The semantics of adjectives

Gradable adjectives are traditionally assumed to denote two-place relations connecting individuals with degrees (see Seuren 1973, Cresswell 1976, von Stechow 1984, Heim 1985, Bierwisch 1989, Klein 1991, Kennedy and McNally 2005a, b). To put it more precisely, a gradable adjective like expensive has the denotation in (103),

where expensive represents a measure function that takes an entity and returns its cost, a degree on the scale associated with the adjective.

(103) [[expensive]] = λdλx. expensive(x) = d

The adjective expensive thus denotes a relation between degrees of cost d and objects x such that the cost of x equals d.

Under such an approach, degree morphology ― in English, comparative morphemes, degree modifiers, measure phrases, and the phonologically null positive degree morpheme pos ― that saturates and imposes restrictions on the degree argument determines the value of the degree argument (see Kennedy and McNally 2005b).⁹

In the following two subsections, we briefly introduce Kennedy and McNally’s (2005a) analysis of degree modifiers and Schwarzchild’s (2004) analysis of measure phrases to explain how degree modifiers and measure phrases restrict the degree argument of the adjectival predicate.

4.3.1.1.1 Degree adverbs

According to Kennedy and McNally (2005a), degree morphemes whose role is to saturate the degree argument of the adjective denote functions from (gradable) adjective meanings to properties of individuals, that is, they are of type <<d, <e, t>>,

<e, t>>. The template in (104), where R is some restriction on the degree argument of

9 Following von Stechow (1984), Kennedy and McNally (2005a: 350) assume that unmodified APs contain a null degree morpheme pos encoding the relation standard, which holds of a degree d just in case it meets a standard of comparison for an adjective G with respect to a comparison class determined by C, a variable over properties of individuals whose value is determined contextually, as shown in (i). Furthermore, the requirements imposed by the standard relation, as Kennedy and McNally (2005a: 350) argue, must vary depending on the lexical features of the adjective.

(i) [[pos]] = λGλx.∃d[standard(d)(G)(C)∧G(d)(x)]

the adjective, is the characterization of the meanings of degree morphemes.

(104) [[Deg(P)]] = λGλx.∃d[R(d)∧G(d)(x)]

It is the value of R that distinguishes different degree morphemes from each other.

Kennedy and McNally (2005a) argue that the distribution and interpretation of degree modifiers are sensitive to the scale structure (open versus closed) and standard value (relative versus absolute) of the expressions they modify. To put it more precisely, proportional degree modifiers are acceptable with closed-scale (or absolute) adjectives while non-proportional ones with open-scale (or relative) adjectives. For example, the proportional modifier half has a denotation along the lines of (105a), where SG represents the scale associated with a gradable adjective G and diff is a function that returns the difference between two degrees, so that the modifier half is compatible only with adjectives that map their arguments onto scales with maximal and minimal elements. The example in (105b), where the adjectival predicate half visible has a denotation like (105c), in which the degree argument of the closed-scale adjective visible is saturated and restricted by the proportional degree adverb half, is therefore grammatical.

(105) a. [[half]] = λGλx.∃d[diff(max(SG))(d) = diff(d)(min(SG ))∧G(d)(x)]

b. The figure was half visible.

c. [[half]]([[visible]])

= λx.∃d[diff(max(Sv))(d) = diff(d)(min(Sv))∧visible(x) = d]

For non-proportional degree modifiers, let us consider very as an example.

According to Klein (1980), a predicate of the form very A is analyzed in essentially

the same way as its simple, unmodified counterpart, with one important difference:

whereas the regular contextual standard is a degree that exceeds a norm or average of the relevant property calculated on the basis of an arbitrary, contextually determined comparison class, the very standard is a norm or average calculated in the same way but just on the basis of those objects to which the unmodified predicate truthfully applies (see von Stechow 1984, and Kennedy and McNally 2005a). For example, in a context in which the standard of comparison for the adjective (phrase) tall is the average degree of height for the comparison class basketball players, the standard of comparison for the AP very tall is an average of height for just the tall basketball players.

Kennedy and McNally (2005a) implement Klein’s (1980) analysis by analyzing very in terms of the standard relation which requires the degree argument of an adjective G to exceed a norm for a comparison class that has the property G in the context of utterance, as made explicit in (106), which specifies the denotation of very relative to a context c.

(106) [[very]]^c = λGλx.∃d[standard(d)(G)(λy.[[pos(G)(y)]]^c)∧G(d)(x)]

The reason for the restriction of very to relative adjectives is that modification by very has the effect of raising the standard for relative adjectives while it has absolutely no semantic effect for absolute adjectives whose standard is always fixed to the appropriate endpoint of the scale regardless of comparison class. The example in (107a), where the adjectival predicate very expensive has a denotation like (107b), in which the degree argument of the open-scale adjective expensive is saturated and restricted by the non-proportional degree adverb very, is therefore grammatical.

(107) a. The coffee at the airport is very expensive. up to Bill’s. A measure phrase can be used to tell us what the size of that gap is. If Bill is 3 inches taller than John, then it is a three-inch gap. This fact can be expressed with the formula in (108b).¹⁰

(108) a. Bill is [3 inches] taller than John.

b. ∃ hb∃h_j hb = UpLim({d: tall’(b, d)}) ∧ hj = UpLim({d: tall’(j, d)}) ∧ 3-inches’([UpLim({d: tall’(j, d)}), UpLim({d: tall’(b, d)})])

A measure phrase is a predicate of a set of degrees, in the case of the comparative this set is just the gap between the two degrees quantified over by the comparative.

Measure phrases can appear with non-compared adjectives: 5 feet tall.¹¹ Like event modifiers in extended NPs and in VPs, the measure phrase predicates of a degree argument of the adjective. But given the kind of meaning a measure phrase

10 There are at least two ways to understand tall’(x, d) corresponding to the two glosses in (i), the latter following a suggestion in Kamp (1975).

(i) a. tall’(x, d) “x’s height is exactly d”

b. tall’(x, d) “x’s height exceeds d”

Given the exceeds reading adopted by Schwarzschild (2004), for any x, tall’(x, d) is satisfied by many degrees: all those that lie below x’s height. It is the upper limit for this set that is relevant to the comparative. So a formula as in (108b) is needed.

11 Liu (2007: 69-71) points out that, like English, the ability of an adjective in Mandarin Chinese to combine directly with a measure phrase for forming a “measure phrase adjective” pattern turns out to be lexically idiosyncratic because only adjectives like gao ‘tall/high’, kuan ‘wide’, shen ‘deep’, hou

‘thick’, da ‘old’, chang ‘long’, and zhong ‘heavy’ form such patterns.

must have to do its job in comparatives, it is not of the right type to directly predicate of a degree argument of an adjective. Schwarzchild (2004) proposes a lexically governed type-shift which applies to some adjectives allowing them to combine with a measure phrase. Specifically, Schwarzchild (2004) proposes that some adjectives must undergo a lexical rule that produces homonyms and these homonyms must have interval arguments (sets of degrees) in place of degree arguments. Such a rule is given in (109).

(109) Homonym Rule: from degrees to intervals

If A has meaning A’ (i.e. A1’) that relates individuals to degrees, then A has a secondary meaning (i.e. A₂’) relating individuals to sets of degrees (intervals).

The secondary meaning is given by: λI.λx. I = {d: A’ (x, d)}

Homonym Rule applies to tall, wide, deep, thick, old, long, high.

Given the Homonym Rule, example (110a) has a semantic structure in (110b), which is equivalent to (110c).

(110) a. John is [5 feet] tall.

b.∃I [tall2’(j, I)∧5 feet’(I)]

c. 5 feet’({d: tall1’(x, d)})