<u>int</u> -> <u>Q</u>
-<<<<<<< HEAD
-We'll need a number of classes of functions to help us maneuver in the
-presence of box types. We will want to define a different instance of
-each of these for whichever box type we're dealing with. (This will
-become clear shortly.)
-=======
As semanticists, you are no doubt familiar with the debates between those who insist that propositions are sets of worlds and those who insist they are context change potentials. We hope to show you, in coming weeks, that propositions are (certain sorts of) Kleisli arrows. But this doesn't really compete with the other proposals; it is a generalization of them. Both of the other proposed structures can be construed as specific Kleisli arrows.
## A family of functions for each box type ##
We'll need a family of functions to help us work with box types. As will become clear, these have to be defined differently for each box type.
->>>>>>> ecff6bbae7c00556584b51913b934bdade0cff40
Here are the types of our crucial functions, together with our pronunciation, and some other names the functions go by. (Usually the type doesn't fix their behavior, which will be discussed below.)
<code>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
-<code>join: <span style="border-bottom: 3px double;">P</span> -> <u>P</u></code>
-
-
-Test1: <span style="border-bottom: 3px double"><d>P</d></span>
-
-Test2: <span style="text-decoration: overline"><u>P</u></span>
-
-Test3: <span style="font-style: italic;">XX</span>
-
-Test4: <span class="double">YY</span>
+<code>join: <span class="double">P</span> -> <u>P</u></code>
The menagerie isn't quite as bewildering as you might suppose. Many of these will
if there is a `map` function defined for that box type with the type given above. This
has to obey the following Map Laws:
- LAWS
+ TODO LAWS
* ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
if there are in addition `map2`, `mid`, and `mapply`. (Given either
of `map2` and `mapply`, you can define the other, and also `map`.
Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
-have to obey the following MapN Laws:
+ have to obey the following MapN Laws:
+
+ TODO LAWS
* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
-Here are some interdefinitions: TODO. Names in Haskell TODO.
+Here are some interdefinitions: TODO
+
+Names in Haskell: TODO
+
+The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
## Examples ##
To take a trivial (but, as we will see, still useful) example,
-consider the identity box type Id: `α`. So if `α` is type `bool`,
+consider the Identity box type: `α`. So if `α` is type `bool`,
then a boxed `α` is ... a `bool`. In terms of the box analogy, the
Identity box type is a completely invisible box. With the following
-definitions
+definitions:
mid ≡ \p. p
mcomp ≡ \f g x.f (g x)
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
-Id is the favorite monad of mimes.
+The Identity Monad is favored by mimes.
To take a slightly less trivial (and even more useful) example,
consider the box type `α list`, with the following operations:
= foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
= [c | b <- g a, c <- f b]
-These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises).
+The last three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises TODO).
In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
each `β` in that list is fed to `f`, which returns a list of `γ`s. The
# safe_div 12 0;;
- : int option = None
# safe_div (safe_div 12 2) 3;;
-# safe_div (safe_div 12 2) 3;;
~~~~~~~~~~~~~
Error: This expression has type int option
but an expression was expected of type int
*)
</pre>
-This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
+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:
<pre>
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));;
+ | None -> None
+ | Some x ->
+ (match v with
+ | Some 0 -> None
+ | Some y -> Some (x / y));;
(*
val safe_div2 : int option -> int option -> int option = <fun>
*)
</pre>
-Beautiful, just what we need: now we can try to divide by anything we
-want, without fear that we're going to trigger any system errors.
+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:
<pre>
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);;
+ | (None, _) -> None
+ | (_, None) -> None
+ | (_, Some 0) -> None
+ | (Some x, Some y) -> Some (x / y);;
</pre>
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 triggered a
+aware of the possibility that one of their arguments has already triggered a
presupposition failure:
<pre>
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. The standard way in OCaml,
-Haskell, and other functional programming languages, is to use the monadic
-`bind` operator, `>>=`. (The name "bind" is not well chosen from our
-perspective, but this is too deeply entrenched by now.) As mentioned above,
-there needs to be a different `>>=` operator for each Monad or box type you're working with.
+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 `>>=` or `bind` operator is not pre-defined, but we will
+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 `bind` operation by hand:
+For now, though, we will define our `>>=` and `map2` operations by hand:
<pre>
-let bind (u: int option) (f: int -> (int option)) =
+let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
match u with
- | None -> None
- | Some x -> f x;;
+ | None -> None
+ | Some x -> j x;;
-let safe_add3 (u: int option) (v: int option) =
- bind u (fun x -> bind v (fun y -> Some (x + y)));;
+let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
+ u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
-(* This is really just `map2 (+)`, using the `map2` operation that corresponds to
- definition of `bind`. *)
+let safe_add3 = map2 (+);; (* that was easy *)
let safe_div3 (u: int option) (v: int option) =
- bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;
-
-(* This goes back to some of the simplicity of the original safe_div, without the complexity
- introduced by safe_div2. *)
+ u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
</pre>
-The above definitions look even simpler if you focus on the fact that `safe_add3` can be written as simply `map2 (+)`, and that `safe_div3` could be written as `u >>= fun x -> v >>= fun y -> if 0 = y then None else Some (x / y)`. Haskell has an even more user-friendly notation for this, namely:
+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 return (x `div` y)}
+ if 0 == y then Nothing else Just (x `div` y)}
Let's see our new functions in action:
None values or real numbers---those details are hidden inside of the
`bind` function.
-The definition of `safe_div3` shows exactly what extra needs to be said in
-order to trigger the no-division-by-zero presupposition. Here, too, we don't
-need to keep track of what presuppositions may have already failed
+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