Number Seminar

Emmon Bach, SOAS, UMass(Amherst)
SOAS: , 2007
contact: ebach@linguist.umass.edu
Copyright 2007 by Emmon Bach. All rights reserved.
Link for these notes.

The Semantics of Number

...the true difference between languages is not in what may or may not be expressed but in what must or must not be conveyed by the speakers.

Roman Jakobson, 1959

Prelude: Some Puzzles and Questions

Here are some sentences illustrating various questions about the interpretation of plurals in natural languages:

1. Three girls invited two boys to two parties.
How many meanings does this sentence have? Under what situations could it be true?
2. Kim and Pat carried the groceries upstairs.
3. Kim carried the groceries upstairs and Pat carried the groceries upstairs.
4. Kim and Pat each carried the groceries upstairs.
Can (2) mean the same as (3) and (4)?
5. Kim carried all the groceries upstairs and Pat carried all the groceries upstairs.
6. Unicycles have wheels.
How many wheels does a unicyle have?
7. If you called a horse's tail a leg, how many legs would horses have?
8. How many cows do you own?
9. I own three cows.
10. Each cow has four legs, so I own 12 legs.
11. ??How many cattle do you own?
12. I own three head of cattle.
13. There are three muds in my garden.
14. There's alluvial mud, Thames Mud, and Muswell Hill mud
15. We'll have three beers, please.
16. Sally, Alice, and Hortense defeated Fred, Ted, and Charlie respectively.

One way to approach questions like those illustrated here is to set up models or situations and to test your intuitions against the various configurations in the models. For sentence (1) for example, we could have these situations:

1. Sally, Alice, and Hortense (as a group) invited both Fred and Ted to two parties A and B.
2. Sally invited Fred to party A, Alice invited Fred and Ted to party B, Hortense invited Ted to party B.
and so on...

If you follow out this way of trying to get at the meanings of linguistic expressions in a consequential way you are in danger of doing MODEL THEORETIC SEMANTICS.

A frequent question: if we can think of umpteen zillion different situations that might verify a sentence do these reflect genuinely different readings, meanings, denotations, etc.?

The Name of the Game: Model Theoretic Semantics

Model theoretic semantics -- sometimes misleadingly called formal semantics -- makes the assumption that expressions of language refer to things that are not language: my cat is not a concept but a thing in the world, if a sentence is true it is verified by things and relations in the world not things in my head. Model theoretic semantics proceeds by positing model structures and associating things in the model structures with expressions of a language. For example, a simple model structure for the propositional calculus, consists of just truth values and functions relating them. The predicate calculus adds a domain of individuals, sequences of individuals, and so on. To get serious with natural languages you need a model structure of at least this complexity. And in addition for tenses and modality you need something like possible worlds and times. Richard Montague's work on the semantics of natural language from the 1970's was the starting point for a rich and varied stream of research. Montague's best known paper (1973) was called "The proper treatment of quantification in English" (PTQ). PTQ is strictly limited to a simple grammar with just enough details of English to allow Montague to make his main points: a uniform treatment of DP's as generalized quantifiers, proper names, intensionality, mood and tense, and quantification. (See Appendix for a formal layout of the classical model structure used by Montague and others.)

Extending the coverage of PTQ

History: As soon as linguists like Barbara Partee, Michael Bennett started using the ideas they found in the work of Montague, David Lewis, Max Cresswell and others to interpret English and other natural languages, there was much work devoted to extending the coverage of PTQ to reach a better approximation to the whole language. At first, this work was carried out in a very conservative way. For example, the extension of PTQ to cover plurals by Partee's student Michael Bennett used machinery that was already implicit in Montague's model: plurals were interpreted as sets. Where PTQ interpreted the (singular) common noun unicorn as denoting the set of individuals that were unicorns, the plural unicorns denoted the set of sets of unicorns, and so on. As time went on, however, people began to make more creative use of the general setup of Montague semantics.

Plurals

Given the model structure outlined in the Appendix, the most immediate idea is to model plural meanings with sets built from A, the set of individuals. A set in that sort of a structure is modeled by functions from entities to truth values (type <e, t>). Michael Bennett added plurals to the fragment, incorporating a new category for them and interpreting them as set counterparts of the singular terms and common nouns of PTQ. (Bennett 1976 is a somewhat more accessible source than 1974.) A justification for this move might come from the existence of predicates and other items that seem to select for plural arguments:

17. They are friends.
18. *They are a friend.
19. *He is friends.
(But cf. He is friends with me.)
20. The boys dislike each other.
21. *The boy dislikes each other.
22. Arsenal are winning.
23. ?Arsenal admire each other.

Note that these new items under this analysis make demands on the category and type hierachy by proliferating the types all the way up the line. (Montague's theory requires a functional mapping from syntactic categories to semantic types.) The examples show that some predications want to have plural subjects, some singular. Moreover, some perfectly ordinary English sentences require special treatment or are predicted to be bad. Since plurals are treated directly in the categorial and type theory, it looks like what seems to be a single verb like know has to have four variants, one for singular subjects and singular objects, one for plural subjects and singular objects, and so on... The same goes for attributive adjectives: big^singular, big^plural and so on and on...

24. Kim knows the answer, and so do the other students.
25. I know the answers.

Sorts

Technically, much of the work that we will consider in this and subsequent sections relies on the idea that A, the domain of individuals, can be taken to be a union of Sorts, perhaps related (as in the case of plural individuals to be considered in a moment), but all logically still first-class members of A, hence all of type e. For this approach to be useful we need to be able to notate all sorts of expressions according to the Sorts that they go with: individual variables, predicates and so on. We will do this with superscripts, so x^pl might indicate a variable that ranges over the Sort of plural entities (whatever that might be in our implementation of the idea). People often save ink by requiring such an indication only on the first occurrence of the expression in a formula. For example:

26. ∀x^pl[x = x]

Here it is understood the x's in the formula are "really" x^pl. But in all such situations you should let clarity and perspicuity be your guiding lights.

Sorts actually came into semantics for rather different reasons. Consider this sentence:

27. Caesar is a prime number.
28. Caesar is not a prime number.

Is this (27) true or false? It is certainly not true, but if it is false then its negation (28) should be true. But both sentences seem to be strange, suffering from a category mistake of some kind. Many would say that (27) is neither true nor false. But then it has to have some third truth value. And indeed theories of sortal incorrectness often go with a system of three truth values: the third one perhaps "undefined" or zilch or some other "ugly object." (Two references for theories of sortal incorrectness are Thomason 1972 and Waldo 1979.)

Plurals II: Linkian structures

Godehard Link's (1983) paper on mass terms and plurals inspired a lot of work not only on the domains he considered but also event structures (Bach 1986). Like the work on generics that spun off from Greg Carlson's work on generics innovations did not change the basic model structure per se, but began to build some special structure into the domain of individuals. Link's way goes like this: start with the set A of (ordinary) individuals. Allow an operation to create "plural individuals" from the singulars, and on up. But these new individuals are still of type e. Now, if the join operation is non-associative, we will have this full structure:

So suppose we have a model with three horses: Attila, Blair, and Charlemagne. With the join operation (denoted by ⊕ "circle plus") we can make this structure:

〚horses〛 =   {        (a ⊕ b ⊕  c)

   (a ⊕ (b ⊕ c)) ((a ⊕ b) ⊕ c) ((a ⊕ c) ⊕ b) 


            (a ⊕ b) (b  ⊕ c)  (a  ⊕ c) } 
     ____________________________________________________________
             
〚horse〛 =       { a           b           c }(OR)
                  

For Link, the join operation is associative, so the elements on the middle line of the plurals will all be identical and equal to the top element.

The denotation of horse is just the set {a, b, c}, the denotation of horses, on this account, is the plural individuals above the line. The structure is atomic, in that there is a lowest level of atoms, the individual horses. This is in contrast to the structures for mass terms, which are not atomic. In addition there is another construct: the set we get by putting together the singulars and the plurals, which Link called *horse.

Roger Schwarzschild (1991) contrasted two theories about plurals: one a "sets theory" and one a "union" theory. The sets theory uses structures like those of Link, but using just set formation as the basic operation. In addition singleton sets are equated with individuals. Structured sets like this preserve the differences in middle area of the picture: {{a,b},c} ≠ {a,{b,c}}. The union theory gives a much simpler structure, because union formation is associative. (In the diagram above, the middle plural individuals above the line would just drop out.) Schwarzschild defends the union view (see also his 1990 paper "Against Groups").

The difference between the two theories can be easily seen in the interpretation of DP's like Ray and the boys or Ray and Tess and Jess. Assume the boys are just Fred and Ted:

1. Ray and the boys
   sets theory: {Ray, {Fred, Ted}}
   union theory: {Ray, Fred, Ted}
2. Ray and Tess and Jess
   sets theory: {Ray, {Tess, Jess}}
   {{Ray, Tess} , Jess}
   {Tess, {Ray, Jess}}
   {Ray, Tess, Jess}
   union theory: {Ray, Jess, Tess}

A good bit of Schwarzschild's thesis goes to countering arguments based on examples like the following, which seem to favour the sets interpretation (and Link's theory as well). Assume a model where all the animals are just cows and pigs. Look at these sentences (Schwarzschild 1991: 69 -- new numbering here, in the source the numbers are 101 - 103):

29. a. The cows and the pigs were separated.
    b. The young animals and the old animals were separated.
30. a. The cows and the pigs talked to each other.
    b. The young animals and the old animals talked to each other.
31. a. The cows and the pigs were given different foods.
    b. The young animals and the old animals were given different foods.

On the union theory the subject phrases in the (a) and (b) versions denote exactly the same sets, on the sets (and groups) theory they do not.

On the other hand, look at these examples (op.cit. 71: 104 - 105):

32. a. The animals filled the barn to capacity.
    b. The cows and the pigs filled the barn to capacity.
    c. The young animals and the old animals filled the barn to capacity.
33. a. The animals were sleeping in the barn.
    b. The cows and the pigs were sleeping in the barn.
    c. The young animals and the old animals were sleeping in the barn.

On the union approach, the examples in each triple will have the same truth values, but not in the sets theory. Schwarzschild goes on to explore the further stipulations and shows they reduce to the union theory. He then develops a theory which puts the burden of explaining some of the apparent evidence for the structured sets approach into a context-sensitive "cover" or partition option in the predicate to account for cumulative and distributive understandings of sentences.

Proper Groups.

Most theories of plurals, treat group nouns like team, department, executive committee as denoting true groups. Note that coextensive groups like these have to be kept separate: "The department meeting is now adjourned. The executive committee will now vote on this proposal." (OK even if exactly the same individuals are in the two groups.)

Mass terms and Material Identity

Link also proposed an influential theory about mass terms: a separate kind of structure models mass terms which is not atomic, and there is a mapping from both kinds of items to their "stuff" or material. (We can come back to discussion of this if there is time.) This mapping in the count domain means that two things can be materially identical but still distinct: for example, Terry's ring and the gold constituting the ring. Note: the ring can be new and the gold old.

Some Other Languages and Applications

Kiowa Number.

The Tanoan languages offer a fascinating system of number in the nominal and pronominal systems. You can find a short introduction in Mithun, 1999: 81-82 (data Jemez). Watkins (1984) gives an extended description of Kiowa (not to be confused with Kiowa Apache, an Athapaskan language). I will cite Kiowa here from Laurel Watkins' work (Watkins 1984).

The most notable feature of the system is inverse number. Nouns are divided into four classes. There are three number categories: singular, dual, plural. According to its class membership, each noun has basic or inherent number or numbers.

1. singular/dual inverse: plural primarily animate
2. dual/plural inverse: singular
3. dual inverse: singular/plural
4. (nouns in this class do not use the inverse suffix

(I omit discussion of the Class IV nouns.) As you can see, the inverse picks the complement meaning with respect to plurality. Disambiguation of the choices (e.g. dual/plural) comes about by combinatorics with number marking in pronominal affixes on verbs and other elements.

Examples:

34. tógúl `young man, two young men'
35. tógú:gɔ́ `(more than two) young men'

But in combination with an Intransitive Prefix èͅ- on a predicate, the first word must be interpreted as dual. This prefix marks intransitives as predicates over sets of entities with cardinality of two.

36. gú: ribs (dual/plural)
37. gú:gɔ̀ rib

In combination with an Intransitive Prefix èͅ-, the first word must be interpreted as dual. The second word is singular and takes inverse agreement, while to get the plural the intransitive prefix gyà- is required on an agreeing intransitive verb. You are invited to consult the sources for more details of this complex system.

Now a question: is there any way to assign a uniform semantic value to the inverse suffix?

Here's a try. Let's adopt the kind of structured domain proposed by Link and others. For any common noun we have the set of all groups formed from the atoms or basic atoms of the domain. For languages like the Tanoan languages, for any noun N we have D(N) = the union of the singletons (or atoms) I(N), the pairs II(N), and the pluralities III+(N). The denotation of an uninflected Class I noun is just the union of the singletons and the pairs, for Class II just the union of the pairs and the plurals, for Class III the pairs. Now the inverse can be interpreted as an operation that takes the denotation of the bare noun and delivers the complement of that denotation within the whole domain of the common noun. This treatment requires that we have available a denotation that is not directly associated with any of the various forms of the noun itself. The effect of the various inflections on other elements that disambiguate the expressions then can be achieved by intersecting the denotation of the nominal expression with cardinality sets, as in the example above.

Obligatory vs Optional Categories: Plurality in Haisla (etc)

About Haisla: a Northern Wakashan language (along with Heiltsuk, Ooweky'ala and a number of languages lumped together as Kwakw'ala -- Franz Boas's Kwakiutl). Haisla is spoken in Kitamaat Village in northern British Columbia in the far west of Canada.

Some Haisla examples (e = ə):

38. begʷánem
person, people
39. bíbegʷanem
people
40. t̓íxʷa
black bear(s)
41. t̓ít̓exʷa
black bears
42. ketá
shoot
43. kiketá
shoot "plural"
44. ketátlnugʷa
I am going to shoot.
45. ketátlnis / kiketátlnis
we (inclusive) are going to shoot
46. kiketátlnugʷa.
I am going to shoot repeatedly / several times
47. kiketátlnugʷaʼi.
I am going to shoot them.
48. kiketá begʷánemax̄i t̓íxʷix̄i.
The man/men shot/repeatedly the bear/bears (repeatedly).

Note: there are a number of different reduplicative and ablaut forms, a number of which are used for marking plurality etc.

Like many North American languages (and in fact probably languages around the world), plurality is an optional category in Haisla and other Wakashan languages. As one might expect not every word has a definite plural form and the common hierarchy applies: nouns near the high end of an animacy scale tend to have plurals; near the low end, they don't.

Here's how I would model a system like this: a plain noun like t̓íxʷa denote sets from *〚t̓íxʷ〛a the big domain covering atoms and sets all the way up.

You can tell a straighforward Gricean story about the interpretations of plain and plural forms in such a language, as opposed to a language in which plurality is enforced.

Obligatory choice: The interlocutor had to make a choice so I will interpret the choice as telling me something important!

Optional choice: There may be no particular reason for making or not making this choice, so I can't really conclude anything. If its important context will probably tell me.

Number enters into Haisla and a number of neighboring languages in a different way at the level of lexical choice. Some predicates are specialized as to shape or other characteristics of the subject of object, among them number. So for example Haisla hena means `to be located (somewhere): of a long cylindrical object.' Coast Tsimshian baa `run singular' k̓oł `run plural.'

Languages with no singular/plural distinction

I take Japanese as an example of a language that has no plurality as such in nouns, requires classifiers for counting, and allows bare nouns as arguments. In Japanese, for example, inu can be used to refer to a single dog or several. To construe the word you need to put it together with a counter or classifier:

49. inu
dog, dogs
50. sambiki no inu
three dogs: 3 classifier (of) dog
51. *san inu

In such languages (cf. Rullman and You on Mandarin), we can take the denotation of nouns like inu to be the whole star-domain: *inu'. The classifier delivers up atoms of the domain that can be counted.

Pirahã: a language with no numbers, number, quantification, etc.?

According to Dan Everett (2005) Pirahã, all of the above. But the language distinguishes mass and count ideas. Read Hale 1975 on this kind of question. [Discussion if time.]

Numbers and Nouns: Yes, Virginia, there is grammar!

What happens when we put together numbers and nouns, as in English two sheep. In English we generally have no choice: if the number is 1, then we use the singular, otherwise plural, even when it makes no good sense:

52. There were three cows in the barn.
53. There are zero cows in the barn.
54. The average family has 1.5 children.
55. There is half (of) a pie in the fridge.
56. No, actually it's one and a half pies.
57. German: Tausend und eine Nacht
58. Ich habe hundert und einen Brief geschrieben.
I have written 101 letters. (gratia Regine Koroma)

Many more examples of this sort of thing in Corbett.

Two principles in language: redundancy and laconicity. Suppose there is some distinction that operates in a grammar, such as number, and suppose this can be expressed lexically as well. Two diametrically opposed impulses: (1) mark it twice! (2) No need to mark it since its already there!

Some Conclusions

The Big Question: Do different languages show different "metaphysicses" as Whorf would have it? No: in the sense that the general model structures are probably something like a universal toolbox or outfitted kitchen. But maybe Yes at the level of choice of details and assignments to lexical classes and grammatical choices.

Appendix: A Model Structure for Interpreting Natural Languages

1. M₁: a candidate model structure for natural languages

1. A: a set of individuals: type e
2. I: a set of worlds
3. J: a set of times, ordered by some relation such as ≤ (transitive, antisymmetric)
4. BOOL= {0, 1} a set of truth values: type t
5. F: all functions built out of these ingredients: type <a,b>, where a and b are types

Sets (or individuals) are modeled by characteristic functions of type <e,t>, sets of sets of individuals are <<e,t>,t>.

M₁ is basically the setup used by Richard Montague in his ground-breaking paper "The proper treatment of quantification in ordinary English" (PTQ: Montague 1973: for this and other references see References at end of these notes). An interpretation of a language L takes such a model structure, and a function D (for Denotation) and assigns elements of M to expressions of L with the help of an assignment g of values to variables. In PTQ D is relativized to a world - time pair and an assignment, and is carried out via a translation of the disambiguated language L' derived from L into an intensional logic IL: so the steps in getting an interpretation look like this:

T(John) = j etc (T a translation)
D_i,j,g(α) = ....(actually 〚α〛_i,j,g = ...) where i is a world, j a time.

An alternative to using a disambiguated language is to assign sets of interpretations to expressions of L (Robin Cooper 1983).

For his grammar (syntax) Montague used a kind of extended categorial grammar and insisted on a functional mapping from syntactic categeries to semantic types.

Note: For Montague, IL is strictly eliminable and is not to be considered as a significant level of representation in the linguist's sense. Some linguists have pursued this line in a consequential way, notably Polly Jacobson in a number of papers under the rubric "direct compositionality" (book by that name just published, edited by Chris Barker and Pauline Jacobson (2007)). On the other hand, Mark Steedman, who like Jacobson works with a variable free syntax and semantics, posits a linguistically significant level of interpretation, his Logical Form. (Not = LF in Chomsky's sense!?) Steedman (2000) and Jacobson (1999, 2000) and some others use a version of categorial grammar -- Combinatory Categorial Grammar -- that deserves a course on its own. It builds on Scho̎nfinkel 1924, Quine 1960, and especially Curry and Feys 1958.

References

Waugh, Linda R, and Monique Monville-Burston, eds. 1990. On Language: Roman Jakobson . Cambridge, Massachusetts / London: Harvard University Press.

References.