1 <!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
2 <!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
7 The [[tradition in the functional programming
8 literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
9 introduce monads using a metaphor: monads are spacesuits, monads are
10 monsters, monads are burritos. These metaphors can be helpful, and they
11 can be unhelpful. There's a backlash about the metaphors that tells people
12 to instead just look at the formal definition. We'll give that to you below, but it's
13 sometimes sloganized as
14 [A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
15 Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
18 The closest we will come to metaphorical talk is to suggest that
19 monadic types place values inside of *boxes*, and that monads wrap
20 and unwrap boxes to expose or enclose the values inside of them. In
21 any case, our emphasis will be on starting with the abstract structure
22 of monads, followed by instances of monads from the philosophical and
23 linguistics literature.
25 > <small>After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in category theory:
26 [1](http://en.wikipedia.org/wiki/Outline_of_category_theory)
27 [2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/)
28 [3](http://en.wikibooks.org/wiki/Haskell/Category_theory)
29 [4](https://wiki.haskell.org/Category_theory), where you should follow the further links discussing Functors, Natural Transformations, and Monads.</small>
32 ## Box types: type expressions with one free type variable ##
34 Recall that we've been using lower-case Greek letters
35 <code>α, β, γ, ...</code> as type variables. We'll
36 use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
37 type variables. For instance, we might have
46 A *box type* will be a type expression that contains exactly one free
47 type variable. (You could extend this to expressions with more free variables; then you'd have
48 to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
52 (α, R) tree (assuming R contains no free type variables)
55 The idea is that whatever type the free type variable `α` might be instantiated to,
56 we will have a "type box" of a certain sort that "contains" values of type `α`. For instance,
57 if `α list` is our box type, and `α` is the type `int`, then in this context, `int list`
58 is the type of a boxed integer.
60 Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `α`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> α` is a box type.
62 Also, for clarity: the *box type* is the type `α list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `α`. We'll often write boxed types as a box containing what the free
63 type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write:
65 <code><u>int</u></code>
67 for the type of a boxed `int`.
73 A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Don't worry about why they're called that, or if you like go read some Category Theory. All we need to know is that these are functions whose type has the form:
75 <code>P -> <u>Q</u></code>
77 That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`.
78 For instance, the following are Kleisli arrows:
80 <code>int -> <u>bool</u></code>
82 <code>int list -> <u>int list</u></code>
84 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
86 Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where if `α list` is our box type, we can write the second type as:
88 <code><u>int</u> -> <u>int list</u></code>
90 Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
94 \x. prime_factors_of x
97 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 arrow types.
100 ## A family of functions for each box type ##
102 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.
104 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.)
106 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
108 > In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
110 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
112 > In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
114 <code>mid (/εmaidεnt@tI/): P -> <u>P</u></code>
116 > In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `𝟭`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`.
118 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
120 > We'll use `m$` as an infix operator, reminiscent of `$` which is just ordinary function application (also expressed by mere juxtaposition). In the class presentation Jim called `m$` `●`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`.
122 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
124 > In Haskell, this is `Control.Monad.<=<`.
126 <code>>=> (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
128 > In Haskell, this is `Control.Monad.>=>`. In the class handout, we gave the types for `>=>` twice, and once was correct but the other was a typo. The above is the correct typing.
130 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
132 <code>=<< (flip mbind, should we call it mdnib?) (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
134 <code>join: <span class="box2">P</span> -> <u>P</u></code>
136 > In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
138 Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, but it's too deeply entrenched to change. We've at least preprended an `m` to the front of "bind".
140 Haskell's names "return" and "pure" for `mid` are even less well chosen, and we think it will be clearer in our discussion to use a different name. (Also, in other theoretical contexts this notion goes by other names, anyway, like `unit` or `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.)
142 The menagerie isn't quite as bewildering as you might suppose. Many of these will be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡ \a. (j a >>= k)</code>. We'll state some other interdefinitions below.
144 We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
145 is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more
146 closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
147 that its types flow more naturally from left to right.
149 These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
151 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
152 if there is a `map` function defined for that box type with the type given above. This
153 has to obey the following Map Laws:
155 <code>map (id : α -> α) == (id : <u>α</u> -> <u>α</u>)</code>
156 <code>map (g ○ f) == (map g) ○ (map f)</code>
158 Essentially these say that `map` is a homomorphism from the algebra of `(universe α -> β, operation ○, elsment id)` to that of <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>. That might be hard to digest because it's so abstract. Think of the following concrete example: if you take a `α list` (that's our <code><u>α</u></code>), and apply `id` to each of its elements, that's the same as applying `id` to the list itself. That's the first law. And if you apply the composition of functions `g ○ f` to each of the list's elements, that's the same as first applying `f` to each of the elements, and then going through the elements of the resulting list and applying `g` to each of those elements. That's the second law. These laws obviously hold for our familiar notion of `map` in relation to lists.
160 > <small>As mentioned at the top of the page, in Category Theory presentations of monads they usually talk about "endofunctors", which are mappings from a Category to itself. In the uses they make of this notion, the endofunctors combine the role of a box type <code><u>_</u></code> and of the `map` that goes together with it.</small>
163 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
164 if there are in addition `map2`, `mid`, and `mapply`. (Given either
165 of `map2` and `mapply`, you can define the other, and also `map`.
166 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
167 have to obey the following MapN Laws:
169 1. <code>mid (id : P->P) : <u>P</u> -> <u>P</u></code> is a left identity for `m$`, that is: `(mid id) m$ xs = xs`
170 2. `mid (f a) = (mid f) m$ (mid a)`
171 3. The `map2`ing of composition onto boxes `fs` and `gs` of functions, when `m$`'d to a box `xs` of arguments == the `m$`ing of `fs` to the `m$`ing of `gs` to xs: `((mid ○) m$ fs m$ gs) m$ xs = fs m$ (gs m$ xs)`.
172 4. When the arguments are `mid`'d, the order of `m$`ing doesn't matter: `fs m$ (mid x) = (mid ($ x)) m$ fs`. In examples we'll be working with at first, order _never_ matters; but down the road, sometimes it will. This Law states a class of cases where it's guaranteed not to.
173 5. A consequence of the laws already stated is that when the functions are `mid`'d, the order of `m$`ing doesn't matter either: TODO
176 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
177 is in addition an associative `mcomp` having `mid` as its left and
178 right identity. That is, the following Monad Laws must hold:
180 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) == mcomp j (mcomp k l)
181 mcomp mid k (that is, mid <=< k) == k
182 mcomp k mid (that is, k <=< mid) == k
184 You could just as well express the Monad laws using `>=>`:
186 l >=> (k >=> j) == (l >=> k) >=> j
190 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others. Also, with these functions you can define `m$` and `map2` from *MapNables*. So with Monads, all you really need to get the whole system of functions are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
192 In practice, you will often work with `>>=`. In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators. This looks similar, but doesn't have the same symmetry:
194 u >>= (\a -> k a >>= j) == (u >>= k) >>= j
198 Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.
200 > <small>In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `μ`) and `mid` (which they call `η`). These are assumed to be "natural transformations" for their box type, which means that they satisfy these equations with that box type's `map`:
201 > <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
202 > The Monad Laws then take the form:
203 > <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>
204 > The first of these says that if you have a triply-boxed type, and you first merge the inner two boxes (with `map join`), and then merge the resulting box with the outermost box, that's the same as if you had first merged the outer two boxes, and then merged the resulting box with the innermost box. The second law says that if you take a box type and wrap a second box around it (with `mid`) and then merge them, that's the same as if you had done nothing, or if you had instead wrapped a second box around each element of the original (with `map mid`, leaving the original box on the outside), and then merged them.<p>
205 > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
206 > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre></small>
209 As hinted in last week's homework and explained in class, the operations available in a Mappable system exactly preserve the "structure" of the boxed type they're operating on, and moreover are only sensitive to what content is in the corresponding original position. If you say `map f [1,2,3]`, then what ends up in the first position of the result depends only on how `f` and `1` combine.
211 For MapNable operations, on the other hand, the structure of the result may instead by a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say `map2 f [10,20] [1,2,3]`, what ends up in the first position of the result depends only on how `f` and `10` and `1` combine.
213 With `map`, you can supply an `f` such that `map f [3,2,0,1] == [[3,3,3],[2,2],[],[1]]`. But you can't transform `[3,2,0,1]` to `[3,3,3,2,2,1]`, and you can't do that with MapNable operations, either. That would involve the structure of the result (here, the length of the list) being sensitive to the content, and not merely the structure, of the original.
215 For Monads (Composables), you can perform more radical transformations of that sort. For example, `join (map (\x. dup x x) [3,2,0,1])` would give us `[3,3,3,2,2,1]` (for a suitable definition of `dup`).
218 Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the primitive Monad operations? For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`.
222 ## Interdefinitions and Subsidiary notions##
224 We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that:
227 f : α -> β; g and h have types of the same form
228 also sometimes these will have types of the form α -> β -> γ
229 note that α and β are permitted to be, but needn't be, boxed types
230 j : α -> <u>β</u>; k and l have types of the same form
231 u : <u>α</u>; v and xs and ys have types of the same form
233 w : <span class="box2">α</span>
236 But we may sometimes slip.
238 Here are some ways the different notions are related:
241 j >=> k ≡= \a. (j a >>= k)
242 u >>= k == (id >=> k) u; or ((\(). u) >=> k) ()
243 u >>= k == join (map k u)
245 map2 f xs ys == xs >>= (\x. ys >>= (\y. mid (f x y)))
246 map2 f xs ys == (map f xs) m$ ys, using m$ as an infix operator
247 fs m$ xs == fs >>= (\f. map f xs)
249 map f xs == mid f m$ xs
250 map f u == u >>= mid ○ f
254 Here are some other monadic notion that you may sometimes encounter:
256 * <code>mzero</code> is a value of type <code><u>α</u></code> that is exemplified by `Nothing` for the box type `Maybe α` and by `[]` for the box type `List α`. It has the behavior that `anything m$ mzero == mzero == mzero m$ anything == mzero >>= anything`. In Haskell, this notion is called `Control.Applicative.empty` or `Control.Monad.mzero`.
258 * Haskell has a notion `>>` definable as `\u v. map (const id) u m$ v`, or as `u >> v == u >>= const v`. This is often useful, and `u >> v` won't in general be identical to just `v`. For example, using the box type `List α`, `[1,2,3] >> [4,5] == [4,5,4,5,4,5]`. But in the special case of `mzero`, it is a consequence of what we said above that `anything >> mzero == mzero`. Haskell also calls `>>` `Control.Applicative.*>`.
260 * Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. map const u m$ v`. For example, `[1,2,3] <* [4,5] == [1,1,2,2,3,3]`. You might expect Haskell to call `<*` `<<`, but they don't. They used to use `<<` for `flip (>>)` instead, but now they seem not to use `<<` anymore.
262 * <code>mapconst</code> is definable as `map ○ const`. For example `mapconst 4 [1,2,3] == [4,4,4]`. Haskell calls `mapconst` `<$` in `Data.Functor` and `Control.Applicative`. They also use `$>` for `flip mapconst`, and `Control.Monad.void` for `mapconst ()`.
268 To take a trivial (but, as we will see, still useful) example,
269 consider the Identity box type: `α`. So if `α` is type `bool`,
270 then a boxed `α` is ... a `bool`. That is, <code><u>α</u> == α</code>.
271 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
275 mcomp ≡ \f g x.f (g x)
277 Identity is a monad. Here is a demonstration that the laws hold:
279 mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
283 mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
287 mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
288 ~~> mcomp (\x.j(kx)) l
289 ≡ (\fgx.f(gx)) (\x.j(kx)) l
290 ~~> \x.(\x.j(kx))(lx)
292 mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
293 ~~> mcomp j (\x.k(lx))
294 ≡ (\fgx.f(gx)) j (\x.k(lx))
295 ~~> \x.j((\x.k(lx)) x)
298 The Identity monad is favored by mimes.
300 To take a slightly less trivial (and even more useful) example,
301 consider the box type `α list`, with the following operations:
306 mcomp : (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
307 mcomp k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
308 = foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
309 = [c | b <- j a, c <- k b]
311 In the first two definitions of `mcomp`, we give the definition first in Haskell and then in the equivalent OCaml. The three different definitions of `mcomp` (one for each line) are all equivalent, and it is easy to show that they obey the Monad Laws. (You will do this in the homework.)
313 In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
314 each `β` in that list is fed to `k`, which returns a list of `γ`s. The
315 final result is the concatenation of those lists of `γ`s.
319 let j a = [a*a, a+a] in
320 let k b = [b, b+1] in
321 mcomp k j 7 ==> [49, 50, 14, 15]
323 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
325 Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
327 let js = [(\a->a*a),(\a->a+a)] in
329 mapply js xs ==> [49, 25, 14, 10]
332 As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to figure out the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the monadic functions of the Option/Maybe box type.
337 Integer division presupposes that its second argument
338 (the divisor) is not zero, upon pain of presupposition failure.
339 Here's what my OCaml interpreter says:
342 Exception: Division_by_zero.
344 Say we want to explicitly allow for the possibility that
345 division will return something other than a number.
346 To do that, we'll use OCaml's `option` type, which works like this:
348 # type 'a option = None | Some of 'a;;
352 - : int option = Some 3
354 So if a division is normal, we return some number, but if the divisor is
355 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
358 let safe_div (x:int) (y:int) =
361 | _ -> Some (x / y);;
364 val safe_div : int -> int -> int option = fun
366 - : int option = Some 6
368 - : int option = None
369 # safe_div (safe_div 12 2) 3;;
371 Error: This expression has type int option
372 but an expression was expected of type int
376 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
377 just the behavior we were hoping for. But we want to be able to use
378 the output of the safe-division function as input for further division
379 operations. So we have to jack up the types of the inputs:
382 let safe_div2 (u:int option) (v:int option) =
388 | Some y -> Some (x / y));;
391 val safe_div2 : int option -> int option -> int option = <fun>
392 # safe_div2 (Some 12) (Some 2);;
393 - : int option = Some 6
394 # safe_div2 (Some 12) (Some 0);;
395 - : int option = None
396 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
397 - : int option = None
401 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
402 want, without fear that we're going to trigger system errors.
404 I prefer to line up the `match` alternatives by using OCaml's
408 let safe_div2 (u:int option) (v:int option) =
412 | (_, Some 0) -> None
413 | (Some x, Some y) -> Some (x / y);;
416 So far so good. But what if we want to combine division with
417 other arithmetic operations? We need to make those other operations
418 aware of the possibility that one of their arguments has already triggered a
419 presupposition failure:
422 let safe_add (u:int option) (v:int option) =
426 | (Some x, Some y) -> Some (x + y);;
429 val safe_add : int option -> int option -> int option = <fun>
430 # safe_add (Some 12) (Some 4);;
431 - : int option = Some 16
432 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
433 - : int option = None
437 This works, but is somewhat disappointing: the `safe_add` operation
438 doesn't trigger any presupposition of its own, so it is a shame that
439 it needs to be adjusted because someone else might make trouble.
441 But we can automate the adjustment, using the monadic machinery we introduced above.
442 As we said, there needs to be different `>>=`, `map2` and so on operations for each
443 monad or box type we're working with.
444 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
445 symbol and it will calculate from the context of the surrounding type constraints what
446 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
447 give you a library that has definitions for all the standard monads, as in Haskell.
448 For now, though, we will define our `>>=` and `map2` operations by hand:
451 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
456 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
457 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
459 let safe_add3 = map2 (+);; (* that was easy *)
461 let safe_div3 (u: int option) (v: int option) =
462 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
465 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
467 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
468 safe_div3 u v = do {x <- u;
470 if 0 == y then Nothing else Just (x `div` y)}
472 Let's see our new functions in action:
476 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
477 - : int option = Some 2
478 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
479 - : int option = None
480 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
481 - : int option = None
485 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
486 for `safe_add3` shows what it looks like to equip an ordinary operation to
487 survive in dangerous presupposition-filled world. Note that the new
488 definition of `safe_add3` does not need to test whether its arguments are
489 `None` values or real numbers---those details are hidden inside of the
492 Note also that our definition of `safe_div3` recovers some of the simplicity of
493 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
494 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
495 need to keep track of what other presuppositions may have already failed
496 for whatever reason on our inputs.
498 (Linguistics note: Dividing by zero is supposed to feel like a kind of
499 presupposition failure. If we wanted to adapt this approach to
500 building a simple account of presupposition projection, we would have
501 to do several things. First, we would have to make use of the
502 polymorphism of the `option` type. In the arithmetic example, we only
503 made use of `int option`s, but when we're composing natural language
504 expression meanings, we'll need to use types like `N option`, `Det option`,
505 `VP option`, and so on. But that works automatically, because we can use
506 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
507 theory of accommodation, and a theory of the situations in which
508 material within the sentence can satisfy presuppositions for other
509 material that otherwise would trigger a presupposition violation; but,
510 not surprisingly, these refinements will require some more
511 sophisticated techniques than the super-simple Option/Maybe monad.)