Gradable Adjectives, Compounded Scales, Conjunction and Structured Meanings

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1 Gradable Adjectives, Compounded Scales, Conjunction and Structured Meanings Alan Bale Winter, Introduction General Issues: 1. What are the semantic properties correlated with being syntactically categorized as a Gradable Adjective? Are Gradable Adjectives inherently linked to degrees or rather are degrees a creation of the comparative and equative morphemes? 2. How should conjunction be interpreted? Should the interpretation of conjunction be a boolean operator (specifically a boolean meet operator)? Specific Issue: Traditional views on the interpretation of gradable adjectives (as degree functions or relations between individuals and degrees) and the interpretation of conjunction (as boolean meet or intersection) have difficulties accounting for the sentences in (1). (1) a. Jen is more beautiful and intelligent than Morag is. b. Jen is as tall and wide as Morag is. In this talk I will argue that sentences such as those in (1) are inconsistent with interpreting gradable adjectives as involving degrees while maintaining a 1

2 boolean interpretation of conjunction. One of the two interpretations must be given up. I will explore and evaluate two possibilities: 1. Gradable adjectives are interpreted as relations between individuals (much like the classical interpretation of transitive verbs). This interpretation of adjectives can account for the sentences in (1) while maintaining a traditional interpretation of conjunction. 2. Conjoined phrases are interpreted as structured meanings (as proposed by Winter, 1995). This interpretation of conjoined phrases can account for the sentences in (1) while maintaining a traditional interpretation of gradable adjectives. 2 Are Gradable Adjectives Measure Functions? Sentences like those in (2) can easily be given an analysis using degrees (or structured equivalents such as extents and delineations). (2) a. This table is longer than the door-frame is wide. b. This table is longer than the couch is. TWO THEORIES OF DEGREES 1. ADJECTIVES AS MEASURE FUNCTIONS * Kennedy (1999), following Bartsch & Vennemann (1972), suggests that adjectives should be interpreted as functions form individuals to degrees. (3) a. Jen is taller than Morag. b. MAX{d : d T [[tall]](j)} > MAX{d : d T [[tall]](m)} = [[tall]](j) > [[tall]](m) 2

3 2. ADJECTIVES AS RELATIONS BETWEEN INDIVIDUALS AND DE- GREES * Cresswell (1976) proposes that adjectives should be interpreted as relations between individuals and degrees. (4) a. Jen is taller than Morag. b. MAX{d :[[tall]](j, d)} > T MAX{d :[[tall]](m, d)} Simplifying: I will collapse the two theories into one shorthand... x is d-adj. This form will represent both possible interpretations of the adjective. * Measure Function Approach: x is d-adj will translate as d ADJ [[ADJ]](x). * Degree Relation Approach: x is d-adj will translate as [[ADJ]] (x, d). The sentences in (5a) and (6a) can be given the truth conditions in (5b) and (6b). (5) a. Jen is more beautiful than Morag is. b. M AX{d : Jen is d-beautiful} > M AX{d : Morag is d-beautiful} (6) a. Jen is as tall as Morag is. b. MAX{d : Jen is d-tall} MAX{d : Morag is d-tall} BACK TO THE SENTENCES PRESENTED IN (1): the interpretation of the sentences in (1) do not yield a meaning that is consistent with intuitions. (7) a. Jen is more beautiful and intelligent than Morag is. 3

4 b. M AX{d : Jen is d-beautiful and d-intelligent} > M AX{d : Morag is d-beautiful and d-intelligent} (8) a. Jen is as tall and wide as Morag is. b. MAX{d : Jen is d-tall and d-wide} MAX{d : Morag is d-tall and d-wide} The problem is two-fold: 1. By assumption, there are no degrees that are both degrees of beauty and intelligence. Thus the set of degree in (7b) would be empty. MAX would be undefined. The sentence is predicted to be anomalous. * Under both theories of degrees: [[beautiful]] [[intelligent]] = 2. The adjectives [[tall]] and [[wide]] do share a scale of lengths. However, there are still problems for conjoining these adjectives: * {d : Jen is d-tall and d-wide} is equivalent to {d : Jen is d-wide} since like most human beings Jen is taller than she is wide. * Similarly {d : Morag is d-tall and d-wide} is equivalent to {d : Morag is d-wide}. Consequently, (8b) is equivalent to... MAX{d : Jen is d-tall and d-wide} MAX{d : Morag is d-tall and d-wide}. This formula specifies the truth conditions of (9). (9) Jen is as wide as Morag is. BUT this is not what the sentence in (8a) means! 2.1 Reasons to think there is no ellipsis. Sentences with conjoined adjectives have the same truth conditions as sentences with conjoined comparative phrases. This raises the question of whether 4

5 ellipsis might account for the data presented above. (10) a. Jen is more beautiful and more talented than Pat is. b. Jen is more beautiful and talented than Pat is. Reasons to think that there is no ellipsis (gapping). * As (11) demonstrates, with canonical gapping constructions, the felicity of the ellipsis/gap is not affected by the addition of adjuncts between the two conjoined phrases as long as parallelism is maintained. (11) a. John fought with Betty and Fred with Suzan. b. John fought with Betty on Tuesday and Fred with Suzan on Wednesday. c. John fought with Betty in the mountain park on Tuesday and Fred with Suzan in the fountain park on Wednesday. * Also, consider non-canonical gapping constructions. The agreement facts in (12a) suggest that a determiner is elided in the second half of the conjunct. It is reasonable to suspect that (12a) underlyingly has the same syntactic form as (12b). (12) a. This fork and knife are gifts from my mother. b. This fork and this knife are gifts from my mother. * As with the canonical gapping constructions, non-cannonical constructions are not affected by the addition of adjuncts between the first and second half of the conjunct. This is demonstrated in (13). (13) a. This fork and knife that I used this morning are gifts from my mother. 5

6 b. This fork that I used yesterday and knife that I used this morning are gifts from my mother. * HOWEVER, the addition of adjuncts between the two halves DOES affect the conjoined gradable adjectives. (14) a. Jen is more beautiful and intelligent than Morag is. b. Jen is as tall and wide as Morag is. (15) a. Jen is more beautiful than Betty is and more intelligent than Morag is. b.? Jen is more beautiful than Betty is and intelligent than Morag is. (16) a. Jen is as tall as Betty is and as wide as Morag is. b.?? Jen is as tall as Betty is and wide as Morag is. (17) a. Jen is less intelligent and beautiful than Morag is. b. Jen is less intelligent than Betty is and less beautiful than Morag is. c.?? Jen is less intelligent than Betty is and beautiful than Morag is. * FURTHERMORE, in canonical gapping constructions, the comparative and equative morpheme cannot be elided without also eliding the adjective it modifies. (18) a. Morag and Patricia are quite beautiful and intelligent but [Jen is more beautiful than Morag] and [Betty more intelligent than Patricia]. b. Morag and Patricia are quite intelligent but [Jen is more intelligent than Morag] and [Betty than Patricia]. 6

7 c. * Morag and Patricia are quite beautiful and intelligent but [Jen is more beautiful than Morag] and [Betty intelligent than Patricia]. (19) a. Morag and Patricia are quite tall and wide but [Jen is as tall as Morag] and [Betty as wide as Patricia]. b. Morag and Patricia are quite intelligent but [Jen is more intelligent than Morag] and [Betty than Patricia]. c. * Morag and Patricia are quite tall and wide but [Jen is as tall as Morag] and [Betty wide as Patricia]. SUMMARY: An ellipsis or gapping account encounters two problems Constructions with conjoined adjectives do not have the same properties as other constructions with ellipsis or gapping. 2. In other canonical gapping constructions, the comparative can never be elided/gapped without also eliding/gapping the adjective. 3 Are Gradable Adjectives Binary Relations? SOLUTION: A potential solution to this problem involves rejecting the idea that Gradable Adjectives are inherently linked to degrees and adopting a position where adjectives are interpreted as binary relations between individuals... * Transitive and reflexive relations such as [[beautiful]] = { x, y : x has as much beauty as y} 2. [[intelligent]] = { x, y : x has as much intelligence as y} 3. [[wide]] = { x, y : x has as much width as y} 4. [[tall]] = { x, y : x has as much height as y} * These relations are called Pre-orders or Quasi Orders (QOs). 7

8 Consequences: * In such a theory, scales and degrees would be derived from the Gradable Adjectives by the comparative morpheme. * Combining Gradable Adjectives with boolean and allows for the creation of compounded scales that would give the correct truth conditions for the sentences in (1). Aside: Worries about the the QOs leading to semantic circularity: * In constructing scales from QOs, it seems as if one is defining a semantics for comparatives using a relation specified in terms of comparative and equative sentences. * BUT, there is a difference between the concept of comparison and the semantics of how comparative and equative sentences are given truth values. * CONCEPT OF COMPARISON WITHOUT LANGUAGE: The underlying QO does not require an analysis of comparatives to define the relation. All it assumes is that given two individuals, speakers can tell if one has as much of a certain property as the other. * Even those without language (monkeys, cats, dogs) are able to compare two objects or individuals in terms of a certain property (to tell which food bowl has more, or which potential mates are more suitable/beautiful). DETAILS OF THE SOLUTION 1. Deriving Scales From QOs (cf. Cresswell, 1976; Klein, 1991): The quotient structure of the QO can be used as a scale. * Building a quotient structure involves two steps: 8

9 (a) Individuals that are reflexively related are collapsed into equivalence classes. (b) The equivalence classes are ordered in a way that is congruent to the original QO, thus creating a scale. (An equivalence class A is ranked above an equivalence class B iff for all x that are members of A and all y that are members of B, x, y is a member of the QO but not y, x.) (20) Quasi Order Quotient Structure Conventional Representation a {a} d 1 = {a} b c {b, c} d 2 = {b, c} d {d} d 3 = {d} e f g {e, f, g} d 4 = {e, f, g} h i {h, i} d 5 = {h, i} j {j} d 6 = {j} 2. Deriving Compounded Scales: If beautiful and intelligent are both interpreted as QOs that encode relations between individuals, then their intersection would also be a QO (transitivity and reflexivity are preserved under intersection). * The nature of this intersected QO can be described in the following way: 9

10 *** x, y ([[beautiful]] [[intelligent]]) iff ( x, y [[beautiful]]) AND ( x, y [[intelligent]]) ** * This entails that x, y is a member of the intersected quasi-order iff x has as much beauty as y and x has as much intelligence as y. (21) B I = B I a c a b a b c b = c d f f e g d f h e g e g h d h * When this intersected QO is converted into a scale, the equivalence classes (and hence degrees) are ordered with respect to both properties.(an equivalence class X will be greater than Y iff every member 10

11 of X has more beauty and more intelligence than every member of Y.) (22) d a d b d c d f d d d e d g d h 3. Interpreting the Comparative: The comparative morpheme is a function that takes three arguments: a QO, a degree and an individual (symbolized by the variables P, d, and x respectively). The details of such an interpretation are given in (23) and (24). * Representing Degrees as ordered pairs (cf. Cresswell,1976): Any degree d is represented as an ordered pair d n, P, where d n is an equivalence class and P is an ordering relation that ranks equivalence classes in a scale. (23) a. MORE( d 1, P, d 2, Q ) is defined iff P = Q, when defined: MORE( d 1, P, d 2, Q ) = 1 iff (d 1 P d 2 )& (d 2 P d 1 ). 11

12 b. AS( d 1, P, d 2, Q ) is defined iff P = Q, when defined: AS( d 1, P, d 2, Q ) = 1 iff (d 1 P d 2 ). (24) a. [[more]] = λp λd λx MORE( x P /, P /, d) b. [[as]] = λp λd λx AS( x P /, P /, d) 4. THAN-CLAUSES: Than-clause can be interpreted as a degree (the exact degree being determined by the adjective and individual in the clause). * The than-clause in (1) than Morag is can be assigned the degree m (B I )/, (B I )/, where (B I) / is the intersection of the two quasi-orders [[beautiful]] and [[intelligent]] and m B I is the equivalence class containing Morag (m) relative to the intersection. * Likewise the second as-clause in the equative in (1) as Morag is can be assigned the degree m (T W )/, (T W )/, where T W is the intersection of the two quasi-orders [[tall]] and [[wide]]. * NOTE: Like others, I assume that the than-clause and as-clause contain a copy of the AP in the main clause ([beautiful and intelligent] and [wide and tall] respectively). 5. EXAMPLES: The interpretation of the sentences in (1) can be represented as in (25) and (26). (25) a. [[Jen is more beautiful and intelligent than Morag is]] = ( ( ( [[more]] [[beautiful and intelligent]]) [[than Morag is...]] ) [[Jen]] ) = ( ( ( ( (λp λdλx MORE( x P /, P /, d) ) B I ) m (B I )/, (B I )/ )j ) =MORE( j (B I )/, (B I )/, m (B I )/, (B I )/ ) 12

13 (Let the equivalence class containing Jen and the one containing Morag j (B I )/ and m (B I )/ be represented by d j and d m respectively. Let the partial order based on the relation (B I) / be represented by with no subscripts.) =MORE( d j,, d m, ) = 1 iff (d j d m ) & (d m d j ) b. INTERSECTING ADJECTIVES B I B I (B I) / h j h i h i d i j i j k m m d m l n k d k m o l d l d h d j n l n d n o k o d o (26) a. [[Jen is as tall and wide as Morag is]] = ( ( ( [[as]] [[tall and wide]]) [[as Morag is...]] ) [[Jen]] ) = ( ( ( ( (λp λdλx AS( x P /, P /, d) ) T W ) m (T W )/, (T W )/ )j ) =AS( j (T W )/, (T W )/, m (T W )/, (T W )/ ) (Let the equivalence class containing Jen and the one containing Morag j (T W )/ and m (T W )/ be represented by d j and d m respectively. Let the partial order based on the relation (T W ) / be represented by with no subscripts.) 13

14 =AS( d j,, d m, ) = 1 iff d j d m b. INTERSECTING ADJECTIVES T W T W (T W) / h h h d h i l i d i j i j d j k j l d l l n k d k m k m n m n d n o o o d o 4 What About Disjunction? Not only can adjectives be combined with and, they also can be combined with or... (27) a. Jen is more intelligent or beautiful than Morag is. b. Jen is as tall or wide as Morage is. Potential Problem: A boolean interpretation of disjunction cannot account for these sentences. 14 d m

15 * Combining two Quasi Orders with set-union does not always yield a quasi order. Transitivity is not preserved. * Without a Quasi Order, one cannot form a scale. Scales require transitivity. HOWEVER: As many others have pointed out (Alonso-Ovalle, 2006; Larson, 1985; Schwarz, 1999; Simons, 2005, 2006; Zimmermann, 2001), there are many contexts where disjunction seems to take wider scope than its surface position. FURTHERMORE, the wide scope of disjunction cannot be completely reduced to ellipsis/gapping (contrary to the proposal of Schwarz, 1999). The non-boolean interpretations of disjunction can derive the correct truth conditions for the sentences in (27) no matter how the adjective is interpreted. BUT is there an asymmetry between the two types of coordinations? conjunction Boolean but not disjunction? Is 5 Is Conjunction a Boolean Operator? SOLUTION: An alternative solution is to reject that conjunction is interpreted as a boolean meet operator. Winter (1995) advances such a proposal to account for some instances of wide scope interpretation for conjunction. WIDE SCOPE CONJUNCTION: Winter notes the following contrast between OR and AND (28) a. John sold and bought a car. (from Hendriks 1993) Possible Paraphrase: John sold a car and bought a car. b. John sold and bought ten shirts. Possible Paraphrase: John sold ten shirts and bought ten shirts 15

16 (29) a. John sold or bought ten books. Incorrect Paraphrase: John sold ten books or bought ten books. b. John sold or bought every book. Incorrect Paraphrase: John sold every book or bought every book. FOUR ASPECTS OF WINTER S PROPOSAL 1. AND IS MEANINGLESS: The word and is not interpreted! 2. STRUCTURED MEANINGS: Any two syntactic objects of the same type, X and Y can combined to form an ordered pair X, Y. This follows from the structured meanings discussed in Cresswell (1985). 3. POINTWISE COMPOSITION: A function Z of type γ, δ can apply to the structured meaning X, Y (where X and Y are of type δ) pointwise. Thus Z( X, Y ) = Z(X), Z(Y ). * Note: Winter did not discuss cases where Z is of type δ and X and Y of type γ, δ. However pointwise composition would be trivial in this case as well: Z( X, Y ) = X(Z), Y (Z) 4. GENERALIZED CONJUNCTION: At any point in the derivation, a meet operator can apply to the members of a structured proposition. Thus optionally, X, Z can become X, Y, where X, Y = X Y. Winter s proposal accounts for wide scope of conjunction. [[John bought and sold a car]] =[[John]] (([[bounght]] and [[sold]]) [[a car]] =[[John]] (( [[bought]], [[sold]] ) [[a car]]) (by Structured Meanings) =[[John]] ( [[a car]]([[bought]]), [[a car]]([[sold]]) ) (by Pointwise composition) =[[John]] (λy x(car(x) &BOU GHT (y, x)) λy x(car(x) &SOLD(y, x)) (by Generalized Conjunction) = x(car(x) &BOUGHT (j, x)) & x(car(x) & SOLD(j, x)) 16

17 Winter s proposal accounts for the conjunction facts with gradable adjectives. [[ Jen is more beautiful and intelligent than Morag is]] =[[Jen]] ( ([[more]] [[beautiful]], [[intelligent]] ) [[than Morage is]]) =[[Jen]] ( ( [[more beautiful]], [[more intelligent]] ) [[than Morage is]]) =[[Jen]] ( [[more beautiful than Morag is]], [[more intelligent than Morag is]] ) = [[Jen]] ([[more beautiful than Morag is]] [[more intelligent than Morag is]] ) = [[Jen is more beautiful than Morag is]] & [[Jen is more intelligent than Morag is]] PROBLEM #1: Quantifiers in the subject position do not seem to allow for the Wide Scope interpretation of conjunction. The sentences in (30) cannot be true when the ten boys that bought the car and swam are different from the ten boys that sold the car and ran. (30) a. Ten boys bought and sold that car. b. Ten boys ran and swam. In fact, these are classic examples of why ellipsis cannot explain all conjunction facts. * NOTE: The same facts do not hold for disjunction. (31) a. Fewer than ten boys bought or sold that car (I don t remember which). b. Fewer than ten boys ran or swam (I don t remember which). PROBLEM #2: Could a Structured Meaning Hypothesis account for the differences between the following sentences. (32) a. Jen is more beautiful and intelligent than how beautiful and intelligent Morag is. 17

18 b.? Jen is more beautiful and intelligent than how beautiful and how intelligent Morag is. (33) a.? Jen is more beautiful and more intelligent than how beautiful and intelligent Morag is. b. Jen is more beautiful and more intelligent than how beautiful and how intelligent Morag is. PROBLEM #3: There is some evidence that suggest that and and or are not both boolean but with the opposite characterization given by Winter. (See Hulsey, 2006) * OR seems to have a non-boolean interpretation. * AND seems to have a boolean interpretation. 6 Tentative Conclusions? I have argued that one of the following proposals should be adopted Gradable Adjectives are interpreted as binary relations between individuals and do not directly involve degrees. 2. Conjunction is not interpreted as a Boolean operator. References [1] Alonso-Ovalle, L. (2006). Disjunction in Alternative Semantics. PhD dissertation, UMass Amherst. [2] Bartsch, R. and Vennemann, T. (1972). Semantic Structures. Athenaum, Frankfurt. [3] Cresswell, M. (1976). The Semantics of Degree. In: B. Partee (ed.), Montague Grammar. Academic Press, New York. [4] Cresswell, M. (1985). Structured Meanings. MIT Press. 18

19 [5] Hendriks, H. (1993). Studied Flexibility. Ph.D. diss., University of Amsterdam. [6] Hulsey, S. (2006). An Argument from Gapping for a Hamblin Semantics for Disjunction. Presentation given at NELS 37. [7] Johnson, K. (2004). In Search of the English Middle Field. Manuscript. UMASS. Kennedy, C. (1999) Projecting the Adjective. Ph.D Dissertation, UCSC. [8] Klein, E. (1991). Comparatives. In: A. von Stechow, D. Wunderlich (eds.), Semantik/Semantics: An International Handbook of Contemporary Research. Waler de Gruyter. [9] Larson, R. (1985). On the syntax of disjunction scope. Natural Language and Linguistic Theory 3, pp [10] Schwarz, B. (1999). On the syntax of either...or. Natural Language and Linguistic Theory, 17: [11] Simons, M. (2005). Dividing things up: The semantics of or and the modal/or interaction. Natural Language Semantics 13, [12] Winter, Y. (1995). Syncategorematic conjunction and structured meanings. Proceedings of Semantics and Linguistic Theory (SALT) 5. [13] Zimmermann, T.E. (2000). Free choice disjunction and Epistemic Possibility. Natural Language Semantics 8(4), pp