Chapter 2 Literature reviews
2.2 The Semantics of the Positive Form of Adjectives
In the formal semantics literature, it is widely assumed that gradable predicates do not themselves denote properties of individuals; rather, they map objects onto abstract representations of measure (i.e., scales) formalized as sets of values (i.e., degrees) ordered along some dimension (e.g., height, length, width) (see e.g., Cresswell 1977, von Stechow 1984, Heim 1985, Kennedy 1999, Graff 2000, Barker 2002, Kennedy & McNally 2005, Kennedy 2007a and Kennedy 2007b). In such a degree analysis of gradable predicates (in contrast to “the vague predicate analysis”),
a gradable adjective expensive is given a denotation like (7), where tall represents a measure function that takes an individual and returns its value, a degree on the scale associated with the adjective, so that tall(x) represents x’ height2.
(7) [[ tall ]] =λdλx.tall(x) ≥d
Pursuant to Graff (2000), Barker (2002), Kennedy & McNally (2005) and Kennedy (2007a), most gradable predicates have contextually dependent interpretation in the positive form (with a few exceptions). In addition, the positive form of a gradable adjective lacks overt morphology, in contrast to its comparative form (i.e., more expensive and wider).
(8) a. This elephant is small.
b. This ant is big.
(8a) could be judged true if asserted as part of a discussion about the size of elephants, but false in a discussion about the size of an ant versus an elephant. Likewise, (8b) could be judged true if asserted as part of a discussion about the size of ants, but false in a discussion about the size of an ant versus an elephant. One possible explanation
2 In fact, the denotation given in (7) is the relational analysis; under such a view gradable predicates are analyzed as relations between individuals and degrees. On the other hand, some authors noted above treat gradable adjectives as functions from individuals to degrees (e.g., Kennedy 1999), as shown in (i).
(i) [[ tall ]] = λx.tall(x)
As pointed out in Kennedy (2005b:10), the crucial differences between the relational analysis and measure function analysis boil down to the following: “In the former, gradable adjectives introduce degree arguments which must be saturated to generate a property of individuals; while in the latter, gradable adjectives must combine with some other expression (possibly something that introduces a relation and a degree) in order to generate a property of individuals.”
In this thesis, I basically take the relational analysis, though shifting to the measure function analysis for the convenience of demonstrations on some occasion.
for this variability, as Kennedy (2005a, 2007a) and Kennedy & McNally (2005) argues, is to assume a degree morpheme pos (i.e., a covert positive morpheme) with a denotation in (9), where s is a context-sensitive function from measure function to degrees: it returns a contextually significant degree (i.e., the standard of comparison) of the gradable property measured by the adjective g.
(9) [[Deg pos]] = λgλx.g(x) ≥ s(g)
In other words, the positive form of adjectives is evaluated with respect to the context-sensitive function denoted by the covert positive morpheme: a DELINEATION FUNCTION (in the terminology of Kennedy) which maps a measure function to a degree that represents the standard of comparison based on the context of utterance. Furthermore, as pointed out in Graff (2000) and Kennedy (2005a), one fundamental semantic property of the positive form of a gradable adjective is that it is vague, and this vagueness leads to borderline cases: the cases in which it is not clear whether the predicate holds for the object or not (i.e., crisp judgment).
Most importantly, Kennedy (2005a) uses this semantic characteristic of the positive form to divide comparison in natural languages into two different modes, namely, explicit comparison and implicit comparison. Crucially, it is the latter that involves borderline cases (i.e., the cases leading to crisp judgment) but not the former.
The definitions of explicit and implicit comparison is illustrated in (10), and the relevant examples are demonstrated in (11) and (12) respectively (see also Kennedy, 2007a and 2007b).
(10) a. Implicit comparison
Establish an ordering relation between object x and y with respect to
gradable property g using the positive form by manipulating the context in such a way that the positive form true of x and false of y.
b. Explicit comparison
Establish an ordering relation between objects x and y with respect to gradable property g using special morphology (e.g.,, more/-er, less, or as) whose conventional meaning has the consequence that the degree to which x is g exceeds the degree to which y is g.
(11) Context 1: A 600-word essay and a 200-word essay (Kennedy 2005a:11) a. This essay is longer than that one.
long(e1) > long(e2)
b. Compared to that essay, this one is long.
long(e1) > s[e2](long)
(12) Context 2: A 600-word essay and a 590-word essay a. This essay is longer than that one.
long(e1) > long(e2)
b. ??Compared to that essay, this one is long.
long(e1) > s[e2](long)
Explicit comparison in (12a) simply requires an asymmetric ordering relation between the degrees to which two objects possess the relevant property (i.e., the length of essay), the crisp judgments thus are not problematic.
However, implicit comparison in (12b) requires the first novel to have a degree of length that is significant relative to the region of the length scale whose lower bound is the length of the second essay. In other words, the differences between the
two degree values of length (i.e., the differences between the length of 600 words and the length of 200 words), as shown in Context 1, must be significantly greater than some contextually determined threshold specifying the degrees of length of that essay.
Before leaving this section, I want to mention another common view on the meaning of comparative constructions in many recent analyses: the definite description of degrees (e.g., von Stechow 1984, Heim 1985, Kennedy 1999, Kennedy 2005a and Kennedy 2007b). The basic idea behind the view of definite description of degrees is that it presupposes an exactly-reading for the degree variable. That is, “Mary is d-tall”
abbreviates that Mary has exactly the degree d on the tallness scale. In an at least-reading for the degree variable the uniqueness presuppositions would not be satisfied.
(13) a. Mary is taller than Bill (is).
b. [the d: Mary is d-tall] > [the d’: Bill is d’-tall]
In order to derive definite descriptions of degrees, a maximality operator is introduced as an essential component of meaning of comparatives in many recent analyses.
(14) a. Mary is taller than Bill (is).
b. max [d: Mary is d-tall] > max [d’: Bill is d’-tall]
To be brief, in this paper, following some common views on gradable predicates in the formal semantics literature, I adopt a degree analysis of gradable predicates (i.e., specifically, a relational analysis). Furthermore, I assume a maximality operator as a basic component in an analysis of meaning of comparatives. In the next section, I briefly discuss the positive morpheme and the adjectival structure in Mandarin, by
reviewing the work of Liu (2010a).