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
A (disjunction/alternatywa), and
C (implication/implikacja). We pick both
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
b the intermediate truth value
CUAUU CUF F