Are there any interesting logics with only finitely many sentences total? Is there a reason why potential logics with this property would be trivial or badly behaved?
I'm looking for a reference to read more or an example of such a logic that's interesting as a mathematical object.
I was recently looking at some examples of interesting logics that are fragments of better-known logics, such as two-variable logic and a list of intermediate logics that are interesting logics between intuitionistic propositional logic and classical propositional logic.
One thing that I haven't come across yet, though, is a modern logic (possibly described as a fragment of another logic) with structural constraints on well-formed formulas that are so strong that only finitely many sentences are possible or only finitely many sentences up to renaming of variables are possible. Traditional term logic, even with the addition of non for modifying predicates, has finitely many sentences up to renaming of variables.
I can think of a few weird properties of such finite-number-of-sentences logics off the top of my head like the fact that ordinary introduction rules like $A, B \vdash A \land B$ will sometimes fail if the conclusion is too big to be well-formed.
NOTE: removed reference to $\alpha$-equivalence, replaced with up to renaming of variables which is more clear.