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 | None -> None | 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 =So now, wherever before our operations expected an `int`, we'll instead have them accept an `int option`. A `None` input signals that something has gone wrong upstream. 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 before. 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. But you will need to explicitly specify which Monad you mean to be deploying. 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)} You can read more about that here: * [Haskell wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation) * [Yet Another Haskell Tutorial on do-notation](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Do_Notation) 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 *)Our definition for `safe_add3` shows what it looks like to equip an ordinary operation to survive in dangerous presupposition-filled world. We just need to `mapN` it "into" the Maybe monad, where `N` is the function's adicity. Note that the new definition of `safe_add3` does not need to test whether its arguments are `None` values or genuine 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. So what the monadic machinery gives us here is a way to _separate_ thinking about error conditions (such as trying to divide by zero) from thinking about normal arithmetic computations. When we're adding or multiplying, we don't have to worry about generating any new errors, so we would rather not force these operations to explicitly track the difference between `int`s and `int option`s. A linguistics analogy we'll look at soon is that when we're giving the lexical entry for an ordinary extensional verb, we'd rather not be forced to talk about possible worlds. In each case, we instead just have a standard way of "lifting" (`mapN`ing) the relevant notions into the fancier type environment we ultimately want to work in. 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. To illustrate some of the polymorphism, here's how we could `map1` the `is_even` function: # let is_even x = (x mod 2 = 0);; val is_even : int -> bool =