My question is this: If I know a deductive system for a logic is there a simple way to derive the appropriate algebra for that system?
For example: I start with the natural deduction system for intuitionistic logic, I perform this theoretical procedure, and I get out the laws of Heyting algebras.
The more meta question is: how do you even go about finding out what the appropriate algebra is? Do people just use intuition, or try repeatedly deriving laws and throwing them into the algebra till they get soundness and completeness. Or is there more systematic way of going about it that I haven't heard about?
Thanks!
Addendum: if there is such a procedure, what does it work on and what doesn't it work on? For example, it might work on all natural deduction systems, or all sequent calculi, etc. And for non-applicability, I can imagine non-transitive logics cannot be turned into algebras based on lattices.