I have a category theory homework problem which asks: "In first-order logic, why does $\forall$ not have a right adjoint?" The typical argument is that:
- If for some operator $\cdot$ it could be asserted $\forall \dashv \cdot$, then $\cdot$ would be a kind of coproduct.
- If $\forall$ preserves coproducts, then $\forall y(\phi y \vee \psi y) \dashv \vdash \forall y \phi y \vee \forall y \psi y$.
- Since 2 does not hold, $\forall$ must not preserve coproducts, and therefore $\forall$ does not have any kind of right adjoint.
I have a few questions about this:
- Why exactly does 2 not hold? Is it because the expression is equivalent to $\forall y(\phi y \vee \psi y) \dashv \vdash \forall y_1 \phi y_1 \vee \forall y_2 \psi y_2$?
- The meaning of $\dashv$ and $\vdash$ are unclear to me in the context of expressions. I understand what $* \dashv \forall$ means, but I need help understanding adjointness in terms of expressions.
- How can the relation of $\forall$ to the particular coproduct $\vee$ be generalized to all coproducts, such that all possible right adjoints to $\forall$ are ruled out?