First order logic and (the lack of) plural predication

sentation of sentence meaning. However, standard first-order predicate logic cannot properly account for plural meaning. The normal interpretation of first order logic predicates, for example, is to take them to be sets of individuals. This allows us to translate (167a) as (167b):

(167) a. Andrea is a student. b. _student(Andrea).

Translating (168a) is slightly trickier, but possible. The common way of doing so is to first translate it into (168b); this, now, can be straightforwardly interpreted as (168c):

(168) a. Andrea and Amy are students.

b. Andrea is a student and Amy is a student. c. student(Andrea) &student(Amy)

The reason (168b) is an acceptable translation of (168a) is because the property of being a student is distributive. If several entities are students, that means that each of them is a student. Since all first order predicates are sets of individuals, they are by nature distributive.

However, the same is not true of all natural language predicates. Take (169a):

(169) a. Andrea and Amy are classmates.

b. # Andrea is a classmate and Amy is a classmates. c. #classmate(Andrea) &classmate(Amy)

How can (169a) be translated into predicate logic? Translating it into (169b) is clearly the wrong way to go. It makes no sense to talk of a set of classmates, that Amy and Andrea happen to be members of. Note that if there was a set of such classmates, than (170a) would be a valid inference, as demonstrated by the validity of (170b):

(170) a. Andrea and Amy are classmates, and Dan and Violeta are classmates. Therefore, Andrea and Violeta are classmates.

b. _classmate(Andrea) & _classmate(Amy) & _classmate(Dan) &

classmate(V ioleta)⇒

classmate(Andrea) & classmate(V ioleta)

One possible solution is to treat classmate as a relation. Instead of being a set of individuals, it is possible to treat it as a set of pairs of individuals, and translate (169a) as (171):

(171) classmate(Andrea)(Amy)

But what should be done then of (172)?

(172) Andrea, Amy and Marcos are classmates.

(173) _classmate(Andrea)(Amy) & _classmate(Amy)(M arcos) &

classmate(Andrea)(M arcos)

But (173) can be true even if (172) is not; say that Andrea and Marcos are both taking semantics together, Amy and Marcos are both attending a statistics class, and that Andrea and Amy are both taking an afternoon knitting class together (and no other classes are being taken by any of them). Any two of them are classmates, so that (173) is true, but it is not true that all three of them are classmates.

One possible solution is to take classmate to be ambiguous between a two-way relation and a three-way relation. But of course, it is possible to add more and more people:

(174) a. Andrea, Amy, Marcos and Tuuli are classmates.

b. Andrea, Amy, Marcos, Tuuli and Jason are classmates. c. Andrea, Amy, Marcos, Tuuli, Jason and Kara are classmates.

d. Andrea, Amy, Marcos, Tuuli, Jason, Kara and Kevin are classmates. e. ...

As this is a potentially infinite progression, classmate would have to be infinitely ambiguous. And not just classmate, but many other predicates; meet, dance, lift a piano, and so forth. This is unappealing from a parsimony viewpoint. One solution to the problem of infinite ambiguity is allowing the logic to contain predicates with variable poliadicity. However, this introduces its own problems. For example, take the following sentences (the situation in question may be some sort of party game, where several people lie on a bedsheet and their friends attempt to lift them all at once):

(175) a. Andrea and Amy lifted Marcos.

b. Andrea and Amy lifted Marcos and Tuuli.

d. Andrea, Amy, Kara and Marcos lifted Tuuli, and Jason. e. ...

In (175), both lifters and liftees can be arbitrarily larger collections. This means that in addition to allowing _lift to take a variable amount of arguments, the logic must afford a mechanism that distinguishes which of these arguments are lifters and which are liftees, such that it is possible to distinguish (175c) and (175d). In addition, it is not clear how either an ambiguity or a variable polidacity account can handle sentences such as (176):

(176) All the girls are classmates.

Here too, a common way to interpret plural subjects like the girls in standard predi- cate logic is by distributing over the girls:

(177) a. All the girls are students. b. ∀x[girl(x)→student(x)]

But obviously, doing the same for classmates returns us to square one:

(178) ∀x[_girl(x)→_classmate(x)]

The relation-based account of classmate, then, is incapable of handling quantified subjects.

From the examples above it can be seen that the proper treatment of plurals is out of the reach of classic predicate logic. What is necessary is to extend predicate logic to include plurals. There have been two main approaches to how to do so.

The first approach seeks to solve the problem while keeping the basic structure of predicate logic intact. Since predicate logic is designed to work with singular ob- jects, this approach reduces all statements about plurals to statements about abstract entities that are themselves singular, but can stand in for the plural. The second ap-

proach is more radical. It abandons first order predicate logic completely in favor of second order predicate logic. This approach no longer needs an intermediary between the plural and its predicate, but its increased power comes at the cost of simplicity.

4.1.2

Reducing plural predication to singular predication