let safe_div (x:int) (y:int) = match y with | 0 -> None | _ -> Some (x / y);; (* an Ocaml session could continue with OCaml's response: val safe_div : int -> int -> int option = fun # safe_div 12 2;; - : int option = Some 6 # safe_div 12 0;; - : int option = None # safe_div (safe_div 12 2) 3;; ~~~~~~~~~~~~~ Error: This expression has type int option but an expression was expected of type int *)This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`, just the behavior we were hoping for. But we want to be able to use the output of the safe-division function as input for further division operations. So we have to jack up the types of the inputs:

let safe_div2 (u:int option) (v:int option) = match u with | None -> None | Some x -> (match v with | Some 0 -> None | Some y -> Some (x / y));; (* an Ocaml session could continue with OCaml's response: val safe_div2 : int option -> int option -> int option =Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we want, without fear that we're going to trigger system errors. I prefer to line up the `match` alternatives by using OCaml's built-in tuple type:# safe_div2 (Some 12) (Some 2);; - : int option = Some 6 # safe_div2 (Some 12) (Some 0);; - : int option = None # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);; - : int option = None *)

let safe_div2 (u:int option) (v:int option) = match (u, v) with | (None, _) -> None | (_, None) -> None | (_, Some 0) -> None | (Some x, Some y) -> Some (x / y);;So far so good. But what if we want to combine division with other arithmetic operations? We need to make those other operations aware of the possibility that one of their arguments has already triggered a presupposition failure:

let safe_add (u:int option) (v:int option) = match (u, v) with | (None, _) -> None | (_, None) -> None | (Some x, Some y) -> Some (x + y);; (* an Ocaml session could continue with OCaml's response: val safe_add : int option -> int option -> int option =This works, but is somewhat disappointing: the `safe_add` operation doesn't trigger any presupposition of its own, so it is a shame that it needs to be adjusted because someone else might make trouble. But we can automate the adjustment, using the monadic machinery we introduced above. As we said, there needs to be different `>>=`, `map2` and so on operations for each monad or box type we're working with. Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that symbol and it will calculate from the context of the surrounding type constraints what monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will give you a library that has definitions for all the standard monads, as in Haskell. For now, though, we will define our `>>=` and `map2` operations by hand:# safe_add (Some 12) (Some 4);; - : int option = Some 16 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);; - : int option = None *)

let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option = match u with | None -> None | Some x -> j x;; let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option = u >>= (fun x -> v >>= (fun y -> Some (f x y)));; let safe_add3 = map2 (+);; (* that was easy *) let safe_div3 (u: int option) (v: int option) = u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;Haskell has an even more user-friendly notation for defining `safe_div3`, namely: safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int safe_div3 u v = do {x <- u; y <- v; if 0 == y then Nothing else Just (x `div` y)} Let's see our new functions in action:

(* an Ocaml session could continue with OCaml's response: # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);; - : int option = Some 2 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);; - : int option = None # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);; - : int option = None *)Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition for `safe_add3` shows what it looks like to equip an ordinary operation to survive in dangerous presupposition-filled world. Note that the new definition of `safe_add3` does not need to test whether its arguments are `None` values or real numbers---those details are hidden inside of the `bind` function. Note also that our definition of `safe_div3` recovers some of the simplicity of the original `safe_div`, without the complexity introduced by `safe_div2`. We now add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't need to keep track of what other presuppositions may have already failed for whatever reason on our inputs. (Linguistics note: Dividing by zero is supposed to feel like a kind of presupposition failure. If we wanted to adapt this approach to building a simple account of presupposition projection, we would have to do several things. First, we would have to make use of the polymorphism of the `option` type. In the arithmetic example, we only made use of `int option`s, but when we're composing natural language expression meanings, we'll need to use types like `N option`, `Det option`, `VP option`, and so on. But that works automatically, because we can use any type for the `'a` in `'a option`. Ultimately, we'd want to have a theory of accommodation, and a theory of the situations in which material within the sentence can satisfy presuppositions for other material that otherwise would trigger a presupposition violation; but, not surprisingly, these refinements will require some more sophisticated techniques than the super-simple Option/Maybe monad.) ## Scratch, more... ## We've just seen a way to separate thinking about error conditions (such as trying to divide by zero) from thinking about normal arithmetic computations. We did this by making use of the `option` type: in each place where we had something of type `int`, we put instead something of type `int option`, which is a sum type consisting either of one choice with an `int` payload, or else a `None` choice which we interpret as signaling that something has gone wrong. The goal was to make normal computing as convenient as possible: when we're adding or multiplying, we don't have to worry about generating any new errors, so we would rather not think about the difference between `int`s and `int option`s.