Classes of Modifiers in Formal Semantics

Recall from Section 2.2.1 that, in formal semantics, modification is modeled as function application, and a common noun is modeled as the characteristic function of a set of entities in the universe U (Heim and Kratzer (1998)). Naturally, in a compositional model of semantics, we would like such an interpretation to hold uniformly for all noun phrases. Therefore, the interpretation of a modified noun M H should similarly be a set of entities inU.

JM HK={e∈ U |e is aM H} (2.2)

Traditionally, modifiers (specifically, adjectives) have been classified taxonomically accord- ing to the set-theoretic relationship between the denotation of the head noun H and that of the modified noun M H. In this way, Kamp and Partee (1995) categorizes modifiers as either subsective or non-subsective. Beyond this top-level distinction, some subsective modifiers can be defined more specifically asintersective, and all non-subsective modifiers can defined more specifically as either privative or plain non-subsective. These standard classes of modifiers are described below and summarized in Figure 5.

Privative Non-Subsective (e.g. fake)

Plain Non-Subsective (e.g. alleged)

Subsective (e.g. good)

good

cars

cars

thieves

alleged _thieves

fake

guns

guns

Intersective (e.g. red)

red

cars

red

cars

Figure 5: Classes of modifiers in formal semantics.

Subsective Modifiers

Subsective modifiers are modifiers which pick out a subset of the set denoted by the un- modified noun; that is:

M is subsective⇔_JM HK⊂JHK (2.3)

Subsective modifiers are formalized as a functions of type << e, t >, < e, t >> which take as input the set denoted by the head noun H and return the set denoted by the modified noun M H, i.e. the subset of elements in _JH_Kfor which the modifier holds:

JM HK=JMK(JHK) ={e∈JHK|eisM} (2.4)

We will discuss computational ways to operationalize Equation 5.2 in Chapter 5.

Note that, often, the meaning of a modifier is dependent on the meaning of the head noun being modified. For example, “good” is subsective: if eis a “good student”, eis not necessarily a “good person” in general. This example illustrates the difficulty in modeling

whether “e is good” in an absolute sense, independent of the context H in which e is being considered. In general, determining whether or not “eis M” can be deeply context dependent. We discuss this issue further in Section 2.3.3 and in Chapters 4 and 5.

Intersective Modifiers. Some subsective modifiers are consideredintersective. Intersec- tive modifiers can be interpreted as the characteristic function of a set, in the same way that a noun can. That is, intersective adjectives can naturally be formalized as functions of type < e, t >. For intersective adjectives, the denotation of M H is simply the intersection of the set denoted by M and the set denoted byH:

M is intersective⇔_JM HK=JMK∩JHK (2.5)

For example, “red” is intersective in the context of“car”, so if e is a “red car”, e is both “red” and a “car”. Note that intersective modifies are just a special case of subsective modifiers as defined in Equation 5.2 in which the question of whether “e isM” equates to determining whether e∈_JM_K.

Non-Subsective Modifiers

For non-subsective modifiers, in contrast, the denotation of M H is not a subset of H. Thus, formalizing the denotation of M as a function of type << e, t >, < e, t >> is less straightforward, as determining the domain of this function presents challenges: if e ∈ JM HKdoes not guarantee e∈JHK, we cannot constrain the domain of JMKto be JHK, as

we could when M was subsective (Eq. 5.2). Formalizing the denotation of _JM_K in general is beyond the scope of this thesis. Rather, for non-subsective modifiers, we will focus only on the entailment relations between the set denoted by _JM H_Kand that denoted by _JH_K.

Plain Non-Subsective Modifiers. When M is plain non-subsective, there is no guar-

denoted by the noun alone, i.e.:

M is plain nonsubsective⇔(JM HK6⊂JHK)∧(JM HK∩JHK6=∅) (2.6)

In other words, if e ∈ _JM H_K, there are possible worlds in which e ∈ _JH_K and there are possible worlds in which e6∈_JH_K. The same is true ife6∈_JM H_K: there are possible worlds in which e∈_JH_K and there are possible worlds in whiche6∈ _JH_K. The modifier “alleged” is quintessentially plain non-subsective, since, for example, an “alleged thief ” may or may not be a true “thief ” and a true“thief ” may or may not be an an“alleged thief ”.

Privative Modifiers. Privative modifiers are defined as modifiers for which the set de- noted by the modified noun is completely disjoint from the set denoted by the unmodified noun: i.e.:

M is privative⇔_JM H_K∩_JH_K=∅ (2.7) “Fake”is considered to be a quintessentially privative adjective since, by the usual definition of “fake”, a “fake gun” is expressly not a“gun”.

While the notion of privative adjectives as described above is widely accepted and often applied in NLP tasks (Amoia and Gardent (2006, 2007); Boleda et al. (2012); McCrae et al. (2014); Angeli et al. (2015)), recent linguistic theories have taken the position that in fact privative adjectives should be considered as simply another type of subsective adjective (Partee (2003); McNally and Boleda (2004); Abdullah and Frost (2005); Partee (2007)). Under this argument, the denotation of the noun H should be expanded to include those entities which belong to M H, so that the domain of JMK can be simply JHK, as when M is subsective. This expanded denotation is used to account for the acceptability of the sentence “Is that gun real or fake?”, which is difficult to analyze if e ∈ _J“gun”_K entails

e 6∈ _J“fake gun”_K, as holds under the traditional definition of privatives. In more recent theoretical work, Del Pinal (2015) has argued that common nouns have a “dual seman- tic structure” and that non-subsective adjectives modify part of this meaning (e.g. the

functional features of the noun) without modifying the extension of the noun. Under this analysis, we can interpret a “fake gun” as having many, but not all, of the properties of a “gun”. Within NLP, there have been similar efforts to characterize privative modifiers more robustly. Nayak et al. (2014) categorize non-subsective adjectives in terms of the proportion of properties that are shared betweenH andM H and Pustejovsky (2013) focus on syntactic cues for exactly which properties are shared.