-Motivating types; motivating the Maybe monad
-============================================
+Motivating types; our first glimpse of the Maybe monad
+======================================================
In [[a long footnote discussing Russell|readings/kaplan-plexy.pdf]],
Kaplan 1989:496 poses a problem for the interaction of structured
denoted by *center*, and so on. It doesn't matter precisely what that
structure is, as long as it has distinct internal parts.
-Kaplan names this complex object "Plexy". On Kaplan's assumptions,
+Kaplan names this complex object "Plexy". Simplifying Kaplan's presentation, let's suppose
the name *Plexy* is directly referential, and refers to the complex
object that represents the meaning of *the center of mass of the solar
-system*.
+system*. Now consider:
2. Plexy is a point.
-Kaplan observes that sentences (1) and (2) have very different
+Kaplan observes that as far as native speaker intuitions are concerned, sentences (1) and (2) have very different
meanings. The sentence in (1) attributes a property to a location in
space, and the setence in (2) attributes the same property to the
referent of *Plexy*, which is a structured meaning. Since meanings
and (2) false. As Kaplan puts it, the two sentences "speak about
radically different objects".
-The problem is that on a structured meaning theory, the meaning of the
-sentence contains Plexy in the position corresponding to the argument
-of the predicate. Likewise, we might suppose that the contribution of
+The problem is that on a simple-minded structured meaning account, the meaning of (1)
+contains Plexy in the position corresponding to the argument
+of the predicate (since, by assumption, Plexy is the meaning of the DP). Likewise, on a simple-minded direct-reference account, the contribution of
a directly referential term is simply the object it refers to.
Combining these two assumptions, we incorrectly predict that (1) and
(2) denote the same structure, and therefore have exactly the same
more complicated expression that, he argues, refers to Plexy, but this
nicety is not crucial to our discussion here.)
+## Types to the rescue
+
Kaplan's solution is, in effect, to impose a type system on his
-grammar in such a way that structured meanings cannot be confused with
+grammar in such a way that complex structured meanings cannot be confused with
the referent of a directly-referential term. He suggests that the
meaning of a directly referential term always be marked by a special
bit of structure that is unique to direct reference. More concretely,
Kaplan suggets that instead of inserting the referent of a directly
referential term directly into the structure of the sentence in which
it occurs, we insert the singleton set containing that referent. As
-long as no complex structured meaning (that is not the meaning of a
-directly referential term) happens to be a singleton set, we have a
-solution. If *P* is Plexy, the meaning of (1) is `<P, point>`, and
-the meaning of (2) is `<{P}, point>`: radically different, as desired.
+long as no complex structured meaning happens to be a singleton set, we have a
+solution. If *P* is Plexy, the meaning of (1) might be `<P, point>`, at the same time that
+the meaning of (2) can be `<{P}, point>`: radically different, as desired.
In terms of the type systems we'll be developing over the next few
-weeks, the type of a DP will be a sum type: the disjoint union of the
+weeks, the type of a DP will be a *sum type*: the disjoint union of the
class of objects that a directly referential term can refer to, and
-the class of objects that can serve as the complex structure
-corresponding to a DP that is not directly referential.
+the class of objects that can serve as complex meaning structures
+corresponding to DPs as in (1) that are not directly referential.
+
+## Motivating Maybe
-Kaplan goes on to use this solution to attack a different problem, the
+At the end of his footnote, Kaplan suggests using his proposal to help with a different problem, the
problem of non-referring names. Russell supposed that if a name had
no referent (e.g., *Santa*), a sentence containing that name would
have no meaning, since there would be no object to insert into the
structure representing the meaning of that sentence. But on Kaplan's
-scheme, there is no problem: *Santa is hungy* would denote `<{},
-hungry>`.
+scheme, there is no problem: *Santa is hungry* would denote `<{},
+hungry>`. This can't be confused with a sentence saying that the empty set is
+hungry, since (supposing we directly refer to the empty set), that would
+denote `<{{}},hungry>`.
This second idea has some obvious flaws. For instance, it predicts
that sentences that differ only in the choice of a non-referring name
will have the same meaning. But it does not seem obvious that the
-sentence *Santa is hungry* does not mean the same thing as *Cupid is
+sentence *Santa is hungry* means the same thing as *Cupid is
hungry*.
Setting aside such objections, we will see over and over again the
-utility of Kaplan's strategy for representing the meaning of
+utility of the general strategy instantiated in Kaplan's strategy for representing the meaning of
directly-referential expressions:
- Kaplan's rule for directly-referential expressions:
- a directly referential expression E contributes either
-
- {} if there is no object that E refers to, or else
- {P} if E refers to P
+> Kaplan's rule for directly-referential expressions: a directly referential expression E contributes either:
+> {} if there is no object that E refers to, or else
+> {P} if E refers to P
-In later weeks, we will call this technique the Maybe monad.
+In later weeks, we will call the general form of this technique the "option" or "Maybe" type, and the general strategy for deploying this type "the Maybe monad." (In OCaml one has types like `int option`; in Haskell they are `Maybe Int`.)
-Kaplan, D. 1989. "Demonstratives. InJ. Almog, J. Perry, & H. Wettstein
+Kaplan, D. 1989. "Demonstratives. In J. Almog, J. Perry, & H. Wettstein
(Eds.), Themes from Kaplan (pp. 481-563)."