1 Motivating types; our first glimpse of the Maybe monad
2 ======================================================
4 In [[a long footnote discussing Russell|readings/kaplan-plexy.pdf]],
5 Kaplan 1989:496 poses a problem for the interaction of structured
6 meanings with direct reference, and then offers a solution based on
9 To grasp the problem, consider a structured-meaning theory of meaning,
10 on which the meaning of natural language expressions are represented
11 by objects that can have internal structure.
13 1. The center of mass of the solar system is a point.
15 Further suppose that the meaning of the DP in subject position in (1),
16 namely, *the center of mass of the solar system*, has internal
17 structure. For instance, suppose that the meaning of this DP is a
18 structure that contains within it an object representing the meaning
19 of *the solar system*, an object representing the relational concept
20 denoted by *center*, and so on. It doesn't matter precisely what that
21 structure is, as long as it has distinct internal parts.
23 Kaplan names this complex object "Plexy". On Kaplan's assumptions,
24 the name *Plexy* is directly referential, and refers to the complex
25 object that represents the meaning of *the center of mass of the solar
30 Kaplan observes that as far as native speaker intuitions are concerned, sentences (1) and (2) have very different
31 meanings. The sentence in (1) attributes a property to a location in
32 space, and the setence in (2) attributes the same property to the
33 referent of *Plexy*, which is a structured meaning. Since meanings
34 need not be locations in space, it is easy to imagine judging (1) true
35 and (2) false. As Kaplan puts it, the two sentences "speak about
36 radically different objects".
38 The problem is that on a simple-minded structured meaning account, the meaning of (1)
39 contains Plexy in the position corresponding to the argument
40 of the predicate (since, by assumption, Plexy is the meaning of the DP). Likewise, on a simple-minded direct-reference account, the contribution of
41 a directly referential term is simply the object it refers to.
42 Combining these two assumptions, we incorrectly predict that (1) and
43 (2) denote the same structure, and therefore have exactly the same
46 (Instead of using a sentence like (2), Kaplan constructs a different,
47 more complicated expression that, he argues, refers to Plexy, but this
48 nicety is not crucial to our discussion here.)
50 ## Types to the rescue
52 Kaplan's solution is, in effect, to impose a type system on his
53 grammar in such a way that complex structured meanings cannot be confused with
54 the referent of a directly-referential term. He suggests that the
55 meaning of a directly referential term always be marked by a special
56 bit of structure that is unique to direct reference. More concretely,
57 Kaplan suggets that instead of inserting the referent of a directly
58 referential term directly into the structure of the sentence in which
59 it occurs, we insert the singleton set containing that referent. As
60 long as no complex structured meaning happens to be a singleton set, we have a
61 solution. If *P* is Plexy, the meaning of (1) might be `<P, point>`, at the same time that
62 the meaning of (2) can be `<{P}, point>`: radically different, as desired.
64 In terms of the type systems we'll be developing over the next few
65 weeks, the type of a DP will be a *sum type*: the disjoint union of the
66 class of objects that a directly referential term can refer to, and
67 the class of objects that can serve as the complex structure
68 corresponding to a DP that is not directly referential.
72 Kaplan goes on to use this solution to attack a different problem, the
73 problem of non-referring names. Russell supposed that if a name had
74 no referent (e.g., *Santa*), a sentence containing that name would
75 have no meaning, since there would be no object to insert into the
76 structure representing the meaning of that sentence. But on Kaplan's
77 scheme, there is no problem: *Santa is hungry* would denote `<{},
80 This second idea has some obvious flaws. For instance, it predicts
81 that sentences that differ only in the choice of a non-referring name
82 will have the same meaning. But it does not seem obvious that the
83 sentence *Santa is hungry* means the same thing as *Cupid is
86 Setting aside such objections, we will see over and over again the
87 utility of the general strategy instantiated in Kaplan's strategy for representing the meaning of
88 directly-referential expressions:
90 Kaplan's rule for directly-referential expressions:
91 a directly referential expression E contributes either
92 {} if there is no object that E refers to, or else
95 In later weeks, we will call the general form of this technique the Maybe type, and the general strategy for deploying this type the Maybe monad.
98 Kaplan, D. 1989. "Demonstratives. InJ. Almog, J. Perry, & H. Wettstein
99 (Eds.), Themes from Kaplan (pp. 481-563)."