# Are there useful logics that reject disjunction introduction?

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
``````
• It seems to me that disjunction introduction does hold according to your truth-table, since whenever a is true or b is true, you have Aab true. But because your truth-table for the conditional is not standard, this isn't reflected as a tautology expressible with the conditional. – Ben Dec 12 '18 at 6:57
• @Ben ... `U` is a designated truth value. If I take `a` as a premise, then it can either have the truth value `T` or `U`. `a=T` would allow me to conclude that `Aab`, but `a=U` would not. In this section of the article on three-valued logic: en.wikipedia.org/wiki/… , LP and Kleene's logic have the same truth tables but different designated truth values. – Gregory Nisbet Dec 12 '18 at 7:10
• I'm sorry I assumed U was undesignated. That makes your truth-table for conjunction the same as Priest's in LP, but the disjunction table different - and it does indeed invalidate disjunction introduction (but not I think disjunctive syllogism?). – Ben Dec 12 '18 at 7:15
• @Ben ... Yes, I think disjunctive syllogism still works. If `Aab` is `T/U` but `b` is false, `a` has to be `T`. Interestingly `b` being `U` is also enough to eliminate the disjunction. – Gregory Nisbet Dec 12 '18 at 7:28