Adjectival and Adverbial Modification: The View from Modern Type Theories
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Abstract
In this paper we present a study of adjectival/adverbial modification using modern type theories (MTTs), i.e. type theories within the tradition of MartinLöf. We present an account of various issues concerning adjectival/adverbial modification and argue that MTTs can be used as an adequate language for interpreting NL semantics. MTTs are not only expressive enough to deal with a range of modification phenomena, but are furthermore wellsuited to perform reasoning tasks that can be easily implemented (e.g. in proofassistants) given their prooftheoretic nature. In MTTsemantics, common nouns are interpreted as types rather than predicates. Therefore, in order to capture the semantics of adjectives adequately, one needs to meet the challenge of modeling CNs modified by adjectives as types. To explicate that this can be done successfully, we first look at the mainstream classification of adjectives, i.e. intersective, subsective and nonsubsective adjectives. There, we show that the rich type structure available in MTTs, along with a suitable subtyping framework, offers an adequate mechanism to model these cases. In particular, this modelling naturally takes care of the characterising inferences associated with each class of adjectives. Then, more advanced issues on adjectival modification are discussed: (a) degree adjectives, (b) comparatives and (c) multidimensional adjectives. There, it is shown that the use of indexed types can be usefully applied in order to deal with these cases. In the same vein, the issue of adverbial modification is discussed. We study two general typings for sentence and VP adverbs respectively. It is shown that the rich type structure in MTTs further provides useful organisational mechanisms in giving formal semantics for adverbs. In particular, we discuss the use of \(\varSigma \)types to capture the veridicality/nonveridicality distinction and further discuss cases of intensional adverbs using the type theoretic notion of context (i.e. without resorting to intensional typing). We also look at manner, subject and speech act adverbials and propose solutions using MTTs. Finally, we show that the current proof technology can help mechanically check the associated inferences. A number of our proposals concerning adjectival and adverbial modification have been formalised in the proof assistant Coq and many of the associated inference patterns are checked to be correctly captured.
Keywords
Type Theory Formal Semantic Proof Assistant Gradable Adjective Subtyping Relation1 Introduction
The main subject of study of this paper is the modelling of adjectival and adverbial modification in doing semantics based on modern type theories (or MTTsemantics for short) (Ranta 1994; Luo 2012). It is wellknown that adjectival and adverbial modification this a notoriously difficult issue to tackle adequately. The main reason behind this, is that the adjectival/adverbial classes are largely nonhomogeneous semantic classes where a strict classification according to semantic properties is quite a difficult task. In MTTsemantics, common nouns are interpreted as types rather than predicates. This poses a new challenge: for example, the question of how to model CNs modified by various classes of adjectives as types in an adequate way. One of the aims of this paper is to take up these challenges; for example, we study how to model modified CNs by means of various type constructors in MTTs and hence show that MTTs provide adequate tools for this endeavour. Since MTTs have rich type structures, employing MTTs for formal semantics also provides us with various benefits that are not available in the Montagovian simple typetheoretic setting as we will argue in this paper. Furthermore, prooftheoretically defined, MTTs are the basis of various type theory based proof assistants which provide us the proof technology that can be used for computerassisted reasoning based on MTTsemantics. We shall briefly illustrate this in terms of reasoning about modifications.
In this introduction, we shall first summarize the issues as regards adjectival and adverbial modification as discussed in the formal semantics literature, mostly within the Montagovian tradition. We want, in this respect, to exemplify the complex nature of the problem as well as to provide a background on the phenomena that we will attempt to account for. We shall then briefly discuss the challenge of modelling modification in MTTsemantics, where CNs are interpreted as types rather than predicates.
1.1 Adjectival and Adverbial Modifications: Some Summatory Notes
An example of an intersective adjective will be black. A black man for example is someone who is both black and a man. In a sense the blackness is not contingent on being a man. Thus, a black man is also a black human, a black animal and so on. To the contrary, subsective adjectives are contingent to the noun class they modify. Thus, a skilful surgeon is someone who is skilful as a surgeon, but we do not know if he is a skilful in general. Nonsubsective adjectives on the other hand, involve two distinct categories: privative adjectives where the adjectivenoun property entails the negation of the property of being a noun. Fake is a prototypical privative adjective. Lastly, noncommital adjectives involve adjectives that do not commit us to any of the aforementioned inferences. An example of such an adjective is alleged. An alleged thief might or might not be a thief.
Generalizing to the worst case, in the case of adjectives the use of intensional typing across the board, i.e also for adjectives (as well as adverbs) that are not intensional, is counterintuitive. Furthermore, the use of meaning postulates will be shown not to be needed, at least for intersective and subsective adjectives, if one moves to a rich type theory with subtyping, like the one we use in this paper. This has been already exemplified in Chatzikyriakidis and Luo (2013) and it is going to be discussed in this paper as well.
Another issue concerning adjectives is the way the positive form is connected with the comparative as well as the superlative form.
Examples like above exemplify the latter claim. In this sense, the two classes of adjectives have also different inferential properties. A sentence like John is healthy implies that John is healthy with respect to all “health” dimensions while in a sentence like John is sick, what we get is an entailment that John is not healthy across some dimension.
Adverbial Modification Adverbials, similarly to adjectives, are a largely nonhomogeneous class. Given this observation, it is not surprising that various classifications have been proposed throughout the years. According to one of the most prominent ones, that of Ernst (2002) and Maienborn and Schafer (2011), adverbs receive a tripartite classification with further subclassifications for each class: (a) predicational, (b) participant oriented and (c) functional adverbials.^{3}
Predicational adverbs comprise the main bulk of adverbs. Its main subcategories including sentence and verb related adverbs. Sentence adverbs are further classified into subject oriented adverbs like arrogantly, speaker oriented adverbs that include speech act adverbials like honestly, epistemic adverbials like possibly and domain adverbs like botanically, while verb related adverbials into mental attitude adverbs like reluctantly, manner adverbs like skillfully and degree adverbs like deeply. Participant oriented adverbials on the other hand include adverbials that introduce a new entity/entities to the situation/event described by the proposition in question. Examples of this type of adverbials include cases like with a knife, with a gun etc. Lastly, functional adverbials include adverbials where some kind of quantification is involved like usually, never etc.
The problem here, as already noted in the literature, is that the manner of the event and not the event itself is modified. Thus, one might want to introduce manner in the ontology of types/basic predicates (depending on whether we have a typed or a nontyped system). We believe that this extension can be done very easily within a rich type system as the one we are endorsing in this paper.
Agent oriented adverbs on the other hand, seem to provide commentary with respect to the utterance. An adverb like frankly seems to imply that a sentence of the form frankly P means something like “I frankly tell you that P”. We will try to show, as in the case of manner adverbs, that the elaborate typing mechanisms of MTTs can be used in order to get this finegrained meaning nuances associated with the different kinds of adverbs.
1.2 Adjectival/Adverbial Modifications in MTTSemantics
The major aim of the paper is to study how to deal with adjectival and adverbial modification in MTTsemantics. As briefly mentioned previously, in order to model adjectival modification adequately in MTTsemantics, one has to meet the challenge of how to interpret CNs modified by various classes of adjectives as types rather than predicates.
As we know, in foundational languages of formal semantics, types in type theories are different from sets in set theory, although both represent collections of objects/elements. In a nutshell and very informally, the difference may be summarised by saying that such types are only manageable sets in the sense that some sets and set operations (e.g. intersection and union), are not available in the world of types for, otherwise, some of the salient and important properties of such type theories would be lost (see, for example, Pierce 1991 for more information). For example, in type theories for formal semantics (either the simple type theory STT as used in Montague’s semantics or MTTs in MTTsemantics), type checking is decidable; in layman’s terms, it is mechanically checkable whether any object a is of type A. In contrast, the truth of the membership relation \(s\in S\) in set theory is undecidable since it is just a logical formula in firstorder logic. In STT, this means that one can check mechanically whether an object is of type e of entities, or of type t of propositions, or of a function type \(A\rightarrow B\). For STT, obviously this must be decidable for otherwise the internal higherorder logic would not work properly (e.g. the applications of its rules would become infeasible). This is similarly the case for MTTs for otherwise one would not have a working internal logic that is necessary for formal semantics. Furthermore, these properties have other advantages: for example, given that MTTs are prooftheoretically specified, they can be effectively implemented on computers—the systems called proof assistants. When formal semantics are given using MTTs, they can be directly implemented in a proof assistant which, among other things, provides a tool for natural language inference based on the implemented MTTsemantics.
Interpreting CNs as types, one faces the challenge that there should be enough types to interpret various classes of CNs adequately. This is a difficult challenge because, even in MTTs which have many inductively defined types, there are less types than predicates/sets. In this paper, we shall show that various type constructors in MTTs provide us with adequate mechanisms to model adjectival/adverbial modification of various kinds. For instance, we will show how to employ \(\varSigma \)types for intersective adjectives, disjoint union types for privative adjectives, and polymorphic \(\varPi \)types with universes and \(\varSigma \) types for subsective adjectives.^{4} In general, we shall illustrate that, using the rich type structure of MTTs, we can provide an account of a number of issues in adjectival/adverbial modification.
It may be useful to emphasise that the scope of the current paper is a rather moderate one. Our aim, as already mentioned, is to show the way modification can be treated in a framework where CNs are modelled as types instead of predicates (or funcational sets of type \(e\rightarrow t\)). As such, this paper should not be taken as a paper that tries to compete with the other stateoftheart approaches found in the considerably rich formal semantics literature in all the aspects that this paper discusses. This is a paper that aspires to show the way to use an alternative formal semantics theory for the study of linguistic semantics. On the other hand, we do not want this paper to be seen as framework gymnastics, i.e. a plain exercise in formal semantics using just another framework. MTTs, as already mentioned, have a number of advantages compared to the simple type theory when it comes to their respective computational properties as well as their fitness to support prooftheoretic reasoning (cf., decidability and practical inference in proof assistants, as mentioned above). These two latter properties of MTTsemantics and their potential application in the study of natural language inference not only from a semantic but from a computational point of view, provide additional reasons for us to believe that the ideas put forth in this paper are worth pursuing.
In Sect. 2, we introduce the core features of modern type theories, emphasizing those relevant to this paper, setting up the background knowledge and notation. In Sects. 3 and 4 we deal with a number of aspects of adjectival modification: (a) the traditional classification into intersective, subsective and nonsubsective adjectives in Sect. 3, and (b) gradable and multidimensional adjectives in Sect. 4. In Sect. 5, we look at adverbial modification. There, building on work by Chatzikyriakidis (2014), we show how the rich typing constructs of MTTs can give us a way out with respect to veridicality, intensional adverbs and some aspects concerning manner and Xoriented adverbs. Lastly, in Sect. 6, we check some of the proposals in this paper in terms of their inferential properties. We show how MTTs can be used in this respect both from a theoretical as well as an implementational point of view. With respect to the latter, we implement some of the ideas in this paper in the Coq proof assistant (Coq 2007), showing its potential use to the study of NL inference.
2 MTTs with Coercive Subtyping: Introduction
In this section, we give a brief introduction to formal semantics based on modern type theories (MTTs) (Ranta 1994; Luo 2010, 2012). We will try and introduce MTTs by exemplifying its various features related to linguistic semantics. A modern type theory is a variant of a class of type theories as studied by MartinLöf (1975, 1984) and others, which have dependent types and inductive types, among others. Among MTTs, we are going to employ the Unified Theory of dependent Types (UTT) (Luo 1994) with the addition of the coercive subtyping mechanism (see, for example, Luo 1999; Luo et al. 2012 and below).
2.1 Type ManySortedness and CNs as Types
A difference between MTTsemantics and Montague semantics lies in the interpretation of common nouns (CNs). In Montague (1974) semantics, the underlying logic (Church’s 1940 simple type theory) can be seen as ‘singlesorted’ in the sense that there is only one type e of all entities. The other types such as t of truth values and the function types generated from e and t do not stand for types of entities. In this respect, there are no finegrained distinctions between the elements of type e and as such, all individuals are interpreted using the same type. For example, John and Mary have the same type in simple type theories, the type e of individuals. An MTT, on the other hand, can be regarded as a ‘manysorted’ logical system in that it contains many types. In this respect, in MTTsemantics one can make finegrained distinctions between individuals and use those different types to interpret subclasses of individuals. For example, we can have \(John{:}\mathop {[\![man]\!]}\nolimits \) and \(Mary{:}\mathop {[\![woman]\!]}\nolimits \), where \(\mathop {[\![man]\!]}\nolimits \) and \(\mathop {[\![woman]\!]}\nolimits \) are different types.
An important trait of MTTsemantics is the interpretation of CNs as types (Ranta 1994) rather than sets or predicates (i.e. objects of type \(e \rightarrow t\)) as in Montague semantics. The CNs man, human, table and book are interpreted as types \(\mathop {[\![man]\!]}\nolimits \), \(\mathop {[\![human]\!]}\nolimits \), \(\mathop {[\![table]\!]}\nolimits \) and \(\mathop {[\![book]\!]}\nolimits \), respectively. Then, individuals are interpreted as being of one of the types used to interpret CNs.
The idea of common nouns being interpreted as types rather than predicates has been argued in Luo (2012a) on philosophical grounds as well. There, the second author argues that Geach’s observation that common nouns, in contrast to other linguistic categories, have criteria of identity that enable common nouns to be compared, counted or quantified, has an interesting link with the constructive notion of set/type: in constructive mathematics, sets (types) are not constructed only by specifying their objects but they additionally involve an equality relation. The argument is then that the interpretation of CNs as types in MTTs is explained and justified to a certain extent.^{6}
Interpreting CNs as types rather than predicates has also a significant methodological implication: this is compatible with various subtyping relations one may consider in formal semantics. For instance, in modelling some linguistic phenomena semantically, one may introduce various subtyping relations by postulating a collection of subtypes (physical objects, informational objects, eventualities, etc.) of the type of entities (Asher 2012). It has become clear that, if CNs are interpreted as predicates as in the traditional Montagovian setting, introducing such subtyping relations would cause problems: even some basic semantic interpretations would go wrong and it is very difficult to deal with some linguistic phenomena such as copredication satisfactorily. Instead, if CNs are interpreted as types, as in MTTs, copredication can be given a straightforward and satisfactory treatment (Luo 2010).
\(\varSigma \)types, \(\varPi \)types, indexed types and Universes We shall introduce several dependent types and the notion of type universe.
Dependent \(\varSigma \)types. One of the basic features of MTTs is the use of Dependent Types. A dependent type is a family of types that depend on some values. The constructor/operator \(\varSigma \) is a generalization of the Cartesian product of two sets that allows the second set to depend on values of the first. For instance, if \(\mathop {[\![human]\!]}\nolimits \) is a type and \(male{:}\mathop {[\![human]\!]}\nolimits \rightarrow Prop\), then the \(\varSigma \)type \(\varSigma h{:}\mathop {[\![human]\!]}\nolimits .\ male(h)\) is intuitively the type of humans who are male.
More formally, if A is a type and B is an Aindexed family of types, then \(\varSigma (A,B)\), or sometimes written as \(\varSigma x{:}A.B(x)\), is a type, consisting of pairs (a, b) such that a is of type A and b is of type B(a). When B(x) is a constant type (i.e. always the same type no matter what x is), the \(\varSigma \)type degenerates into product type \(A\times B\) of nondependent pairs. \(\varSigma \)types (and product types) are associated projection operations \(\pi _1\) and \(\pi _2\) so that \(\pi _1(a,b)=a\) and \(\pi _2(a,b)=b\), for every (a, b) of type \(\varSigma (A,B)\) or \(A\times B\).
Further explanations of the above types are given after we have introduced the concept of type universe below.
Indexed types Indexed types are special kinds of dependent types where the type depends on an index. In effect we are dealing with families of types that are indexed by a type parameter. The type parameter is usually a simple one in most cases. Examples of indices include, for instance, the type N of natural numbers, the type Human of human beings, and the type Height of heights (see below). We can think for example that for h : Human, there is a family of types Evt(h) of events that are conducted by h. In the same sense, one can also think that for h : Height, there is a family of types Human(h) of humans with a height parameter. This types will be of great importance in our discussion of gradable adjectives in Sect. 4.
where the \(\varSigma \)type in (23) is the proposed interpretation of handsome man and the disjoint union type in (24) is that of ‘gun’ (the disjoint union of real guns and fake guns—see the discussion in Sect. 3). We can furthermore use partitions of the universe cn that would correspond to more restrictive universes that might be needed by different predicates. For example, one might assume predicates having a restriction which only allows arguments that are of type Human or any of its subtypes. In that case, one can introduce the subuniverse \(\hbox {CN}_{H}\) that consists of type Human and all its sybtypes (Fig. 1). It goes without saying that the universe cn is an open universe, where additional types can be always added.
The first predicate is the type for the noun, given that nouns are considered to be predicates in MG, and the second predicate is the one denoted by the verb. In MTT semantics where common nouns are types and not predicates, the relation is now between a type and a predicate. Typing has to further be polymorphic given that we have a multitude of basic types, and in principle we would want to have quantification with all these types. The solution is to have a polymorphic type extending over the universe cn, i.e. the type in (20). Thus, we first need an argument of type A, \(A{:}\textsc {cn}\). This corresponds to the cn argument. As soon as we get this argument, what we get back is the type \((A\rightarrow Prop) \rightarrow Prop\). Type polymorphy will predict that the type returned will be dependent on the value of A. If \(A=\mathop {[\![man]\!]}\nolimits \) then the type returned will be \((\mathop {[\![man]\!]}\nolimits \rightarrow Prop) \rightarrow Prop\), if \(A=\mathop {[\![human]\!]}\nolimits \), \((\mathop {[\![human]\!]}\nolimits \rightarrow Prop) \rightarrow Prop\) and so on. For example, some human is of type \((\mathop {[\![human]\!]}\nolimits \rightarrow Prop)\rightarrow Prop\) given that the A here is \(\mathop {[\![human]\!]}\nolimits {:}\textsc {cn}\) (A becomes the type \(\mathop {[\![human]\!]}\nolimits \) in \((\mathop {[\![human]\!]}\nolimits \rightarrow Prop)\rightarrow Prop\)). Then, given a predicate like \(walk{:}\mathop {[\![human]\!]}\nolimits \rightarrow Prop\), we can apply some human to get \(\mathop {[\![some\ human]\!]}\nolimits (\mathop {[\![walk]\!]}\nolimits ){:}Prop\). The reader can now realize how the adverb typing in (25) is to be understood.
which have different domain types. This has the advantage of disallowing interpretations of infelicitous examples like the ham sandwich walks.
However, interpreting CNs by means of different types could lead to serious undergeneralizations without a subtyping mechanism. Thus, subtyping is crucial for MTTsemantics. For instance, consider the interpretation of the sentence a man talks in Table 1: for m of type \(\mathop {[\![man]\!]}\nolimits \) and \(\mathop {[\![talk]\!]}\nolimits \) of type \(\mathop {[\![human]\!]}\nolimits \rightarrow Prop\), the function application \(\mathop {[\![talk]\!]}\nolimits (m)\) is only welltyped because we assume \(\mathop {[\![man]\!]}\nolimits \) be a subtype of \(\mathop {[\![human]\!]}\nolimits \).
Coercive subtyping (Luo 1999; Luo et al. 2012) provides an adequate framework to be employed for MTTbased formal semantics (Luo 2010, 2012).^{9} It can be seen as an abbreviation mechanism: A is a (proper) subtype of B (\(A\ {<}_{}\ B\)) if there is a unique implicit coercion c from type A to type B and, if so, an object a of type A can be used in any context \({\mathfrak {C}}_B[\_]\) that expects an object of type B: \({\mathfrak {C}}_B[a]\) is legal (welltyped) and equal to \({\mathfrak {C}}_B[c(a)]\).
Thus, an \(x{:}\mathop {[\![man]\!]}\nolimits \) can be used as an \(x{:}\mathop {[\![human]\!]}\nolimits \), and as such the inference goes through for ‘free’ in a way.
Another important trait of the coercive subtyping mechanism, which will be very important in our discussion of intersective adjectives, is that subtyping also propagates through the constructors. For example, if we have \(\mathop {[\![man]\!]}\nolimits <\mathop {[\![human]\!]}\nolimits \), then we also get \(\varSigma m{:}\mathop {[\![man]\!]}\nolimits .\mathop {[\![handsome]\!]}\nolimits (m) < \varSigma m{:}\mathop {[\![human]\!]}\nolimits .\ \mathop {[\![handsome]\!]}\nolimits (m)\). For more information on subtyping propagation see, for example, Luo (1999).
Coercion Contexts and Local Coercions It is a wellknown fact that word meanings heavily rely on contexts. In this sense, a lot of the times we need to deal with cases like the following classic meaning transfer example from (Nunberg 1995):
Example 1
Assuming that the act of shouting requires that the argument be human, it is obvious that sentence (1) is not wellformed, unless it is uttered by somebody in some special extralinguistic context (e.g. by a waiter in a café to refer to a person who has ordered a ham sandwich).
where what we have is that type ham is coerced into human in this particular setting. The range of coercions we can perform can get morefine grained as it has been exemplified for example in the work of Asher and Luo (2012). The interested reader is referred there for more information on this issue. People interested in seeing the use of contexts for NL semantics, please see Ranta (1994), Boldini (2001) as well as Chatzikyriakidis and Luo (2014c) for the similar notion of signature.
Prooftheoretic Facet of MTTsemantics One of the key features of formal semantics in MTTs is its prooftheoretic facet. It has been pointed out in Luo (2014) that MTTsemantics is both modeltheoretic and prooftheoretic. Without getting into the details, we emphasise here the prooftheoretic characteristics of MTTsemantics: it allows us to understand MTTsemantics in a prooftheoretic way [as logics can be understood prooftheoretically (Kahle and SchroederHeister 2006)] and, furthermore, allows a direct application of type theory based proof assistants such as Coq (2007) in conducting practical inferences based on MTTsemantics. This has amounted to a computational treatment of formal semantics and opens up a new avenue in semanticsbased reasoning in natural language (see Chatzikyriakidis and Luo 2014b and Sect. 6). Discussing this is out of the scope of the current paper. The interested reader may consider reading some related papers including, for example, Luo (2014).

A sentence (S) is interpreted as a proposition of type Prop.

A common noun (CN) can be interpreted as a type.

A verb (IV) can be interpreted as a predicate over the type D that interprets the domain of the verb (ie, a function of type \(D\rightarrow Prop\)).

An adjective (ADJ) can be interpreted as a predicate over the type that interprets

A VP adverb can be interpreted as a function from predicates (\(A\rightarrow Prop\)) to predicates (\(A\rightarrow Prop\)) where the A extends over the universe cn

A quantifier is interpreted as a function from a type \(A{:}\textsc {cn}\) extending over the universe cn to a function from predicates over A to propositions.^{10} the domain of the adjective (ie, a function of type \(D\rightarrow Prop\)).

Modified common nouns (MCNs) can be interpreted by means of \(\varSigma \)types (see below).
Examples in formal semantics
Example  Montague semantics  MTTsemantics  

CN  Man, human  \(\mathop {[\![man]\!]}\nolimits , \mathop {[\![human]\!]}\nolimits {:}e\rightarrow t\)  \(\mathop {[\![man]\!]}\nolimits , \mathop {[\![human]\!]}\nolimits {:}Type\) 
IV  Talk  \(\mathop {[\![talk]\!]}\nolimits {:}e\rightarrow t\)  \(\mathop {[\![talk]\!]}\nolimits {:}\mathop {[\![human]\!]}\nolimits \rightarrow Prop\) 
ADJ  Handsome  \(\mathop {[\![handsome]\!]}\nolimits {:} (e\rightarrow t)\rightarrow (e\rightarrow t)\)  \(\mathop {[\![handsome]\!]}\nolimits {:}\mathop {[\![man]\!]}\nolimits \rightarrow Prop\) 
\(\hbox {ADV}_{ VP}\)  Quickly  \(\mathop {[\![quickly]\!]}\nolimits {:} (e\rightarrow t)\rightarrow (e\rightarrow t)\)  \(\mathop {[\![quickly]\!]}\nolimits {:}\varPi A{:}\textsc {cn}.(A\rightarrow Prop)\rightarrow (A\rightarrow Prop) \) 
Quantifier  Some  \(\mathop {[\![some]\!]}\nolimits {:}(e\rightarrow t)\rightarrow (e\rightarrow t) \rightarrow t\)  \(\mathop {[\![some]\!]}\nolimits {:}\varPi A{:}\textsc {cn}.(A\rightarrow Prop)\rightarrow Prop\) 
MCN  Handsome man  \(\mathop {[\![handsome]\!]}\nolimits (\mathop {[\![man]\!]}\nolimits ){:} e \rightarrow t\)  \(\varSigma m{:}\mathop {[\![man]\!]}\nolimits . \mathop {[\![handsome]\!]}\nolimits (m){:}Type\) 
S  A man talks  \(\exists m{:}e.\ \mathop {[\![man]\!]}\nolimits (m)\) & \(\mathop {[\![talk]\!]}\nolimits (m){:}t\)  \(\exists m{:}\mathop {[\![man]\!]}\nolimits . \mathop {[\![talk]\!]}\nolimits (m){:}Prop\) 
Note that this table shows some basic comparisons to Montague semantics as well as potential ways to interpret these categories. For some of these cases, other ways of interpretation will be pursued. For example, besides the use of a basic predicate type for adjectives, type polymorphy will be used for e.g. cases of subsective adjectives.
3 Intersective, Subsective and Nonsubsective Adjectives
In this section, we look at the traditional formal semantics classification of adjectives (Kamp 1975; Kamp and Partee 1995; Partee 2007) and discuss the solutions using MTTs based on earlier work of ours (Chatzikyriakidis and Luo 2013). Historically, Ranta (1994) was the first to propose the use of \(\varSigma \)types to represent adjectival modification.^{11} However, the proposal was not completely working because there was no proper subtyping mechanism that is essential in order for \(\varSigma \)types to be employed for adjectival modification.^{12} This problem was solved in Luo (2010), where the second author proposed to employ coercive subtyping (Luo 1999; Luo et al. 2012). As a consequence, CNs modified by intersective adjectives can be properly represented by means of \(\varSigma \)types.
Subsective adjectival modification was then studied in Chatzikyriakidis and Luo (2013) where the authors proposed to use the universe cn of common nouns and polymorphism in representations of subjective adjectives like small,large. Nonsubsective adjectives have also been studied: for privative adjectives like fake, (Luo 2011a; Chatzikyriakidis and Luo 2013) propose to use disjoint union types while, for noncommittal adjectives like alleged, belief contexts (Chatzikyriakidis and Luo 2013). All these proposed solutions maintain a lower type for adjectives. The rich typing provided by MTTs in conjunction with coercive subtyping (Luo 1999; Luo et al. 2012) can give us an attractive solution to all adjectives under the traditional classification into intersective, subsective and nonsubsective adjectives.
3.1 Intersective and Subsective Adjectives
Intersective and subsective adjectives can be treated properly using \(\varSigma \)types. Using \(\varSigma \)types to represent adjectival modification, as already mentioned, was originally proposed by Ranta (1994). However, in Ranta’s account, neither of the two classes can be captured successfully: for intersective adjectives, we lack a proper subtyping mechanism and, for subsective adjectives, we need to use the universe of common nouns and polymorphism.
In practical terms and taking black as our example, this means that for every A and B where \(A<B\), we have black \(A<black\) B. In case no relation between A and B exists or if the subtyping relation is reversed, no inference should be possible. Indeed, given the subtyping relations and the fact that subtyping relations propagate through the constructor types, we predict the desired inferences.
The above idea has been proposed by Chatzikyriakidis and Luo (2013) and it is basically an implementation of the intuition that subsective adjectives are only relevant for the particular CN they modify in each case. Thus, a small elephant is only small with respect to elephants, a skilful surgeon is only skilful for a surgeon, and so on. Using the type proposed in (45), we can have different instances of a subsective adjective, say P, depending on the choice of A, with \(A{:}\textsc {cn}\). This account of intersective and nonsubsective adjectives relies on the following assumptions: (a) CNs are types, (b) adjectives are predicates (or lower function types, see discussion in Sect. 3.2.2), (c) predicates may be polymorphic. The welcome result in this approach is that inferential properties are derived via typing only and no extra axioms in the form of meaning postulates are needed.
3.2 Nonsubsective Adjectives
Thus, in fake fur, fur is coerced to include fake furs as well.
Note that in the second example gun is taken to mean real gun. This needs some explanation. According to the Partee explanation, without the coercion of gun to include fake guns, the adjective real would also be redundant (since all guns would be real guns). So, in the above example we take this to mean that a fake gun is not a real gun.
where DATE consists of the triples (y, m, d) where y ranges over integers to represent years, m over Jan to Dec to represent months, and d over the days 1, 2, ... to represent days.
In the above example man basically has the meaning of boyfriend, husband, in effect it is turned into a stage level noun. The way this kind of coercions precisely work is a matter well beyond the scope of this paper. However, coercive subtyping is a mechanism that has been argued to be fit for dealing with a wide range of linguistic coercions, see e.g. Luo (2011b), Asher and Luo (2012). What is relevant here, is that former needs a stage level noun to combine with and in order to combine with individual level nouns, a some kind of coercion into stage level has to be performed.
where \(B(p,A) = \varPi \varGamma _p.\ A\) with \(\varGamma _p\) being the belief context of \(p{:}\mathop {[\![human]\!]}\nolimits \).^{15}
Again, as in the case of former, one should restrict the scope of alleged. This is because the definition in (69) will overgenerate, since it will predict combinations like alleged chair and alleged democracy to be possible. It seems that alleged, similarly to former, can only combine with stage level CNs. In this sense, we update the definition for \(\mathop {[\![A_N]\!]}\nolimits {:}\) \(\hbox {CN}_{s}\).
3.2.1 A Note on Unknown Beliefs
In the above cases, we have two distinct utterers with different versions of John’s belief context, \(G_{j(u1)}\) and \(G_{j(u2)}\). These cases would be impossible to get by assuming one \(G_j\) context. In this sense, the belief context is relativized to the utterer in each case. Thus, in cases of unknown agents, this b elief context might just be the minimal context including \(B(p,A_N)\) in the case of alleged and nothing more. We take this to be a promising way to deal with these cases. However, we have to look at the formal details behind such a proposal. This task cannot be taken up in this paper. Similar considerations apply to the other cases where belief contexts are used in this paper.
3.2.2 A Note on Predicativity
It is a wellknown fact that a number of adjectives cannot be used predicatively but only attributively. We have considered adjectives of this sort so far, e.g. alleged or former are of this type. An explanation for the behaviour of this class of adjectives comes from Coppock (2008) who argues that adjectives that cannot be used predicatively are not semantically predicative. In standard terms, this means that the adjective is not a predicate. It has been pointed out to us that if we assume a lower predicate type for adjectives, problems arise w.r.t nonpredicative adjectives. This is correct. However, we do not assume that all adjectives are predicates. We do assume that there are no higher order types associated with adjectives, but we do assume that adjectives might not be predicates. We have already exemplified the latter point for adjectives like former where the typing for former was \((Time\rightarrow \textsc {cn}_s)\rightarrow \textsc {cn}_s\). In effect, for nonpredicative adjectives the idea is that these are function types that return a CN type rather than a predicate (i.e. returning an element of type Prop). Similar considerations apply to the adjective alleged and generalizing, to all nonpredicative adjectives.
4 Gradable Adjectives and Multidimensional Adjectives
4.1 Gradable Adjectives
Proponents of such an approach can be found in Bartsch and Vennemann (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 nongradable ones. The difference between the two is that gradable adjectives, even though being predicates from individuals to truth values, they further involve partially ordered domains. Gradable adjectives impose a partitioning of this 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), McConnellGinet (1973), Klein (1980), van Benthem (2012). 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 nice and most recent overview of the two approaches is Lassiter (2014).
Note that the value in \(\varGamma \) might be elaborated via context extension. Following Ranta (1994) particular, we define a mapping \(f{:}\varDelta \rightarrow \varGamma \), where everything that is in \(\varGamma \) is also in \(\varDelta \) plus some potentially extra information.^{22} Also, this idea of using TT contexts will be natural in cases the standard value is wayoff the one usually found in the default context. Consider for example the context of all statements pertaining to basketball. In there, and given the nature of the sport, the meaning of tall and short is inevitably different.^{23}
In the universe D, one can find types \(\mathop {[\![Height]\!]}\nolimits ,\mathop {[\![Weight]\!]}\nolimits ,\mathop {[\![Width]\!]}\nolimits {:}D\) among other types.
4.2 Multidimensional Adjectives
The idea here is to \(\mathop {[\![Health]\!]}\nolimits \) as an inductive type, in order to encode all the different dimensions we need. This is one way of dealing with multidimensional adjectives in MTTs. Of course, there are a number of issues in case one wants to further give a full theory of gradable and multidimensional adjectives. Our goal was to show an initial way of approaching these kind of adjectives in MTTs. The interested reader who wants to further investigate the issue of gradability and multidimensionality of adjectives is directed to Lassiter (2014) and references therein for gradability and Sassoon (2012) for multidimensional adjectives.
5 Adverbial Modification
The literature on adverbs in MTTs is rather poor. The only paper specifically dealing with adverbial modification is Chatzikyriakidis (2014).^{25} However, adverbs have been also treated in Chatzikyriakidis and Luo (2014b) as part of a discussion on Natural Language Inference (NLI). There, a first approach of some aspects of adverbial modification like veridicality, nonveridicality, adverbial typing and intensional adverbs among others has been attempted. Here, we extend the approach of Luo (2011a), Chatzikyriakidis (2014), Chatzikyriakidis and Luo (2014b) to further adverbial classes and deepen the analysis given there.
5.1 Veridicality
A very basic distinction in terms of the semantic properties of adverbs, in particular the inference patterns that they give rise to, concerns what has been dubbed as veridicality. Veridicality is found in both VP and sentence level adverb. Veridicality in the case of sentence adverbs means that Adv(P) presupposes P whereas in the case of VP adverbs V(P(x)) presupposes P(x)).^{26}
Taking v to be \(\mathop {[\![John\ went]\!]}\nolimits \), (94) is the semantic representation of (93).
Note that what we have presented here only deals with the veridical property and does not say anything further about the semantics of the adverbs in each case. In order to get into the specifics of each veridical adverb, more information will be introduced, potentially in the form of a conjunction, but this is something that we have not looked at yet.
5.2 Intensional Adverbs
In order to deal with these data, the first author Chatzikyriakidis (2014) uses the type theoretic notion of context similarly to the way used by Ranta (1994) and also here in this paper.
One would expect this sentence not to be correct on the assumption that accidentally is a antonym of intentionally.
A better analysis of the meaning of intentionally would be something like the following: A intentionally P means that A has the intention P and furthermore fulfilled this intention, i.e. P holds. In order to formalize this, we introduce the notion of intention contexts, which represent an agent’s collection of intentions.
Thus, (99) is predicted. On the other hand, in order to prove that Oedipus intentionally married his mother, we need to have M(O, MoO) in the intention context of Oedipus. If we assume 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. If we assume that Oedipus’ intention context is unknown, we cannot prove it nor disprove before this information becomes available.
5.3 Manner Adverbials
Needless to say that the associated veridical inference is captured with the above entry, given that it is included as the first member of the conjunction. In effect, with this entry he wrote always follows from he wrote illegibly.
5.4 Some Notes on Other Classes of Adverbs
A solution in MTTsemantics, that maintains the core of the analysis of manner adverbs, will involve again indexed types. But now, instead of the type of events indexed only by manners, what we have is types of events which are indexed by humans as well as manners. Thus, \(\mathop {[\![human(m)]\!]}\nolimits \) is the type of humans with manner m.
The semantics of speaker oriented adverbs seem more difficult to grasp. Here, we are going to only look at speech act adverbials like honestly, frankly. Such adverbs can be seen as providing commentary with respect to the utterance. In this respect, 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 dating back to Schreiber (1972) gives rise to a way of looking at speech act adverbs that is not that different from manner adverbs. Piñón (2013) provides an interesting account according to which speaker oriented adverbs make reference to individual manners of speaking. We will not go into the details of his proposal. Assuming that this be a reasonable way to look at speaker oriented adverbials, one can sketch an account in MTTs as follows.
According to the account just presented, a speech act adverb is not of type \(Prop \rightarrow Prop\); instead it takes both an utterance event as well as a proposition as arguments and returns a proposition.
With this last remark, we will stop our discussion on adverbial modification, leaving a number of issues unresolved. To recap, we have shown that MTTs can provide us with a rich and expressive typed language in order to deal with a number of aspects pertaining to modification. From a theoretical point of view, we hope that we have presented arguments for using MTTs for NL semantics.
6 Modification and Inference
One attractive characteristic of MTTsemantics is that it can be seen as prooftheoretic (Luo 2014). This means that the judgments in the underlying type theories can be understood by means of their inferential roles. This latter fact constitutes MTTs a good solution w.r.t consequences that the semantics proposed in each case bring about, i.e. inference. This prooftheoretic aspect of MTTs has been the reason that these are widely implemented in computer reasoning systems, i.e. proof assistants. Proof assistant technology has gone a long way since its emergence. The proofassistant Coq is a prime example of the advance reached in the field and a number of remarkable developments have been achieved via its use [e.g. see the proof for the four colour theorem (Gonthier 2005)]. Coq implements the Calculus of Inductive Constructions (CiC), in effect an MTT. Actually, CiC is quite close to the MTT we are using, i.e. UTT with coercive subtyping (Luo 1994; Luo et al. 2012). The fact that Coq ‘speaks’ so to say an MTT, in combination with the fact that it is a powerful reasoning engine, makes Coq suitable to implement and further reason about MTT semantics. The authors have exemplified the use of Coq as a means to deal with NLI in various papers (Chatzikyriakidis and Luo 2014b, a, 2016). There, it was shown that Coq can be used as a NL reasoner formalizing part of the FraCaS test suite examples and reasoning about them. Besides the various practical/computational applications that such an endeavour can lead to, there is an additional side to using Coq, which has to do with the correctness of the accounts one proposes. A correct account of a certain NL phenomenon should be able to derive all the correct consequences associated with it, while on the other hand it should not derive any unwanted consequences. This is basically to say that a correct account is judged by the number of inferences it gives rise to as well as the number of them that it does not. In our case and given the nice interaction of MTTsemantics and the associated prooftechnology, we can actually check whether what the propose derives the proper inferences or not. This is what we are going to see now, by looking into the consequences that a number of our proposals made in this paper give rise to. In what follows we give a short introduction to Coq and then test the predictions of our account using MTT derivations for some cases and also presenting the relevant implementations in Coq along with their explanation in the “Appendix”.
6.1 The Coq Proof Assistant
This says that given an A of type \(\textsc {cn}\) and a predicate over A, there is an x : A such that P holds of x.
Imagine that we want to see the consequences of this definition. For example we may want to check whether John walks implies that some man walks or that some man walks implies that some human walks. We define, following our theoretical assumptions about CNs, man and human to be of type cn and declare the subtyping relation \(\mathop {[\![man]\!]}\nolimits <\mathop {[\![human]\!]}\nolimits \). This is all we need to get the above inferences. These assumptions suffice to prove these inferences in Coq.
6.2 Testing the Theory

\(\varSigma (man,black)< \mathop {[\![man]\!]}\nolimits < \mathop {[\![animal]\!]}\nolimits \) (by first projection as coercion);

therefore, \(\exists x{:}\varSigma (man,black).\ walk(x)\) implies \(\exists y{:}man.\ walk(y)\);

that is, \(\mathop {[\![\textit{a black man walks}]\!]}\nolimits \) implies \(\mathop {[\![\textit{a man walks}]\!]}\nolimits \).
Indeed, for intersective adjectives, this is the case. For example assuming black is of type \(object \rightarrow Prop\), the above can be proven. However, given the typing we have proposed for subsective adjectives, no proof is found. For if we try to prove George is a small animal from George is a small man we are stuck, since we are basically trying to prove small(animal)(George) from small(man)(George). Given that small is relativized to different domains in each case, it seems that no proof can be found.
With privative adjectives like fake, and we assume an analysis as this was sketched in this paper, where these involve coercion of the CN to include fake CN denotations, we proposed a disjoint union type. Using this type we can predict that a fake gun is a gun but it is not a real gun. For example one can prove that all real guns are not fake guns and vice versa in Coq
since, by (90), one obtains from the second projection that walk slowly implies walk.
We have not yet tried the intensional cases of adverbs. We leave this as future work, even though we believe that this will not be difficult to do. In particular, Coq’s Local section mechanism (in effect local contexts) would be useful for implementing the account of domain adverbials. But as we have said, we end the discussion here, leaving these issues for future research.
Footnotes
 1.
Within the simple type theory used in Montague Grammar, e is the type of individuals, t is the type of truthvalues and s the type of worldtime pairs.
 2.
In this paper, we shall use the notation \(\mathop {[\![w]\!]}\nolimits \) for the semantics of w. For example, for the CN human, in MG, \(\mathop {[\![human]\!]}\nolimits {:}e\rightarrow t\), while in MTTsemantics, \(\mathop {[\![human]\!]}\nolimits \) is a type. Sometimes, we shall also use capitalised words for types in MTTs: for example, we might use Human for the type of humans; in this case, \(\mathop {[\![human]\!]}\nolimits \) and Human are the same types.
 3.
 4.
See next section for an the explanation of these types.
 5.
This is of course based on the assumption that the definite NP is of a lower type and not a Generalized Quantifier.
 6.
See Luo (2012a) for more details on this.
 7.
This was proposed for the first time in Luo (2011b).
 8.
There is quite a long discussion on how these universes should be like. In particular, the debate is largely concentrated on whether a universe should be predicative or impredicative. A strongly impredicative universe U of all types (with \(U{:}U\) and \(\varPi \)types) is shown to be paradoxical (Girard 1971) and as such logically inconsistent. The theory UTT we use here has only one impredicative universe Prop (representing the world of logical formulas) together with an infinitely many predicative universes which as such avoids Girard’s paradox (see Luo 1994 for more details).
 9.
It is worth mentioning that subsumptive subtyping, the traditional notion of subtyping that adopts the subsumption rule (if \(A\le B\), then every object of type A is also of type B), is inadequate for MTTs in the sense that it would destroy some important properties of MTTs [see, for example, Section 4 of Luo et al. (2012) for details].
 10.
For a constructive version of generalized quantifiers see Sundholm (1989).
 11.
Ranta (1994) did not consider different classes of adjectives and we think that he mainly had intersective adjectives in mind when considering this.
 12.
 13.
It has to be noted however that in order to do full justice to this kind of adjectives one has to engage into the issue of temporal sensitivity of nouns. In our case, this means that we have to look at the way the temporal index of CNs interacts with the rest of the sentence. Unfortunately, such engagement cannot be done in this paper for obvious reasons of space. This however as well as the more general issue of providing a solid temporal theory using MTTs is one of the things that we are currently looking at. For more information on the temporal sensitivity of CNs, the interested reader is redirected to Enc (1981), Musan (1995), Tonhauser (2002) for thorough discussions and proposals on temporal sensitivity.
 14.
Please see the remark on predicativity at the end of this section.
 15.
This is the analog of a formula that involves existential quantifications. One may turn such types into propositions by means of the following operation: for any type A, \(Exists(A) = \exists x{:}A.True\). Then, with this mechanism, (69) can be represented as the proposition \(\exists p{:}\mathop {[\![human]\!]}\nolimits .\ Exists(B(p,A_N))\).
 16.
Here we do not spell out the type \(\mathop {[\![Height]\!]}\nolimits \). One might take Height to be the type of natural numbers and use 170 to stand for 1.70, etc.
 17.
The transitive properties of comparatives are not encoded in this example for reasons of simplicity. One may very well do so having as a guide the previous entry without measures.
 18.
The definition involves a biimplication, given that if the height of human x is less than the height of another human y, then it is also the case that x is shorter than y. The definition also works as an implication.
 19.
Where \(n{:}\mathop {[\![Height]\!]}\nolimits \), the contextual degree parameter.
 20.
\(<>\) stands for either \({<}\mathrm{or}{>}\). For type D see the following discussion.
 21.
\(K_{sp}\) stands for the speaker’s knowledge context.
 22.
This corresponds to what Boldini (2000) called logical inference between contexts. In particular \(\exists n_1{:}Nat. n_1=n\) will be reduced to \(180=n\) by the \(\exists \) elimination rule.
 23.
Another consequence of this approach is that given the polymorphic type of the function, the n is always relativized to both \(A{:}\textsc {cn}\) and i : D. c.)Thus, for a polymorphic adjective like small, the contexrualized value will be relativized to the type A (e.g. \(\mathop {[\![human]\!]}\nolimits ,\mathop {[\![animal]\!]}\nolimits \) etc.).
 24.
The inductive type Health in cn is the finite type (also called an enumeration type), sometimes written as \(\{Heart,\ Blood\_ pressure,\ Cholesterol\}\).
 25.
One of the reasons for this is that researchers found it difficult to give adverbial typings when CNs are interpreted as types. The first to discuss adverbial typings was Luo (2011a) who proposed to use the universe cn and polymorphism to solve this problem. This proposal was followed in Chatzikyriakidis (2014) and also in this paper.
 26.
With \(P{:}A\rightarrow Prop\), x : A and \(A{:}\textsc {cn}\)
 27.
The same ideas discussed in the case of unknown beliefs in Sect. 3.2.1 are also relevant for unknown intentions.
 28.
We will not get into a discussion on whether this is absolutely correct and some people might have different judgments with respect to this.
 29.
Evidence from passive constructions shows that this is the case. Thus, in the boat was sunk intentionally by the government, the paraphrase we get is that it was intentional on behalf of the government to sink the boat, rather than it was intentional on behalf of the boat to sink itself that a subject oriented interpretation would imply. See Jackendoff (1972), McConnellGinet (1982), Geuder (2000) for more details on agent oriented adverbs.
 30.
Again, note that in MTTs, N(x) will be the type judgment, x : N.
 31.
This is easily proven in Coq (we use dependent record types for encoding the \(\varSigma \) type approach). See “Appendix” for the actual code and example.
 32.
Such propagations of suutyping relations through type constructors is desirable, but it is not implemented in Coq.
 33.
See “Appendix” for the actual code.
 34.
See “Appendix” for the code and example
 35.
The ’...’ is not part of the actual code. It just says that more health dimensions can be added depending on the finegrainedness we want to achieve.
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