4.1. The data

Let us start with a basic similarity and a basic distinction between adverbs and adjectives: both are modifiers of some sort. However, adjectives are noun modifiers, while adverbs are verbal or sentential modifiers. This descriptive difference renders them formally different when trying to give a formal semantic analysis even on the most basic level.

In section 3.3, we have used an early classification for adjectives, according to which these can be roughly classified into three main classes: intersective, subsective and non-subsective, with non-subsective adjectives being further subclassified into privative and non-committal. We used this classification as a case study for MTT-semantics and provided an account of these rough classes of adjectives using MTTs. However, useful as a first approximation, this tripartite distinction is definitely coarsegrained and unable to capture a number of more fine-grained distinctions involved in adjectival modification. To give an example of why this is so, consider the case of adjectives “short” and “excellent”. According to our initial classification, both are subsective adjectives. However, the first one is also a gradable adjective and, thus, can be modified by degree adverbs, as in very short, or form comparative/superlative forms as in (4.1), whereas the second adjective, “excellent”, is not gradable and thus does not exhibit these properties. Given this state of affairs, we have to make some extra assumptions in order to capture the semantics of gradable adjectives, for example in order to take care of inferences involving comparatives like the inferences shown as follows:

(4.1) John is shorter than George ⇒ if George is short, then so is John.

(4.2) John is 170 cm tall ⇒ if George is 160 cm tall, then George is shorter than John.

A standard assumption in the literature is that gradable adjectives involve some kind of measurement Usually, this measurement is assumed to be a degree argument, whose presence or absence is then considered to be the main difference between gradable and non-gradable adjectives. This extra argument has been proposed to be formally encoded in the adjective’s typing as in (Bartsch and Vennermann 1973; Von Stechow 1984; Heim 2000), or not formally encoded as in (Lewis 1970; McConnell-Ginet 1973; Klein 1980; Van Benthem 2013).²

Another issue in adjectival semantics concerns the so-called multidimensional adjectives. The problem with these adjectives is that they involve gradability across more than one dimension. For example, in the case of adjectives such as “narrow” the relevant gradable parameter is width. Adjective “narrow” is gradable across one dimension, monodimensional so to speak. On the other hand, multidimensional adjectives such as “healthy” and “sick” involve more than one gradable parameter across more than one dimension. More precisely, the intuition is that adjectives such as “healthy” quantify over a number of dimensions, e.g. blood pressure and cholesterol (Sassoon 2012). Even adjectives such as “big”, which at a first look seem to be similar to monodimensional adjectives such as “narrow”, are in fact multidimensional. This is because “big” may involve different dimensions such as height and width or even abstract grade parameters such as idiocy in cases such as “big idiot”. Multidimensional adjectives rely heavily on the context with respect to the dimensions that need to be satisfied in order for an utterance to be felicitous. Multidimensional adjectives can be further distinguished into positive and negative (Sassoon 2012): positive multidimensional adjectives include cases such as healthy, while negative multidimensional adjectives include cases such as sick. The difference between the two concerns the form of quantification over dimensions in each case: universal quantification over dimensions for positive multidimensional adjectives (4.3) and existential quantification over dimensions for negative multidimensional adjectives (4.4). This is based on the well-known observation that except phrases are compatible with universal quantifiers such as “all” but not with existential quantifiers such as “some” (Von Fintel 1993):

(4.3) Dan is healthy except with respect to blood pressure.

(4.4) # Dan is sick except with respect to blood pressure.

Given this difference between positive and negative multidimensional adjectives, the two classes also exhibit different inferential properties. A sentence such as “John is healthy” implies that John is healthy with respect to all “health” dimensions while for “sick”, what we get is that John is not healthy across some dimension. The following inferences exemplify the latter claims:³

(4.5) Dan is healthy ⇒ There is nothing wrong with Dan healthwise.

(4.6) Dan is sick ⇒ There is something wrong with Dan healthwise.

One other thing we are going to look at in this chapter with respect to adjectival semantics is context dependency. To get an idea of the issue, consider the following example taken from (Kennedy 2007):

(4.7) Coffee in Rome is expensive.

This example shows that what counts as expensive is highly context dependent and difficult to decide in a number of cases. Context dependency seems to be a more general issue associated with the interpretation of pretty much all linguistic elements, but in the case of gradable adjectives it refers to the fact that there is contextual variability in what it means for something to be an X. Thus, an assertion like (4.7) could be fine in a context comparing Rome to Lisbon but not in a context comparing Rome to Tokyo. The second case is in effect a case where one cannot make judgments of whether something is an X or not. So, it seems that for every gradable predicate there are clear cases where something is an X and clear cases where something is not X, but also cases where this cannot be decided. For example, think of the adjective “tall”: each of us can assign a collection of human beings as belonging to the collection of tall people, a set of people not belonging to that collection, but also borderline cases where this cannot be decided. An added complication arises in case there is interaction with context dependency with respect to a given class (for example “tall” within the context of basketball).

The second main issue we are going to look at in this chapter concerns adverbs. Similarly to adjectives, or even to a greater extent, adverbs seem to resist a homogeneous categorization. It is, thus, not surprising that various classifications have been proposed for adverbs throughout the years. According to one of the most prominent classifications, classification of Ernst and Maienborn (Ernst 2002; Maienborn and Schafer 2011), a tripartite classification can be used for adverbs with further subclassifications for each class: (1) predicational, (2) participant-oriented and (3) functional adverbs.⁴

Predicational adverbs comprise the main bulk of adverbs. Its main subcategories include sentence and verb-related adverbs. Sentence adverbs are further classified into subject or agent-oriented adverbs such as “arrogantly”, speaker-oriented adverbs that include speech-act adverbs such as “honestly”, epistemic adverbs such as “possibly” and domain adverbs like “botanically”, while verb-related adverbs include mental attitude adverbs such as “reluctantly”, manner adverbs such as “skillfully” and degree adverbs such as “deeply”. Participant-oriented adverbs, on the other hand, include adverbs (or adverbials) that introduce new entities to the situation/event described by the proposition in question. Examples of this type include cases such as “with a knife” and “with a gun”. Finally, functional adverbs include adverbs where some kind of quantification over the frequency of some event is involved such as “usually” and “never”. Now, besides this classification, there are a number of inferential properties associated with different adverbs that span across classes and can thus include members of different classes. One very basic property is veridicality. In plain words, a veridical adverb is an adverb that combined with a VP or proposition (depending on whether the veridical adverb is a VP or a propositional adverb) implies the VP or the proposition without the adverb:

(4.8) He opened the door slowly ⇒ He opened the door.

(4.9) Fortunately, he arrived on time ⇒ He arrived on time.

Another common inferential pattern is the one found with intensional adverbs, i.e. adverbs such as “intentionally” or “allegedly”. This type of adverbs are known to create what we call opaque contexts, or contexts exhibiting referential opacity. In simple words, substitution of co-referring terms in referentially opaque contexts does not preserve truth. Depending on whether the adverb is a VP adverb or a propositional modifier, we get opacity only for the object in the first case and for both the object and the subject in the second case. This is exemplified with “allegedly” and “intentionally” as follows:

(4.10) Oedipus intentionally married Jocaste ⇒ The son of Laius intentionally married Jocaste.

(4.11) Oedipus intentionally married Jocaste Oedipus intentionally married his mother.

(4.12) Oedipus allegedly married Jocaste The son of Laius allegedly married Jocaste.

(4.13) Oedipus allegedly married Jocaste Oedipus allegedly married his mother.

(4.10) and (4.11) exemplify the situation found with intensional VP adverbs: opacity is only created for the object (4.10) but not for the subject (4.11). On the other hand, (4.12) and (4.13) exemplify the situation with intensional sentence adverbs: opacity is created for both the subject and the object.

Another set of inferences associated with adverbs concerns event modification. This relies on the Davidsonian assumption (Davidson 1967) that adverbs provide restrictions with respect to the event denoted by the sentence/proposition in each case.⁵ Adverbs according to these approaches are assumed to modify the event in some way. Typical cases exemplifying such a behavior are adverbs such as “beautifully”. Thus, in Davidsonian/neo-Davidsonian semantics manner adverbs are seen as modifiers over events. Informally, a sentence such as “Mary dances beautifully” means that there is a dancing event, of which Mary is the agent, and this event is a beautiful event. The problem here, as already noted in the literature, is that the manner of the event and not the event itself is modified. Thus, we might want to introduce manner in the ontology of types/basic predicates, something that has already been argued for by Dik (1972), Schäfer (2008), among others. If we take this to be a correct way to think of manner adverbs, then we should be able to get the following inferences:

(4.14) Mary dances beautifully ⇒ Mary dances in a beautiful way/manner.

Other types of adverbs that we are going to be looking at in this chapter include subject (or agent) oriented and speech-act adverbs. In the former case, we find adjectives such as “stupidly”, where not only the manner but also the agent of the event is at play:

(4.15) John stupidly opened the door ⇒ the manner that John opened the door was stupid.

On the other hand, speech-act adverbs seem to provide commentary with respect to the utterance. An adverb such as “frankly” seems to imply that a sentence of the form “frankly” P means something such as “I frankly tell you that P”:

(4.16) Frankly, he is an idiot. ⇒ I frankly tell you that he is an idiot.

This completes our introduction to the data concerning aspects of adjectival and adverbial modification we are going to attempt to provide accounts for. We will argue that the mechanisms afforded by MTTs can give us the necessary tools in order to deal with these aspects of adjectival and adverbial modification.

4.2. Gradable adjectives

Adjectives such as “small”/“large”, as already mentioned, are not only subsective (see section 3.3.2 for subsective adjectives), but involve a further distinguishing property, i.e. gradability. In general, by gradable adjectives we mean the class of adjectives that involve some kind of grading property/parameter that allows them to be quantified according to it. For example, in the case of “small”/“large”, the grading parameter is size. Gradable adjectives have comparative and superlative forms and can be further modified by degree adverbs (e.g.“much”). In the literature, and depending on whether the gradable parameter is formally encoded in the definition for gradable adjectives or not, two approaches can be found. Accounts that formally encode this parameter assume that gradable adjectives involve a degree argument, while non-gradable adjectives do not. For example, on the assumption that small and large are given lower predicate types (i.e. e → t), the modified typing in order to deal with gradability will be d → (e → t) (d stands for degree). On this view, the adjective “small” can be given a definition as follows:⁶

(4.17) small = λd : Degree.λx : e.height(x) ≤ d

Proponents of such an approach can be found in (Bartsch and Vennermann 1973; Von Stechow 1984; Heim 2000) among others. The other option for treating gradable adjectives is to assume that they involve the same typing as non-gradable adjectives. Then, the difference is that gradable adjectives, even though being predicates from individuals to truth values, further involve partially ordered domains. Gradable adjectives impose a partitioning on these partially ordered domains. For objects x that fall into the upper side of the domain imposed by adjective A, A(x) is true while for objects y on the lower side of the scale, A(y) is false. This is the approach that Kennedy (1999) calls the vague predicate approach. Proponents of such an approach can be found in (Lewis 1970; McConnell-Ginet 1973; Klein 1980; Van Benthem 2013) among others. The list of accounts for gradable adjectives is quite long to be fully mentioned and the interested reader is redirected to Kennedy (1999) for more information on these accounts and additional references. Another informative and more recent overview of the two approaches is (Lassiter 2014).

In what follows, we are going to propose a generic framework to deal with gradable adjectives using MTTs.⁷ The account is based on earlier treatments proposed by the authors with regard to comparative adjectives (Chatzikyriakidis and Luo 2014, 2017a) and it is furthermore in line with approaches where gradability involves an explicit grade parameter. The account makes use of indexed types, in particular CNs indexed by a degree parameter.⁸ Let us start by taking a look at the adjective “tall”. We assume that heights are measured by a type Height of numbers such as 1.70.⁹ We are then led to consider the family of types Human : Height → Type indexed by heights: Human(n) is the type of humans of height n. Then, we can define height that takes a number i and a human of height i and returns i:

(4.18) height : Πi : Height. Human(i) → Height

(4.19) height(i, h) = i

The type and definition for tall are then given as follows:

(4.20) tall : Πi : Height. Human(i) → Prop

(4.21) tall(i, h) = height(i, h) ≥ n

The above definition specifies that for any i of type Height, tall takes an argument of type Human(i) and returns the proposition that i, the height of the human, is greater than or equal to a natural number n. This natural number n stands for the contextually restricted parameter – humans taller than n are regarded as tall. In a similar fashion, we can define the comparatives, where the RHS of (4.23) is the same as i > j:

(4.22) taller_than : Πi, j : Height. Human(i) → Human(j) → Prop

(4.23) taller_than(i, j, h₁,h₂) = height(i, h₁) > height(j, h.₂)

From this definition, we can easily prove that, for example, if height(i, h₁) ≥ height(j, h₂) and tall(j, h₂), then tall(i, h₁).

Of course, “tall” can be used with types of non-humans: for example, we can talk about a “tall building” or a “tall cat”. On the other hand, uses such as “tall democracy” or “tall mind” do not seem to be felicitous, at least without some sort of contextual coercion. Using either Human(i) as the argument or “tall” or a polymorphic argument based over the universe CN will undergenerate and overgenerate, respectively. One can try to use a subuniverse of CN, CN_PHY that basically includes all physical objects (types PHY and its subtypes). In this respect, we can introduce the universe CN_PHY with the following introduction rule:¹⁰

With this rule and assuming that every physical object has a height, we are now in a position to upgrade the definition for tall (we assume that the argument A is implicit in the definition):

(4.24) tall : ΠA : CN_PHY . Πi : Height. A(i) → Prop

(4.25) tall(i, h) = i ≥ n

A natural question to ask is where do we get this parameter from. In what we have provided so far, it is just a number, which does not depend on anything. A more proper way is to posit that the value is dependent on the noun, the adjective and sometimes even some other contextual information, which in MTT-semantics are represented as a type, a predicate and a context (in type theory), respectively. We use polymorphism and type dependency in MTTs to realize this. First, we introduce the universe of (totally ordered) degree types, Degree. As examples of degrees, one would find in Degree types such as Height, Weight and Width, among many others. The inference rules of CN_G are given below, the second of which says that CN_G(D) is a subtype of CN and the third is an example of an introduction rule for CN_G:

We can now introduce the polymorphic standard, STND, in the following way: first, for any common noun A, let ADJ(A) be the type of syntactic forms of adjectives whose semantic domain is A. For instance, TALL : ADJ(Human), where TALL strands for the syntax of tall. Then, STND takes a degree D, a D-indexed common noun A and (the syntax of) an adjective whose domain is A, and returns the relevant standard for the adjective:

(4.26) STND : ΠD : Degree. ΠA : CN_G(D).ADJ(A) → D

We can now give a revised definition for an adjective such as tall whose type is (4.24):

(4.27) tall(i, h) = height(i, h) ≥ STND(Height, Human, T ALL)

Note that indexing on the noun by means of a degree gives us for free the fact that we are not talking about tallness in general but tallness with respect to the relevant class (represented by the type Human in the above example).¹¹ Furthermore, the polymorphic STND function is a more straightforward interpretation of Kennedy’s context sensitive function from measure functions (adjectives basically) to degrees (Kennedy 2007). We may consider standards that are dependent on contextual information: for example, whether something is regarded as an expensive car might depend on the place such consideration takes place. In that case, the STND function may take an additional parameter of locations that would take this into account.

REMARK.– Meaningful CN subuniverses. We take a meaningful universe (or subuniverse) in the context of linguistic semantics to mean a universe that helps us in the interpretation of NL semantics. For example, we have used the universe CN as the universe that makes polymorphism over all common nouns possible and allows adequate typing for phenomena such as VP adverbs and subsective adjectives to be provided:

(4.28) VP_ADV : ΠA : CN. (A → Prop) → (A → Prop)

(4.29) ADJ_SUBS : ΠA : CN.A → Prop

Furthermore, we have used the universe CN_PHY in our analysis of “tall” in order to restrict the domain of polymorphism to the subuniverse that includes all physical objects and their subtypes. Other similar useful subuniverses can be constructed in order to help us in our semantic representations. Consider, for example, the subsective adjective“skillful”. According to what we have been saying so far, it is of the type given in (4.30). Digging a bit deeper, we can see that “skillful” is not really compatible with arguments that are not of type Human, or at least of type Animal. For example, we cannot talk about a skillful carpet or a skillful democracy. Thus, we could update the definition for “skillful”, taking these issues into consideration. On the assumption that “skillful” is only relevant for human arguments, polymorphism is on the subuniverse CN_H, i.e. the universe including types Human and its subtypes:

(4.30) skillful : ΠA : CN_H.A → Prop

The important question is, of course, how can we decide what the relevant universe is in each case? Well, one way to do it is by linguistic investigation as typically done in formal linguistics, i.e. getting judgments of native speakers that will help us decide the elements of the universe to be formed. Another way to do that is to use existing lexical-semantics resources that might contain such information. For example, Chatzikyriakidis et al. (2017) experiment with a rich lexical-semantic network constructed using GWAPs (Games with a Purpose) (Von Ahn and Dabbish 2008) and JeuxdeMots (Lafourcade 2007) in order to extract information relevant for multi-typed systems, e.g. common noun types, subtyping relations, typings for predicates, etc. We believe that such connections should be explored in future work combining lexical-semantic information drawn from linguistic resources with rich formal semantics formalisms like the one we are describing in this book. However, we need to be careful when constructing universes. Some universes can be formally paradoxical even though they may seem justified from a linguistic perspective. Thus, a better way to put what we have been saying is the following: we construct meaningful universes based either on linguistic intuitions and/or information from lexical/semantic networks, but only when we can formally justify it, i.e. to prove meta-theoretically that the incorporation of the new universe into the original type theory is OK (e.g. logically consistent, among other properties). □

4.3. Gradable nouns

Gradable nouns concern cases where the gradable element is not an adjective, but rather a noun:

(4.31) John is an enormous idiot.

(4.32) He is a big stamp collector.

In (4.31)/4.32), the most natural reading is not one of large physical size but rather of the nominal holding to a high degree. Examples like these have been usually taken as evidence not only for gradability in the nominal domain but also for the existence of adnominal modifiers, i.e. the adjectives enormous and big in (4.31)/4.32) (Paradis 2001; Morzycki 2009). On the other hand, Constantinescu (2013) argues that size adjectives are always associated with size readings that, depending on the noun, can receive concrete or abstract size readings. The relevant arguments put forth by proponents of the adnominal modifier approach (notably Morzycki 2009) that claim gradable nouns should be analyzed on a par with gradable adjectives have a number of problematic aspects. For example, as Solt (2009) correctly notes, there is no well-defined class of adnominal modifiers, in contrast to what we found in the adjectival domain (too, very etc.), a fact noted by other researchers as well, e.g. Constantinescu (2013). What we want to propose here, building on ideas we have been developing throughout this book, is that gradable nouns are interpreted as types, in particular Σ types, where their first projection is an abstract degree parameter, while their second projection a CN type depending on this degree parameter. We assume that adjectives such as enormous are polymorphic predicates that their arguments extend over the universe CN and the indices of these arguments extend over the Degree universe. In what follows, we put these assumptions in action.

4.3.1. Gradable nouns as Σ-types

Indexed types, as already said, are a type of dependent types, i.e. families of types indexed by a parameter whose type is usually a simple one. We have used indexed types so far in our treatment of gradable adjectives. The question is whether we can extend the usage of indexed types to gradable nouns as well. We will argue that this is indeed possible. What we want to propose here is that the distinction between nouns and adjectives is still clear, with adjectives taken to be predicates and nouns types, but, at the same time however, we assume that gradable nouns such as idiot and gradable adjectives such as tall both involve a degree parameter, albeit an abstract parameter in the former case. A natural way to capture this idea, i.e. abstract nouns being types but still involving a degree parameter, is to use Σ-types and assume that the first projection is actually the abstract parameter. To do this, we consider the type family IHuman : Idiocy → Type indexed by idiocy degrees of type Idiocy : Degree, where Idiocy is a type whose objects form a total order and can be compared to each other by, for example, a ≥-relation: Then, idiot can be represented by means of (4.33):

(4.33) Idiot = Σi : Idiocy. IHuman(i) × (i ≥ STND(Idiocy, Human, IDIOT IC))

An idiot is thus a triple (i, h, p) where h is a human whose idiocy degree i is bigger than or equal to the standard of being an idiot. Note that this account has not only similarities with the ideas proposed in (Constantinescu 2013) but also brings out a connection with gradable adjectives in the sense that both gradable adjectives and gradable nouns involve a degree parameter. However, these two constructions are clearly different in terms of their formal status, adjectives being predicates while nouns types.

Let us now consider “enormous idiot”. The interpretation we want to get in this case is one where someone is an idiot to a very high degree. This means that this degree must be (much) higher than the degree of idiocy needed for someone to be considered an idiot (the standard STND(Idiocy, Human, IDIOT IC) in (4.33)). In order to capture that we first propose that enormous can be interpreted as having the following type, where PHY_D : CN_G(D) is the type of physical objects indexed by D:

(4.34) enormous: ΠD : Degree. ΠA : D → CN_G(D). Πd : D. A(D) → Prop

Then we propose the following definition for “enormous”, for D : Degree, A : D → CN_G(D), d : D, and a : A(D):

(4.35) Enormous(D)(A)(d)(a) = d ≥ STND(D, PHY_D,ENORMOUS)

We are now ready to interpret enormous idiot (D and A arguments are implicit):

(4.36) Enormous Idiot = Σh : Idiot. enormous((π₁(h),π₂(π₁(h))) in where STND(D, P HY_D,ENORMOUS) ≥ STND(Idiocy, Human, IDIOT IC)

Enormous idiot is thus a pair, where the first projection consists of a proof of being an idiot h (Idiot itself also a Σ-type, see (4.33)) and the second projection requires that the standard of idiocy associated with the first projection of the second projection of h is greater than the standard for enormous.

4.4. Multidimensional adjectives

Multidimensional adjectives are adjectives that can be quantified across different dimensions. Adjectives such as “sick” and “healthy” fall into this category. Following Sassoon (2012), two different classes of multidimensional adjectives are distinguished: positive and negative. The idea is that every positive multidimensional adjective has a negative counterpart, i.e. its antonym (e.g. healthy and sick). What is different between the two is the form of quantification over dimensions in each case. Positive adjectives involve universal quantification over dimensions, while negative adjectives involve existential quantification. For example, for someone to be considered healthy, he/she must be healthy in all dimensions, whereas sick, it suffices to be for someone to be considered sick across one dimension only. In order for this intuition to be borne out more clearly, the exception phrase except can be used. The interesting bit here is that this phrase is only compatible with universal quantification. As seen below, “healthy” is compatible with “except”, but “sick” is not:

(4.37) Dan is healthy except with respect to blood pressure.

(4.38) # Dan is sick except with respect to blood pressure.

This intuition can be implemented in an MTT setting using an inductive type for multiple dimensions. Consider an adjective such as “healthy”. In order for someone to be considered healthy, one must be able to universally quantify over a number of “health” dimensions: cholesterol, blood pressure, etc. To formalize this, we can introduce the inductive type Health of type Degree as follows:¹²

(4.39) Health : Degree = heart | blood_pressure | cholesterol

We assume that the adjective healthy is of the following type (we use Human as a simple type rather than a type-valued function as used earlier):

(4.40) healthy : Health → Human → Prop

We can now use this parameter as a primitive to define Healthy and Sick as follows:

(4.41) Healthy = λx : Human.∀h : Health. healthy(h, x)

(4.42) Sick = λx : Human.¬(∀h : Health. healthy(h, x))

4.4.1. Multidimensional adjectives: going more fine-grained

What we have presented so far with regard to multidimensional adjectives is the general intuition behind them. However, there are a number of complications that need to be addressed. First of all, the nature of the quantifier associated with positive adjectives does not seem to always be the universal quantifier. Sassoon and Fadlon (2017) define quantificational multidimensional adjectives in the following sense:

Quantificational adjectives like optimistic often involve counting of dimensions. As a default, entities fall under them iff they are classified under sufficiently many (e.g. some, most or all) dimensions. (Sassoon and Fadlon 2017)

Of course, this is not a problem in itself. We can modify the account with respect to different adjectives as involving different quantificational force:

(4.43) Healthy = λx : Human.[SOME,ALL,MOST] h : Health.healthy(h, x)

This sufficiently many though, can be context dependent. One can assume that the quantifier quantifies over relevant dimensions in specific contexts. In this case, we can use the notion of context in type theory to formalize this. Actually, the idea is that the inductive type will be defined differently in different contexts: for example, it will involve all conceivable health dimensions in a medical context and fewer dimensions in every day contexts. Thus, the definition of healthy can be overloaded, picking the relevant dimensions in each case (relevant means available in that context), in the same sense the second author has proposed for cases of homonymy (Luo 2011b).¹³

A further interesting discussion with respect to multidimensionality concerns multidimensional nouns. For example, a noun such as “bird”, at least according to theories such as prototype and exemplar theories,¹⁴ is argued to involve a rich couple of dimensions, i.e. in order for something to count as a bird, a couple of different dimensions (for example, dimensions such as winged, small and can breed) have to be taken into consideration. Then, the idea is that the conceptual structure of a noun such as “bird” will involve an ideal value for each dimension. A similarity measure is mapping entities to degrees, representing how far from the ideal dimensions of the prototype the values for the respective entities are. This is represented as a weighted sum. What is important here, skipping formal details, is the following:

The distances of x from the prototypical values in the different bird dimensions integrate into a unique degree in the given noun by means of averaging operations, like weighted-sums... (Sassoon and Fadlon 2017)

The above passage argues that dimensions integrate (another way of putting it is collapse) into a unique degree, and, thus, are not accessible for quantification as it is the case with multidimensional adjectives. Viewing common nouns as types seems to be compatible with this claim. The idea is as follows: in order for an object to be of a CN type, the standard of membership with respect to the weighted sum of its similarity degrees to the ideal values in the dimensions of the noun has to be exceeded.¹⁵ Actually, Sassoon and Fadlon (2017) revise later on their view, and talk about weighted products in the case of these type of nouns. Somewhere in the middle between this two types of multidimensionality, i.e. multidimensional adjectives such as “healthy” and multidimensional nouns such as “bird”, we find social nouns such as “linguist” and “artist”. These seem to behave like multidimensional adjectives, in that their dimensions seem to be accessible for quantification as pe the following example:

(4.44) He is an artist in many respects.

These examples are argued to represent intermediate cases, where the dimensions are integrated into a single degree, albeit the relevant operation is one of weighted sum and not product. The argument is that these dimensions are made easier available to quantification in these cases. This might then mean that the types become more elaborate in this cases. Consider the case for artist, and consider the inductive type for all its dimensions (we note them here as a₁,a₂,a₃ pending a more serious discussion of what these dimensions really are):¹⁶

(4.45) Inductive Art : D = a₁ | a₂ | a₃

Now, one can think that in cases where social nouns make their dimensions accessible, what happens is that some sort of quantification is at play in the form of a Σ-type, where the first projection is just a type Human and the second projection specifies that all dimensions of artistry hold of this human. Our definition for artist with these assumptions in line is given as follows:¹⁷

(4.46) artist = Σh : Human.∀a : Art.DIM_CN (h, a)

Notice that the above is still a type and not a predicate. We can think that the creation of such types should be in general available, as even non-social nouns, e.g. natural-kind nouns such as dog, can be sometimes, context allowing, used in a way that seems to make their dimensions available. For example, we can imagine a context where the following is true:

(4.47) My dog is a cat in most respects.

Thus, it seems that the operation to turn simple types into Σ-types that make their dimensions available is a more general operation and should be restricted with respect to context and general world-knowledge considerations.

There are far more issues to consider when we look at multidimensional adjectives (and nouns). One thing to be pursued in future work is a unification of the account for gradable adjectives and nouns with that of multidimensional adjectives and nouns by inserting gradability into the analysis of multidimensionality. A number of challenging issues are to be taken care of if such a move is to be made. For example, the definition for the polymorphic standard function we have discussed in section 4.2 will have to be extended so that, besides regular gradable adjectives and nouns, it also works with multidimensional adjectives and nouns. We will not pursue this here, but it will be interesting to see how this can be done formally. However, we cannot go into detail into all these issues here. This topic deserves a separate book in its own right. We direct the interested reader to Sassoon (2012) and Sassoon and Fadlon (2017) for literature review and a detailed exposition of the complexity of the phenomenon in question. However, it seems to us that MTT-semantics is suitable to provide a very promising way of dealing with multidimensional adjectives and nouns in their entirety.

With these ending remarks on multidimensional adjectives, we conclude the part of the chapter concerning adjectival modification and move on to discuss adverbial modification.

4.5. Adverbial modification

In this section, we take up the issue of adverbial modification and discuss ways to treat aspects of the semantics of adverbs within an MTT setting. There is not much research on treating adverbs in MTTs. The only papers specifically dealing with this sort of modification is Chatzikyriakidis (2014) and Chatzikyriakidis and Luo (2017a).¹⁸ However, adverbs have been also treated in Chatzikyriakidis and Luo (2014) as part of a discussion on natural language inference (NLI). There, a first approach of some aspects of adverbial modification such as veridicality, non-veridicality, adverbial typing and intensional adverbs among others has been attempted. In this chapter, we attempt to deal with a number of phenomena related to adverbial modification, either extending ideas found in previous accounts or propose new ideas.

The first question to resolve is the typing of adverbs within an MTT setting that takes CNs to be types rather than predicates. In particular, many-sortedness that goes hand in hand with this view needs to be taken into consideration. The proposals put forth so far are based on the second author’s proposal (Luo 2011a,b), subsequently followed in Chatzikyriakidis and Luo (2014); Chatzikyriakidis (2014); Chatzikyriakidis and Luo (2017a). There, two types are proposed, one for VP adverbs and one for sentence level adverbs, respectively: (1) VP adverbs receive a polymorphic type extending over the universe CN (4.48), and (2) sentences adverbs are just functions from propositions to propositions (4.49):

(4.48) ΠA : CN. (A → Prop) → (A → Prop)

(4.49) Prop → Prop

This is a useful proposal for the typing of adverbs, but does not capture semantic properties associated with classes of adverbs. Furthermore, a couple of type refinements might be needed for other types of adverbs, as we shall see. For the moment, let us start with one very common property exhibited by a number of adverbs: veridicality.

4.5.1. Veridicality

Veridicality is a very common property among adverbs. It is found in both VP and sentence level adverbs. In the case of sentence adverbs, it means that Adv(P) presupposes P, whereas in the case of VP adverbs V (P (x)) presupposes P (x).How are we going to capture this property? There is a straightforward way to do that. Let us exemplify the proposal with sentence adverbs first. What we need is a definition of veridical sentence adverbs according to which, given an argument Q : Prop, which interprets the sentence in question, returns the proposition that interprets the sentence modified by the sentence adverb. This property is defined as follows:

(4.50) VER_Prop(Q) = ∀P :Prop. (P ⇒ Q)

Given (4.50), we can prove the following two theorems:

(4.51) VER_Prop(Q) ⇒ Q, for any Q : Prop

(4.52) ∀P : Prop. (P ⇒ Q) ⇒ (P ⇒ VER_Prop(Q)), for any Q : Prop

Given (4.51), we can, for example, prove “Mary arrived” from “Fortunately, Mary arrived”, while (4.52) says that VER_Prop(Q) is the smallest (i.e. weakest w.r.t. ⇒) among the formulas that imply Q.

VP veridical adverbs can be treated in a similar way by performing the necessary modifications in the definition. Remember that VP adverbs in MTT-semantics are polymorphic higher order predicates extending over the CN universe. This has to be then be taken into consideration. The following definition does exactly that: for any A : CN, Q : A → Prop and x : A,

(4.53) VER_VP (A, Q, x) = ∀P : A → Prop. P (x) ⇒ Q(x).

The usual inferences associated with veridicality are also captured in the VP adverb case. Veridical adverbs, or the veridical property associated with a number of adverbs, seem to have a natural interpretation in MTT-semantics. But what about other semantic aspects of adverbs? At this point, we take a look at a number of other aspects of adverbial modification, taking a look at event adverbs first. This is an informal term we are going using to refer to the following three types of adverbs: manner, agent-oriented and speech-act adverbs. The assumption behind the use of the term is that the accounts we are going to propose for this type of adverbs will involve the use of events in some way or another.

4.5.2. Event adverbs: manner, agent-oriented and speech-act adverbs

We introduce the term “event adverbs” informally here to refer to three types of adverbs that we are going to discuss in this section that crucially rely on assumptions about events, at least in the way we are going to approach them. More specifically, there is a reliance on Davidsonian or neo-Davidsonian assumptions about the way linguistic semantics are to be interpreted. The main idea in Davidsonian or neo-Davidsonian semantics is that sentences involve event quantification, i.e. there is an additional first class citizen argument, an event argument, at play, when we are trying to give the semantics for NL sentences. For Davidson (1967), events are spatio-temporal properties located in space and time. In terms of the actual formal system to be used, Davidson argued for a very simple untyped, restricted version of first-order logic that consists of a collection of conjunctive statements for the semantics of NL sentences. To give an example, consider the following neo-Davidsonian analysis of the sentence “Mary buttered the bread with a knife”:

(4.54) ∃v. buttered(v, Mary, bread) ∧ with(v, knife)

One note is in order, before we proceed. Most of the accounts we are going to look at in this subsection are basically neo-Davidsonian rather than Davidsonian. The natural question to ask is, of course, what is new in neo-Davidsonian semantics. There are at least two differences compared to Davidsonian semantics, both already discussed in detail in Parsons (1990): (1) the introduction of an inventory of semantic roles, e.g. agent and patient, and (2) turning predicate arguments into arguments of semantic roles. Consider the neo-Davidsonian translation of example 4.54:

(4.55) ∃v. buttered(v) ∧ Agent(v, Mary) ∧ Patient(v, bread) ∧ With(v, knife)

To see an example of how neo-Davidsonian ideas have been used to analyze event adverbs, let us take the case of manner adverbs. Manner adverbs, a subcategory of predicational adverbs, are VP adverbs that constrain the way/manner the subject performs the action denoted by the VP. Classic treatments of manner adverbs usually assume some sort of event modification is at play. According to this view, an adverb such as “illegibly” will be interpreted as positing that the event under consideration is an illegible one. However, it is the manner of the event rather than the event itself that is illegible. Thus, such a proposal will not really capture the semantics we want. This consideration has led researchers to argue for the inclusion of manners in their semantic ontology (as semantic roles) as well (Dik 1972; Schäfer 2008). With this inclusion of manners, a more reasonable account can be put forth, as for example is done in Schäfer (2008) for the sentence “John wrote illegibly” (slightly modified):

(4.56) ∃v. Agent(john, v) ∧ write(v) ∧∃m. Manner(m, v) ∧ illegible(v)

So how are we going to treat this type of adverbs in MTTs? The first thing to look at is ontology. More specifically, the way manners are to be represented. The idea we will pursue here is based on dependent event types that has been proposed by Luo and Soloviev (2017) and will be discussed in section 7.2. The main intuition is that events can depend on one or more thematic relations. For example, for a : Agent, we can form Evt_A(a) the type of events dependent on agents. The same can be done for manners, assuming that manner can be seen as a thematic relation as well. Thus, for m : Manner, we can form Evt_M (m) the type of events dependent on manners. How would predicates look like in such a setting? A normal predicate in a non-dependent event setting can be assumed to be of type A → Event → Prop (with A : CN). Thus, a predicate with an event dependent on manners will be of type (A → Evt_M (m) → Prop) and a manner adverb will be a polymorphic predicate modifier with the following type:

(4.57) ADV_manner : Πm : Manner. ΠA : CN. (A → Evt_M (m) → Prop) → (A → Evt_M (m) → Prop)

Putting all these together, an adverb such as “illegibly” will be defined as follows: for any P : A → Evt_M (m) → Prop, x : A and E : Evt_M (m),

(4.58) illegibly(P, x, E) = P (x, E) ∧ illegible(E),

where illegible : Evt_M (m) → Prop. Needless to say, the veridical inference associated with manner adverbials is captured with the above entry. For example, the above definition will predict that a sentence such as “he wrote” always follows from “he wrote illegibly”.

Subject-oriented adverbs, or better agent-oriented adverbs given that these involve the agent and not the subject,¹⁹ are traditionally looked at on a par with manner adverbs. One of the main reasons for this is the existence of ambiguous readings, i.e. manner/agent-oriented, with a number of adjectives. Agent-oriented adverbs are always assumed to involve a property of the agent, rather than the manner associated with the action. Thus, the sentence “John stupidly called Mary” means that John’s act of calling Mary was stupid. In the literature, we find accounts like McConnell-Ginet (1982) where manner adverbs are treated as arguments of the verb and agent-oriented adverbs as predicate modifiers. Event based accounts take agent-oriented adverbs to involve some additional structure compared to manner adverbs. For example, Rexach (1997) mentions that a way to capture the difference between the two classes is to assume that, in manner adverbs, the adjective related to the adverb (e.g. for “illegibly”, “illegible”, and so on) is predicated of the event only, while in the case of agent-oriented adverbs, of both the event and the agent:

(4.59) ADVmanner = λP.λx.∃e. P(e, x) ∧ AdjADV1(e)

(4.60) ADVagent = λP.λx.∃e. P(e, x) ∧ AdjADV2(e, x)

Keeping the core of the analysis for manner adverbs, we want to propose the use of dependent event types for agent-oriented adverbs as well. In this case, however, instead of events dependent on manners only, we will use events dependent on both manners and agents. For a : Agent and m : Manner, Evt_AM (a, m) is the type of events whose agent and manner are a and m, respectively. With these assumptions in place, we can reflect the differences in the two classes of adjectives as shown below, with m : Manner, A : CN and a : Agent arguments suppressed, Adj_ADV1 : Evt_M (m) → Prop and Adj_ADV2 : Evt_AM (a, m) → Prop:

(4.61) ADV_manner = λP : (A → Evt_M (m) → Prop).λx : A. λE : Evt_M (m). P (x, E) ∧ Adj_ADV1(E)

(4.62) ADV_agent = λP : (A → Evt_AM (a, m) → Prop).λx : A. λE : Evt_AM (a, m).P (x, E) ∧ Adj_ADV2(E)

Lastly, we look at speech-act adverbs. This type of adverbs includes cases such as “honestly” and “frankly”. One way to approach their semantics is to think of them as providing commentary with respect to the utterance. Thus, the sentence “frankly, I do not know what to say”, roughly means “I frankly tell you that I do not know what to say”. This paraphrase that dates back to Schreiber (1972) can give us a way of looking at speech-act adverbs, which will also relate to the previous treatments given for manner and agent-oriented adverbs. One interesting account within this line of reasoning is given by Piñón (2013), who analyzes speech-act adverbs as making reference to individual manners of speaking. Based on this intuition, we can formalize an account within MTTs. First, we need to define the type of utterance events. We do that in the following sense: for u : Utterer and m : Manner, Evt_UM (u, m) is the type of events with utterer u and manner m. With these assumptions in line, the type of frank can be given as follows:

(4.63) frank : Πu : Utterer. Πm : Manner. (Evt_UM (u, m) → Prop)

Then, the adverb frankly can now be given the following definition: for any u : Utterer, m : Manner and v : Evt_UM (u, m):

(4.64) frankly(u, m, v) = λP : Prop. P ∧ frank(u, m, v)

and frankly thus defined is of the following type:

(4.65) frankly : Πu : Utterer Πm : Manner. Evt_UM (u, m) → Prop → Prop

In fact, all speech-act adverbs can be defined as above. According to the account just presented, a speech-act adverb is not of type Prop → Prop; instead it takes both an utterance event and a proposition as arguments, and returns a proposition.

4.5.3. Domain, evaluative adverbs

A subcategory of speech-act adverbs concerns domain adverbs. The assumption is that these adverbs restrict the propositional argument’s domain of application. For example, a sentence such as “botanically, tomato is a fruit”, means that the proposition tomato is a fruit holds in the context/domain of botanology. The meticulous reader might already have guessed that we can provide a treatment by using type theoretic contexts. Indeed, it has already been suggested by Chatzikyriakidis (2014) and Chatzikyriakidis and Luo (2017a), proposing the following definition for domain adverbs, according to which they are functions from propositions to propositions and specify that the propositional argument holds at the relevant domain that is considered to be a type theoretic context:

(4.66) ADV_dom = λP : Prop. Γ_domP

Such a treatment is however prone to hyperintensionality. The way out is to use a treatment similar to the one found in section 3.3.4 for non-committal adjectives, but at the same time avoids hyperintensional problems. We repeat in brief the account that was proposed for alleged in section 3.3.4: for any human being h : Human and any non-committal adjective Adj, the modal operator H_h,Adj is introduced, a predicate over propositions:

(4.67) H_h,Adj : Prop → Prop

One can think of H_h,Adj as a set of propositions. This idea can be naturally extended to domain adverbs in the following sense: for any domain adverb ADV_DOM, the modal operator Δ_ADVDOM is a predicate over propositions:

(4.68) Δ_D,ADVDOM : Prop → Prop.

An adverb such as “mathematically” can be defined using Δ_{MATHEMATICALLY} P, “botanically” using Δ_BOTANICALLY P and so on. Note that here the collection of propositions is not dependent on a human but on a domain in these types of adverbs. Hyperintensionality does not arise here in the following way: Δ_{MATHEMATICALLY} P is true iff P is in Δ_{MATHEMATICALLY} P . On the other hand, we cannot deduce Δ_{MATHEMATICALLY} Q from Δ_{MATHEMATICALLY} P if P is not definitionally equal to Q. If they are, then P and Q are the same thing. This idea will become even more clearer in a bit, when we will discuss a similar treatment for intensional adverbs.

This treatment can be naturally extended to evaluative adverbs as well. Evaluative adverbs express the stance or attitude of the speaker with respect to a proposition and include cases such as “fortunately” and “amazingly”. We can, thus, deal with evaluative adverbs in the following sense: for any h : Human and any evaluative adverb ADV_eval, define a predicate over propositions H_h,ADVeval such that H_h,ADVeval : Prop → Prop represents the collection of propositions relating to the speaker’s specific type of attitude denoted by the adverb. For example, for H_{h,fortunately}, this corresponds to the collection of propositions considered fortunate by the agent h.

4.5.4. Intensional adverbs

In this last section, we are going to have a look at a number of intensional adverbs and we will show how to deal with their associated opacity. It has been noted that epistemic adverbs, e.g. “allegedly”, create opaque contexts for both the subject and the object, while adverbs such as “intentionally” create opaque contexts only for the object:

(4.69) Oedipus allegedly married Jocaste.

(4.70) Oedipus intentionally married Jocaste.

From (4.69), we obtain:

(4.71) Oedipus allegedly married Jocaste the son of Laius allegedly married Jocaste.

(4.72) Oedipus allegedly married Jocaste Oedipus allegedly married his mother.

From (4.70), on the other hand, we obtain:

(4.73) Oedipus intentionally married Jocaste ⇒ The son of Laius intentionally married Jocaste.

(4.74) Oedipus intentionally married Jocaste Oedipus intentionally married his mother.

Chatzikyriakidis and Luo (2017a) proposed an analysis of the meaning of “intentionally” as follows: A intentionally P means that A has the intention P and furthermore fulfilled this intention, i.e. P holds. Given this assumption, the authors propose to introduce intention contexts, which represent an agent’s collection of intentions: an agent p’s intentional context is taken to be a number of entries x:A (A : Prop) that correspond to this agent’s intentions:

(4.75) D_p = x₁ : A₁, ..., x_n : A_n(x₁, ..., x_n−1)

Then, a generalized intention operator is built:

(4.76) I_pA = ΠD_p.A = Πx₁ : A₁...Πx_n : A_n(x₁, ..., x_n−1).A

With these in line, the proposal by Chatzikyriakidis and Luo (2017a) for intentionally is as follows:

(4.77) Intentionally = λP : Human → Prop. λx : Human.I_x(P (x)) ∧ P (x)

In the case of “Oedipus intentionally married Jocaste”, we get a paraphrase that Oedipus had the intention of marrying Jocaste and he did so. In (4.73), we see that the x is bound. If we assume Eq(Person, O, SoL) in the global context, then substituting O for x and then SoL for O, we get the following(M stands for married and J for Jocaste):

(4.78) Intentionally O (M(J)) = I_SoL(M(J(SoL)) ∧ M(J)(SoL)

Chatzikyriakidis and Luo (2017a) further claim Oedipus intentionally married his mother does not follow. They say that this is because we need to have M(O, MoO) in the intention context of Oedipus in order to do this. Assuming that the intention context of Oedipus is known and according to the standard reading of the story does not involve the aforementioned intention, then this does not follow. And, if we assume that Oedipus’ intention context is unknown, we cannot prove it nor disprove before this information becomes available. However, as in the case of domain adverbs, the same problems of hyperintensionality arise. The issue is as follows: the classical treatment of beliefs by Ranta (1994), which is used by Chatzikyriakidis and Luo (2017a) for intentions in the way sketched here, is prone to hyperintensional problems as Luo (2017) shows. The problem is that beliefs in this treatment are closed under derivability, and, thus, for example, if one believes P, he/she believes every proposition that is logically equivalent to P . This further extends to intentional contexts as defined here. Thus, if M(O, J) is part of Oedipus’ intentional context, then one can derive that Oedipus intentionally married his mother given that M(O, J) = M(O, MoO). The same holds in case M(O, J) = M(O, MoO) is available in the global context Γ. In order to take care of this issue, we define intentional contexts advocating the same idea used in this book for non-committal adjectives and domain/evaluative adverbs. We introduce a predicate over propositions H_h,ADVINTS : Prop → Prop, for any h : Human and any intensional adverb ADV_INTNS. For example, for intentionally, Hh,_{intentionally} corresponds to the collection of agent h’s intentions. Then, we can redefine Intentionally in the following sense:

(4.79) Intentionally = λP : Human → Prop. λx : Human.Hx,intentionally(P (x)) ∧ P (x)

For cases of opaque to both the subject and the object adverbs, we need a slightly different analysis. For example, allegedly can be defined as follows:

(4.80) Allegedly = λP : Prop. ∃h:Human, Hh,alleged(P )

As mentioned above, similarly to the discussion on “alleged” in section 3.3.4, H_h,alleged represents the collection of allegations made by a human h.

4.6. Final remarks on modification: vagueness

The discussion on the context dependency of adjectival modification is part of a more general discussion on vagueness. In simple words, vagueness makes deciding what counts for something to be an X, where X is a gradable predicate (usually an adjective), difficult. There are three main problems associated with vagueness, the first one already mentioned and addressed in this paper: (1) context dependency, (2) the existence of borderline cases and (3) the fact that vague adjectives (and predicates in general) give rise to the sorites paradox. In the way our account stands, we cannot capture vagueness. We believe that this kind of problem needs to involve some kind of probabilistic reasoning. Indeed, a couple of researchers have pointed this out and have produced a body of research to this direction (Cooper et al. 2015; Goodman and Lassiter 2015; Lassiter and Goodman 2017; Bernardy et al. 2018, 2019). It is not clear to us at this moment whether a system putting MTT-semantics and Bayesian reasoning is even feasible, and we definitely do not have the time or space to discuss this in this book. For the moment, we end the discussion here, pending more research in this direction, where insights from Bayesian semantics, machine learning and logical semantics can be combined in a meaningful way to provide answers to problems, and aspects of them, but not their entirety, can be described well by any of the three approaches.