Are there useful logics that reject disjunction introduction?

Question

Are there logics with fairly natural semantics that don't have disjunction introduction? Is there a good way of thinking about what "flavor of system" you're in when you don't take disjunction introduction? Systems without double negation elimination seem to have a "constructive" feel to them. Is there a nice "family resemblance" of sorts between logical systems lacking disjunction introduction?

Here's a somewhat artificial example of a three-valued logic constructed not to admit disjunction introduction while still having fairly symmetrical truth tables and agreeing with classical logic when the intermediate truth value U is ignored.

Here's a three-valued logic that doesn't admit disjunction introduction, and has the connectives K (conjunction/koniunkcja), A (disjunction/alternatywa), and C (implication/implikacja). We pick both T and U as our designated truth-y truth values.

Kab      b        Aab       b             Cab    b
      F  U  T            F  U  T              F  U  T
  (F) F  F  F        (F) F  F  T          (F) T  T  T 
a (U) F  U  U      a (U) F  F  T        a (U) F  T  T
  (T) F  U  T        (T) T  T  T          (T) F  T  T

So, we can see that disjunction introduction doesn't hold by demonstrating that CaAab is a non-tautology. Let's assign both a and b the intermediate truth value U.

CUAUU
CUF
F

Answer 1

In some paraconsistent and relevant logics, there's a distinction between intensional and extensional disjunction. Disjunction introduction fails in those logics for intensional disjunction.

The motivation is that these logics reject the principle according to which A, ~ A entails an arbitrary sentence B. The argument for this principle involves assuming A, using disjunction introduction to get A v B, then using disjunctive syllogism with ~A to get B.

One way to resist this argument is to reject disjunctive syllogism (this is the course most relevant and paraconsistent logics take). But another way is to reject disjunction introduction instead - this is the way that gets you "intensional disjunction".

There is a brief discussion of this in the following section of the SEP article on disjunction: https://plato.stanford.edu/entries/disjunction/#DisjSyllAddi

There are probably helpful points in the rest of that entry too.