Was wondering if there was a theory already out there that considers the "category $\text{Prop}$ of propositions". It is a preorder (at most one arrow between two propositions), in which $A \to B$ means that if the proposition $A$ is proved, assumed, or defined true then $B$ is a (similarly) true proposition. Arrows compose associatively when the objects take on positive truth values. And so a category is formed.

Note that the category contains not just true propositions but *any* proposition. An arrow will connect two possibly false propositions when assuming one would prove the other true. I guess this would come in handy when doing a long proof-by-contradiction.

**Question.** Was wondering what theory most closely mimics this situation, since usally in proof assistants $A \implies B \implies C$ is read right-associatively: $A \implies (B \implies C)$. Or in other words we can't speak of a category where $\implies$ is data of an arrow (since associativity of composition is dropped for (mostly) philosophical reasons).

In $\text{Prop}$, given two propositions $A, B$, we have that ($A$ and $B$) is their categorical product and that similarly ($A$ or $B$) is their categorical coproduct. Thus this category has finite products and coproducts.

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