# Motivating types; our first glimpse of the Maybe monad

In a long footnote discussing Russell, 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 meanings 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. Our more systematic type-theoretic treatment permits us to dispense with worries about whether the meaning of the DP in (1) might itself be the empty set.

## 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 proposal for representing the meaning of directly-referential expressions:

Kaplan's rule for directly-referential expressions: a directly referential expression E contributes either:
{} (or what we'll call None or Nothing) if there is no object that E refers to, or else
{P} (or what we'll call Some P or Just P) if E refers to P

We will call the type used here an "option" or "Maybe" type (from OCaml and Haskell, respectively). And we'll call the general strategy for deploying this type "the Option/Maybe monad."

Kaplan, D. 1989. "Demonstratives. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes from Kaplan (pp. 481-563)."