+Extensional types Intensional types Examples +------------------------------------------------------------------- + +S s->t s->t John left +DP s->e s->e John +VP s->e->t s->(s->e)->t left +Vt s->e->e->t s->(s->e)->(s->e)->t saw +Vs s->t->e->t s->(s->t)->(s->e)->t thought ++ +This system is modeled on the way Montague arranged his grammar. +There are significant simplifications: for instance, determiner +phrases are thought of as corresponding to individuals rather than to +generalized quantifiers. If you're curious about the initial `s`'s +in the extensional types, they're there because the behavior of these +expressions depends on which world they're evaluated at. If you are +in a situation in which you can hold the evaluation world constant, +you can further simplify the extensional types. Usually, the +dependence of the extension of an expression on the evaluation world +is hidden in a superscript, or built into the lexical interpretation +function. + +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 intensional 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). + +The intenstional types are more complicated than the intensional +types. Wouldn't it be nice to keep the complicated types to just +those 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: + + type s = int;; + type e = char;; + type t = bool;; + +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 'a intension = s -> 'a;; + let unit x (w:s) = x;; + + let ann = unit 'a';; + let bill = unit 'b';; + let cam = unit 'c';; + +In our monad, the intension of an extensional type `'a` is `s -> 'a`, +a function from worlds to extensions. Our unit will be the constant +function (an instance of the K combinator) that returns the same +individual at each world. + +Then `ann = unit 'a'` is a rigid designator: a constant function from +worlds to individuals that returns `'a'` no matter which world is used +as an argument. + +Let's test compliance with the left identity law: + + # let bind u f (w:s) = f (u w) w;; + val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =

-Extensional types Intensional types Examples -------------------------------------------------------------------- - -S s->t s->t John left -DP s->e s->e John -VP s->e->t s->(s->e)->t left -Vt s->e->e->t s->(s->e)->(s->e)->t saw -Vs s->t->e->t s->(s->t)->(s->e)->t thought -- -This system is modeled on the way Montague arranged his grammar. -There are significant simplifications: for instance, determiner -phrases are thought of as corresponding to individuals rather than to -generalized quantifiers. If you're curious about the initial `s`'s -in the extensional types, they're there because the behavior of these -expressions depends on which world they're evaluated at. If you are -in a situation in which you can hold the evaluation world constant, -you can further simplify the extensional types. Usually, the -dependence of the extension of an expression on the evaluation world -is hidden in a superscript, or built into the lexical interpretation -function. - -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 intensional 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). - -The intenstional types are more complicated than the intensional -types. Wouldn't it be nice to keep the complicated types to just -those 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: - - type s = int;; - type e = char;; - type t = bool;; - -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 'a intension = s -> 'a;; - let unit x (w:s) = x;; - - let ann = unit 'a';; - let bill = unit 'b';; - let cam = unit 'c';; - -In our monad, the intension of an extensional type `'a` is `s -> 'a`, -a function from worlds to extensions. Our unit will be the constant -function (an instance of the K combinator) that returns the same -individual at each world. - -Then `ann = unit 'a'` is a rigid designator: a constant function from -worlds to individuals that returns `'a'` no matter which world is used -as an argument. - -Let's test compliance with the left identity law: - - # let bind u f (w:s) = f (u w) w;; - val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =