From a0bd7698972a1d3602bffe49c101360ad57efb71 Mon Sep 17 00:00:00 2001 From: jim Date: Mon, 6 Apr 2015 09:35:35 -0400 Subject: [PATCH] removed --- topics/_week8_intensionality.mdwn | 227 -------------------------------------- 1 file changed, 227 deletions(-) delete mode 100644 topics/_week8_intensionality.mdwn diff --git a/topics/_week8_intensionality.mdwn b/topics/_week8_intensionality.mdwn deleted file mode 100644 index c96d3d3c..00000000 --- a/topics/_week8_intensionality.mdwn +++ /dev/null @@ -1,227 +0,0 @@ -Now we'll look at using monads to do intensional function application. -This is just another application of the Reader monad, not a new monad. -In Shan (2001) [Monads for natural -language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that -making expressions sensitive to the world of evaluation is conceptually -the same thing as making use of the Reader monad. -This technique was beautifully re-invented -by Ben-Avi and Winter (2007) in their paper [A modular -approach to -intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf), -though without explicitly using monads. - -All of the code in the discussion below can be found here: [[code/intensionality-monad.ml]]. -To run it, download the file, start OCaml, and say - - # #use "intensionality-monad.ml";; - -Note the extra `#` attached to the directive `use`. - -First, the familiar linguistic problem: - - Bill left. - Cam left. - Ann believes [Bill left]. - Ann believes [Cam left]. - -We want an analysis on which the first three sentences can be true at -the same time that the last sentence is false. If sentences denoted -simple truth values or booleans, we have a problem: if the sentences -*Bill left* and *Cam left* are both true, they denote the same object, -and Ann's beliefs can't distinguish between them. - -The traditional solution to the problem sketched above is to allow -sentences to denote a function from worlds to truth values, what -Montague called an intension. So if `s` is the type of possible -worlds, we have the following situation: - - -
-Extensional types              Intensional types       Examples
--------------------------------------------------------------------
-
-S         t                    s->t                    John left
-DP        e                    s->e                    John
-VP        e->t                 (s->e)->s->t            left
-Vt        e->e->t              (s->e)->(s->e)->s->t    saw
-Vs        t->e->t              (s->t)->(s->e)->s->t    thought
-
- -This system is modeled on the way Montague arranged his grammar. -There are significant simplifications compared to Montague: for -instance, determiner phrases are thought of here as corresponding to -individuals rather than to generalized quantifiers. - -The main difference between the intensional types and the extensional -types is that in the intensional types, the arguments are functions -from worlds to extensions: intransitive verb phrases like "left" now -take so-called "individual concepts" as arguments (type s->e) rather than plain -individuals (type e), and attitude verbs like "think" now take -propositions (type s->t) rather than truth values (type t). -In addition, the result of each predicate is an intension. -This expresses the fact that the set of people who left in one world -may be different than the set of people who left in a different world. - -Normally, the dependence of the extension of a predicate to the world -of evaluation is hidden inside of an evaluation coordinate, or built -into the the lexical meaning function, but we've made it explicit here -in the way that the intensionality monad makes most natural. - -The intensional types are more complicated than the extensional -types. Wouldn't it be nice to make the complicated types available -for those expressions like attitude verbs that need to worry about -intensions, and keep the rest of the grammar as extensional as -possible? This desire is parallel to our earlier desire to limit the -concern about division by zero to the division function, and let the -other functions, like addition or multiplication, ignore -division-by-zero problems as much as possible. - -So here's what we do: - -In OCaml, we'll use integers to model possible worlds. Characters (characters in the computational sense, i.e., letters like `'a'` and `'b'`, not Kaplanian characters) will model individuals, and OCaml booleans will serve for truth values: - - type s = int;; - type e = char;; - type t = bool;; - - let ann = 'a';; - let bill = 'b';; - let cam = 'c';; - - let left1 (x:e) = true;; - let saw1 (x:e) (y:e) = y < x;; - - left1 ann;; (* true *) - saw1 bill ann;; (* true *) - saw1 ann bill;; (* false *) - -So here's our extensional system: everyone left, including Ann; -and Ann saw Bill (`saw1 bill ann`), but Bill didn't see Ann. (Note that the word -order we're using is VOS, verb-object-subject.) - -Now we add intensions. Because different people leave in different -worlds, the meaning of *leave* must depend on the world in which it is -being evaluated: - - let left (x:e) (w:s) = match (x, w) with ('c', 2) -> false | _ -> true;; - left ann 1;; (* true: Ann left in world 1 *) - left cam 2;; (* false: Cam didn't leave in world 2 *) - -This new definition says that everyone always left, except that -in world 2, Cam didn't leave. - -Note that although this general *left* is sensitive to world of -evaluation, it does not have the fully intensionalized type given in -the chart above, which was `(s->e)->s->t`. This is because -*left* does not exploit the additional resolving power provided by -making the subject an individual concept. In semantics jargon, we say -that *leave* is extensional with respect to its first argument. - -Therefore we will adopt the general strategy of defining predicates -in a way that they take arguments of the lowest type that will allow -us to make all the distinctions the predicate requires. When it comes -time to combine this predicate with monadic arguments, we'll have to -make use of various lifting predicates. - -Likewise, although *see* depends on the world of evaluation, it is -extensional in both of its syntactic arguments: - - let saw x y w = (w < 2) && (y < x);; - saw bill ann 1;; (* true: Ann saw Bill in world 1 *) - saw bill ann 2;; (* false: no one saw anyone in world 2 *) - -This (again, partially) intensionalized version of *see* coincides -with the `saw1` function we defined above for world 1; in world 2, no -one saw anyone. - -Just to keep things straight, let's review the facts: - -
-     World 1: Everyone left.
-              Ann saw Bill, Ann saw Cam, Bill saw Cam, no one else saw anyone.              
-     World 2: Ann left, Bill left, Cam didn't leave.
-              No one saw anyone.
-
- -Now we are ready for the intensionality monad: - -
-type 'a intension = s -> 'a;;
-let unit x = fun (w:s) -> x;;
-(* as before, bind can be written more compactly, but having
-   it spelled out like this will be useful down the road *)
-let bind u f = fun (w:s) -> let a = u w in let u' = f a in u' w;;
-
- -Then the individual concept `unit ann` is a rigid designator: a -constant function from worlds to individuals that returns `'a'` no -matter which world is used as an argument. This is a typical kind of -thing for a monad unit to do. - -Then combining a predicate like *left* which is extensional in its -subject argument with an intensional subject like `unit ann` is simply bind -in action: - - bind (unit ann) left 1;; (* true: Ann left in world 1 *) - bind (unit cam) left 2;; (* false: Cam didn't leave in world 2 *) - -As usual, bind takes a monad box containing Ann, extracts Ann, and -feeds her to the extensional *left*. In linguistic terms, we take the -individual concept `unit ann`, apply it to the world of evaluation in -order to get hold of an individual (`'a'`), then feed that individual -to the extensional predicate *left*. - -We can arrange for a transitive verb that is extensional in both of -its arguments to take intensional arguments: - - let lift2' f u v = bind u (fun x -> bind v (fun y -> f x y));; - -This is almost the same `lift2` predicate we defined in order to allow -addition in our division monad example. The difference is that this -variant operates on verb meanings that take extensional arguments but -returns an intensional result. Thus the original `lift2` predicate -has `unit (f x y)` where we have just `f x y` here. - -The use of `bind` here to combine *left* with an individual concept, -and the use of `lift2'` to combine *see* with two intensional -arguments closely parallels the two of Montague's meaning postulates -(in PTQ) that express the relationship between extensional verbs and -their uses in intensional contexts. - -
-lift2' saw (unit bill) (unit ann) 1;;  (* true *)
-lift2' saw (unit bill) (unit ann) 2;;  (* false *)
-
- -Ann did see bill in world 1, but Ann didn't see Bill in world 2. - -Finally, we can define our intensional verb *thinks*. *Think* is -intensional with respect to its sentential complement, though still extensional -with respect to its subject. (As Montague noticed, almost all verbs -in English are extensional with respect to their subject; a possible -exception is "appear".) - - let thinks (p:s->t) (x:e) (w:s) = - match (x, p 2) with ('a', false) -> false | _ -> p w;; - -Ann disbelieves any proposition that is false in world 2. Apparently, -she firmly believes we're in world 2. Everyone else believes a -proposition iff that proposition is true in the world of evaluation. - - bind (unit ann) (thinks (bind (unit bill) left)) 1;; - -So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave). - - bind (unit ann) (thinks (bind (unit cam) left)) 1;; - -But in world 1, Ann doesn't believe that Cam left (even though he -did leave in world 1: `bind (unit cam) left 1 == true`). Ann's thoughts are hung up on -what is happening in world 2, where Cam doesn't leave. - -*Small project*: add intersective ("red") and non-intersective - adjectives ("good") to the fragment. The intersective adjectives - will be extensional with respect to the nominal they combine with - (using bind), and the non-intersective adjectives will take - intensional arguments. - - -- 2.11.0