The systems of the λ-cube have the axiom $\star:\square$.


I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and $\square$ in each interpretation?


$t : T : \star : \square$


**Programs:** t is a program of type T. (Possibility: T is a program of type $\star$?)


**Proofs:** t is a proof of theorem T. It's hard to see T as a proof of $\star$, though.


**Set elements:** t is a member of set T. (Possibility: T is a member of the universe $\star$ of sets. Then it seems difficult to assign a meaning to $\square$ that avoids the membership $\square : \star$.)


I'd like to fill out this table both vertically and horizontally, with both further interpretations and the missing descriptions of $\star$ and $\square$, and possibly meanings of $T : \square$ for $T \neq \star$.


Thank you!



What is the intuitive meaning of star and box in a pure type system?
$\star$ is a *kind*, which classifies *types*. $\square$ is a sort, and it classifies *kinds*. So this is a 4-layer deep classification. Once you get to have type-constructors, kinds get really useful. Eventually, you wish for kind-constructors too, and then you need sorts.


Turns out that you really rarely ever need to get deeper than that (even though Coq and Agda have infinitely many such levels). I am not sure I have ever read a good Curry-Howard explanation of kinds and sorts. I would hazard a guess that classical mathematics rarely worries about kinds/sorts, I would tend to dig into $n$-categories to find a good relation.


