Are you talking about base suppositions about truth tables or truth tables in themselves. What I mean is, are the classes you list (i.e. tautology, etc.) master categories of the established rules (i.e. conjunction, disjunction, etc.)

like instead of taking the basic truth tables as is, you suppose there is a grander scale. so instead of saying;

* conjunction
* disjunction
* conditional
* biconditional

you would suppose that these would come after the classifications you list (i.e tautology, etc.) so it would looks something like this (the x means one of the classifications you suppose)

* (x) -- > conjunction
* (x) -- > disjunction
* (x) -- > conditional
* (x) -- > biconditional

More elaboration please!

Well, grander might not be the right word for it. Allow me to show what I'm referring to:
Definition:
A formula A is a contingency if and only if $(a)=0 and $(a)=1.




I was wondering if there are more that I know nothing about.

There are logics which take certain chains as axioms or patently valid by the axioms, but it is not really pertinent to logic to root through the infinite possible predicates to try to create groups of patently valid, invalid and contingent lines, since you quickly end up getting diminishing returns. Those predicates which turn up most often are generally considered patently valid or invalid or contingent and they generally suffice under ordinary cirumstances. In some very rare cases where one must find the truth table of a very laborious predicate sentence, one just has to crank through using the atoms and identifying patently valid or invalid pieces to shorten up the process.

I think the sort of project that you are proposing would be more suited to formal ontology.

What project am I proposing?