between these two facets? Awodey in his Category Theory book also hints to a two-faceted correspondence: "The exact correspondence is given by mutually inverse constructions between Heyting algebras and IPCs.We briefly indicate one direction of this correspondence, leaving the other one to the reader’s ingenuity." page 133. Is the "other direction" of this correspondence what I'm looking for?
Another use of Heyting/Boolean algebras is in topos theory, where the structure of a subobject classifier in a topos is generally a Heyting algebra. Is it a Heyting algebra as in the first notion i.e the Lindenbaum algebra of the internal logic or in the second notion i.e as a model/domain of possible truth values of the internal logic. I'd say its the second one but I'm not so sure.
Finally, how does this mesh with Quantum logic? Quantum logic is supposed to arise from the orthocomplemented lattice of closed subspaces of a Hilbert space. So is this lattice the model of some logic? Or is the Lindenbaum-Tarski algebra of this logic? My hunch is that the lattice arising from a specific Hilbert space, is a model that corresponds to a specific quantum system, but then what is the logic whose model that is? Every discussion about quantum logive I've read deals with the lattice part but I can't find anything on the actual "logic", as in the actual formal language with proposition symbols, connectives etc.
 A: About Many-Valued Logic you can see :

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*Merrie Bergmann, An Introduction to Many-Valued and Fuzzy Logic (2008) :


Ch.4.1 Numeric Truth-Values for Classical Logic
Ch.4.2 Boolean Algebras and Classical Logic
Ch.5 Three-Valued Propositional Logics: Semantics
Ch.9 Alternative Semantics for Three-Valued Logic : where the abstract algebraic structures characterizing the operations of three-valued logical systems (Kleene’s “Strong” Three-Valued Logic and Lukasiewicz’s Three-Valued Logic), are explored.


For Quantum Logic you can see also :

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*Miklós Rédei, Quantum logic in algebraic approach (1998).


See also : J.Michael Dunn & Gary Hardegree, Algebraic Methods in Philosophical Logic (2001).
A: You could read one of the foundational papers in Logic proving the completeness and universality of an m-valued logic, by Emil Post in 1921.
Emil Post, “Introduction to a General theory of Elementary Propositions ”, American Journal of Mathematics 43: 163–185, 1921.
For the connection with Quantum Logic I have had some reflections on this topic in two papers
Zeno Toffano, "Eigenlogic in the spirit of George Boole" ,  arXiv:1512.06632 [cs.LO]
Francois Dubois, Zeno Toffano, "Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems" , arXiv:1607.03509 [quant-ph]
Zeno TOFFANO
CentraleSupelec
France
