I know that the monoidal category $(\text{Set},\times)$ doesn't provide duals. Is it meaningful in any way to ask how duals would look like if we forced them into existence there? Is there a canonical way to equip a cartesian closed category with duals?
Would it make sense to assert that in such a category we would have rational cardinalities? (Somehow this feels so natural to me.)
I assume that what you mean by duals is, for each object $A$, a dual $A^\bot$ such that $\text{Hom}(A,B)\cong A^\bot\times B$ (natural in $B$). If you adjoined such duals to the category of sets, then for every set $A$ you'd have $1\cong \text{Hom}(A,1)\cong A^\bot\times 1$. If $1$ is to remain the monoidal unit, this gives $1\cong A^\bot$ for all $A$. That is, all the newly adjoined dual-sets would have to be isomorphic to singletons. as a result, the required equation $\text{Hom}(A,B)\cong A^\bot\times B$ will fail most of the time.