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1X Basic Properties of Number Words

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1X Basic Properties of Number Words NUMERICALS :COUNTING , MEASURING AND CLASSIFYING SUSAN ROTHSTEIN Bar-Ilan University 1x Basic properties of number words In this paper, I discuss three different semantic uses of numerical expressions. In their first use, numerical expressions are numerals, or names for numbers. They occur in direct counting situations (one, two, three… ) and in mathematical statements such as (1a). Numericals have a second predicative interpretation as numerical or cardinal adjectives, as in (1b). Some numericals have a third use as numerical classifiers as in (1c): (1) a. Six is bigger than two. b. Three girls, four boys, six cats. c. Hundreds of people gathered in the square. In part one, I review the two basic uses of numericals, as numerals and adjectives. Part two summarizes results from Rothstein 2009, which show that numericals are also used as numerals in measure constructions such as two kilos of flour. Part three discusses numerical classifiers. In parts two and three, we bring data from Modern Hebrew which support the syntactic structures and compositional analyses proposed. Finally we distinguish three varieties of pseudopartitive constructions, each with different interpretations of the numerical: In measure pseudopartitives such as three kilos of books , three is a numeral, in individuating pseudopartitives such as three boxes of books , three is a numerical adjective, and in numerical pseudopartitives such as hundreds of books, hundreds is a numerical classifier . 1.1 x Basic meanings for number word 1.1.1 x Number words are names for numbers Numericals occur bare as numerals in direct counting contexts in which we count objects (one, two, three ) and answer questions such as how many N are there? and in statements such as (1a) 527 528 Rothstein and (2). They name numbers and semantically are analogous to proper names which name individuals: (2) a. Two, four, six and eight are the first four even numbers. b. Two is the only even prime number. c. Two times two is four. / Two plus two is four. I assume that numbers are abstract entities and numerals are proper names for numbers at type n. (1a) and (2) shows that numbers, like other individuals, have properties. (2c) shows that plus , and times denote operations on numbers and are of type <n,<n,n>> as illustrated in (3). (3) times : λnλn.n £ n times two : λn.n £ 2 two times two : 2 £ 2 The singular agreement in the verb in (2c), as opposed to the plural agreement in (2a) is an indication of the fact that two times two is a complex numeral, denoting the number 4. Numerals can also be conjoined with and as in (4a-c). In this case, plural agreement on the verb is common in English, although this is not an absolute requirement. In other languages (e.g. Dutch) agreement must be singular (4d). This means that and in English is not normally an expression of type <n,<n,n>>, but denotes a function from pairs of numbers into pluralities. (4) a. "Two and two make five." (Orwell, 1984 : Part III, chapter 4) b. "Two and two are four, four and four are eight…"(Danny Kay, Inchworm ) c. "Now one and one is two mama, two and two is four," (Robert Johnson: Sweet Home Chigago ) d. Twee en twee is vier. (Dutch) Two and two is-SG four. Numericals also have a nominal interpretation at the predicate type and behave like normal nouns. In (5a), the N predicate four combines with a determiner, and in (5b) twos is pluralized and modified by a cardinal adjective. In this too, they look like proper names, see (6): (5) a. Move the four on the right of the equation to the left hand-side. b. Two twos are four, three twos are six. c. "What are twelve sevens?" (Roald Dahl, Matilda ) (6) There are two Johns in the class. The John I am talking about sits on the right. Numericals like hundred, thousand, million must combine with another numerical in order to form a numeral, as shown in (7): (7) a. Two/one hundred people stood in line. b.*Hundred people stood in line. Note that there is a contrast between two hundred which is a numeral, and two hundreds , analogous to two twos in (5b), which counts instances of one hundred. This shows up in (8): Numericals: Counting, Measuring and Classifying 529 (8) a. Nine hundreds are nine hundred, ten hundreds are a thousand. b. Nine hundred is nine hundred While both the examples in (8) are tautological, (8a), with plural agreement on the verb, is informative in the same way that The Morning Star is the Evening Star is informative, while (8b), where the verb has singular agreement, is uninformative, analogous to The Morning Star is the Morning Star . There is thus good evidence that numerals are grammatically names for individual numbers, and are analogous to proper names which denote singular individuals. I assume that a number n (i.e. the denotation of a numeral) is an equivalence class of entities with cardinality 2, i.e. entities constructed out of two atomic parts: 2 = {x: │x│= 2}. A numeral n has the denotation in (9a),1 and thus a simple numeral, such as two , has the interpretation in (9b). The corresponding predicate nominal interpretation at type <d,t> is the set {x: │x│= 2}. There are also complex numerical expressions such as hundred , which are of type <n,n> and combine with a numeral to form a complex numeral denoting the number n £ 100, as in (9c). We will call these 'multiplicative numerals', and discuss them further in section 3. (9) a. n = {x: │x│= n} = λx. │{y: y vATOM x} │= n The numeral n denotes the set of entities with n non-overlapping atomic parts. b. 9 two 0 = {x: │x│= 2} = λx. │{y: y vATOM x} │= 2 c. 9 hundred 0 = λn. {x: │x│= 100 £ n} 1.1.2 Numericals as cardinal adjectives Numericals have a predicative use as cardinal adjectives, exploiting the meaning in (9): (10) a. The guests are two. b. The two guests arrived. c. Two guests arrived. d. The musicians are two of our friends. e. Two of the guests arrived. Following Landman 2003, we assume that in (10c), the numerical is an adjective which raises to determiner position if there is no determiner. The adjective two has a predicate denotation derived from (9), and given in (11a): it denotes the property an entity has if the cardinality of the set of its atomic part is 2. Raising to determiner position induces type-shifting to the type of generalized quantifiers, as in (11b): (11) a. two <d,t> : λx. │x│= 2 = λx. │{y: y vATOM x} │= 2 b. two <<d,t>,<<d,t>,t>>: λQλP. ∃x[P(x) ˄ Q(x) ˄ │x│= 2] The meanings in (11) are used in the interpretations of (10-c) as follows: 1 More properly, the numeral n at type n denotes the individual correlate of the equivalence class, or set of entities ∩ with n atomic parts, in the sense of Chierchia 1984, Chierchia and Turner 1988. So two denotes {x: │x│= 2} and ∪∩ {x: │x│= 2}= {x: │x│= 2}. I work out the semantics of this explicitly in work in progress. 530 Rothstein (12) a. The guests are two : │σ{x: GUESTS(x)} │ = 2 "The cardinality of the unique maximal sum of the set of guests is two." = "Being two is a property that the maximal sum of guests has." b. The two guests arrived: ARRIVED( σ{x: GUESTS(x) ˄ │x│= 2}) "The maximal sum of guests, whose cardinality is two, arrived". c. Two guests arrived ∃x[GUESTS(x) ˄ ARRIVED(x) ˄ │x│= 2] "There is a plural individual in GUESTS with two atomic parts who arrived." Rothstein 2010a shows that a numerical in a partitive construction such as (10d,e) has its standard adjectival interpretation. Two of the guests denotes the set of parts of the plural entity denoted by the guests which have two atomic parts. See Rothstein 2010a for details. 2x Number interpretation in pseudopartitives: 2.1 x Counting vs. measuring The main use of a cardinal adjective is to modify a count (but not a mass) noun: (13) three flowers/four books/*three flour(s) . However, cardinal adjectives also modify classifiers as in (14): (14) a. Container classifiers: two cups of flour/two cups of beans/two glasses of water b. Measure classifiers: two kilos of flour/two kilos of beans/two glasses of water Selkirk 1977, Doetjes 1997, Chierchia 1998, Landman 2004, Rothstein 2009 and others have all showed that classifier phrases like two glasses of water are ambiguous between an ‘individuating’ reading, illustrated in (15a) and a measure reading in (15b): (15) a. Mary, bring two glasses of water for our guests! b. Add two glasses of water to the soup! Despite the surface similarity of these pseudopartitive constructions, Rothstein 2009, 2010b shows that different grammatical constructions are associated with each reading, and that the numerical is interpreted differently in each case. In (15a) two glasses of water is interpreted with its individuating or container reading, and denotes pluralities of glasses whose cardinality is 2 which contain water. On this reading, two is a predicate expression, giving the cardinality property of the plurality of glasses. (15b) illustrates the measure reading. Here two glasses of water denotes quantities of water, whose measure on the scale of volume, calibrated in terms of glass-measures, is two. Two is a numeral interpreted at type n, and denotes a number, indicating a value on the scale. Rothstein 2009 shows that these readings can be disambiguated by a variety of tests.
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