I've been intending to write a post or two on truth values and modal operators, but I am currently rather more busy than I expected to be, with grading, revising, and other things. So instead here are some ideas about modal operators and truth values, without much comment, that I've been toying with recently.
1. There is no fundamental difference between modal operators and truth values. You can have a logic with no truth values, using True and False as modal operators, and you can have a modal logic with no modal operators, only an expanded range of truth values. That is, you can have your truth values as purely operational, or you modal operators as part of your truth table. There is no significant underlying difference. You can go back and forth, if you want; this, in a sense, is what we do, since we treat T and F as truth values and other modalities as operators.
2. Negation can be treated as a modal operator not distinct from False.
3. Connectives are not constant between different modal systems; the only part of their truth tables that is invariant across such systems is the purely alethic part, i.e., the part using True and False.
4. Truth values admit of being multiplied (conjoined) or summed (disjoined). Thus, you can have not only T and F as truth values, even in a purely alethic table, but also TT, TF, FT, FF, TTTT, etc. In classical logic, any string of T's only is equivalent to T; any string of F's or F's and T's where there is an even number of F's is equivalent to T; any string of F's or F's and T's where there is an odd number of F's is equivalent to F. T+T is equivalent to T, F+F is equivalent to F, F+T trivially applies to everything. There is no particular need to have a classical truth table, though. You can deny that FF is equivalent to T, for instance, and that is like rejecting double negation; you can deny that F+T trivially applies to everything, and that is like rejecting excluded middle. You can deny commutativity (i.e., that TF is equivalent to FT) and that gets you something else again. There are modal logics where we in fact do things analogous to any and all of these. You could also reject associativity; I'm not sure if I've ever come across a modal logic that does this. You can do stranger things with truth tables than anyone seems to have done before.
5. Truth tables need not be deterministic. That is, one could have a deterministic truth table in which, for instance, (Tp & Tq) always has the truth value T (by standard conjunction rules). But you could also have a truth table in which the truth value of (Tp & Tq) is T or ◊. This 'or' is naturally interpreted as additive conjunction: that is, you can have T, or you can have ◊, as you deem fit. In fact, most of the modal systems we use make use of nondeterministic truth tables in this way. It is possible that we can think of truth values as resources, and describe them by means of linear logic; for instance, you can have a logic where Necessary linearly implies True which linearly implies Possible. If that's genuinely possible, that would, I think, be a more interesting example of the value of linear logic than the usual vending machine examples.
6. You could focus on truth values almost entirely, treating propositions as simply ways of indexing truth values.
7. Of course, I've put it all in terms of truth tables, but, again, with regard to (1), you could instead do it all in terms of modal operators and the rules governing them. For some things that we might like to do in either case, you would need to have variables into which we could plug modal operators. For instance, for a given proposition p you might not know its truth value; is it necessary, true, possible, false, etc.? So we can use a variable. Likewise, we can, with the right information, solve for such variables in exactly the way you might solve for any variable.
8. If you can do any of this, there seems no good reason to deny that you can quantify over truth values / modal operators. And why wouldn't you want to play that game! But it would seem to be tricky business.
