The Adjunction $\_\times A\dashv (\_ )^A$ for Preorders: The Deduction Theorem.

Question

The following is from Turi's Category Theory Lecture Notes.

Definition 11.11 Let $A$ be an object of a category $\mathbb{C}$ with binary products. The right adjoint of $\_\times A:\mathbb{C}\to\mathbb{C}$, if it exists, is denoted by $$(\_ )^A:\mathbb{C}\to\mathbb{C}$$ and is called the exponential functor[.]

[. . .]

One way to denote exponentials $X^A$ is as $A\Rightarrow X$. This stems from a logical reading of the adjunction. Indeed, in a preorder $P$ with meets $\wedge$, if we interpret $\le$ as logical entailment $\vdash\,\,\,$ [. . .] and $\wedge$ as conjunction, then the above adjunction is nothing but the well-known deduction theorem: $$\frac{a\wedge b\vdash c}{a\vdash (b\Rightarrow c)}.$$

I would Love to find out more about this and similar phenomena. However, what with a plethora of Logic articles and the like on the Deduction Theorem, I'm not having much luck searching for more information. Help me out, please :)

I have a few pdfs on Categorical Logic I've run some searches through but - besides some cool stuff about the "state monad induced by the adjunction $(\cdot )\times S\dashv (\cdot )^S:\text{Set}\to\text{Set}$" - I got nothing that resembles the above.

Answer 1

You can have a look at the end of Chapter I of Sheaves in Geometry and Logic by MacLane and Moerdijk. If you're interested in categorical fallouts in logic, it is worth reading further this book (which is basically an introduction to topos theory).

Edit. I also once read some lecture notes of Steve Awodey which covers quantifiers as adjunction : chapter 9, section 9.4 and 9.5.

Answer 2

It seems that I was right in thinking that this is simply categorical logic; particularly, categorical proof theory. These notes look ideal.

[Please do correct me if I'm wrong. Further information is welcome of course.]