Going off the definition of logical validity that depends on true premises going to true conclusions, there is no imperative logic. However, it seems that some "imperative arguments" look regular enough:
- Go to the store.
If you go to the store, get milk.
C. Therefore, get milk.
As far as I know, the mainstream(?) formal way of representing imperatives in logic is by underlining an assertive sentence, the underline being elliptical for a "Make it the case that" operator. This allows us to stay in proposition-land while dealing with imperatives. However...
Let's define the basic operator for the word, "Do," as in, "Do this." Let's have the symbol be ±, so ±A means "Do A." Then ±(¬A) goes to, "Do not A" = -A. So we have ±A and -A. In natural language we also have expressions like, "Do be nice," which fact allows us to have any imperative represented with the ±/- operators.
This might seem uninformative, and it pretty much by itself is, but I did notice that expressions like, "Don't do that," give rise to mathematical (if simple, nothing fancy like algebra or calculus I guess?) equivalences, e.g. if you preface A by an odd number of minuses, you end up with, "Don't," and so even numbers of minuses cancel out. By implication, an infinite sequence of minuses = ±A, and an infinite alternating sequence of ±/-(A) = ±A.
Now, again, imperatives aren't truth-apt. However, there is a transyntactic unity of assertoric and prescriptive logic, namely via the erotetic function as in, "Why do this?" "J is why do this," or, "J is why," are truth-apt (at least, they've got a subject and a predicate standing in the assertoric relation). So, erotetic logic unites assertoric and prescriptive functionality as such. Anyway, we'll let this fact be signed J:±A as, "J is why do A."
With this in mind, let's correlate simple logical forms of imperatives with deontic operators:
- J:±A = A is obligatory (there is a reason to A).
- J:-A = A is forbidden (there is a reason not to A).
- ¬J:±A & ¬J:-A (there is no reason to do A and no reason to not do A) = A is indifferently optional.
- J:±(A v ¬A) (there is a reason to do (A or not A)) = A is differently optional.*
*[This is a distinction I picked up from McNamara's article on deontic logic in the SEP.]
... So, with these formulas in mind, I was trying to come up with one for supererogation, and came up with:
- J:±(A v (A & B) v (A & B & C) v ... v (A & ... & N))
In this case, A is "the least that can be done" (maybe, anyway) and the entire sequence (where B, C, etc. are not negations of A) abstracts over possible supererogatory value.
So, this would be why there is no suberogation operator: the parallel/inverse construction for J-(A ... N) doesn't "compute." "Don't do A or don't do (A and B)," always rules out A, and ruling out A here means ruling out all cases conjoined to A (I think). It's not like we're ruling out B by itself; B is not "suberogatory."
I wish I remembered how I put it long ago, because it was much better put. I also wish I knew if this has all already been said/argued/w/e before. So, fellow Sophiaphiles, (second) wish granted?