Resources, references, or examples for logics with finitely many sentences 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.

All automated (computer based) theorem provers involve some logic with a finite number of sentences.
I would consider some of those automated theorem provers of interest, since they have gotten used to solve open problems in mathematics and logic, such as when the Robbins problem got solved.  Some others might not have solved any open problem, but have the ability to solve open problems with human assistance (or one might think of the computer as the human's assistant).  Also, model finders with only a finite number of possible sentences, I would think of interest, since they save people time, as computers work faster than humans in some situations.
