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+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
+meanings with direct reference, and then offers a solution based on
+types.
+
+To grasp the problem, consider a structured-meaning theory of meaning,
+on which the meaning of natural language expressions are represented
+by objects that can have internal structure.
+
+ 1. The center of mass of the solar system is a point.
+
+Further suppose that the meaning of the DP in subject position in (1),
+namely, *the center of mass of the solar system*, has internal
+structure. For instance, suppose that the meaning of this DP is a
+structure that contains within it an object representing the meaning
+of *the solar system*, an object representing the relational concept
+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". 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*. Now consider:
+
+ 2. Plexy is a point.
+
+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
+need not be locations in space, it is easy to imagine judging (1) true
+and (2) false. As Kaplan puts it, the two sentences "speak about
+radically different objects".
+
+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
+meaning.
+
+(Instead of using a sentence like (2), Kaplan constructs a different,
+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 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 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
+class of objects that a directly referential term can refer to, and
+the class of objects that can serve as complex meaning structures
+corresponding to DPs as in (1) that are not directly referential.
+
+## Motivating Maybe
+
+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 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* means the same thing as *Cupid is
+hungry*.
+
+Setting aside such objections, we will see over and over again the
+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
+
+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. In J. Almog, J. Perry, & H. Wettstein
+(Eds.), Themes from Kaplan (pp. 481-563)."