A First Order Semantic Approach to Adjectival Inference

A first order semantic approach to adjectival inference Marilisa Amoia Claire gardent INRIA/Universite´ de Nancy 1 & CNRS/Loria University of the Saarland Campus Scientifique BP 239 Saarbrucken,¨ Germany 54506 Vandoeuvre-les-Nancy, France [email protected] [email protected] Abstract der automated reasoners that could be put to use to reason about the meaning of higher-order formulae. As shown in the formal semantics litera- In this paper, we present a semantics for adjec- ture, adjectives can display very different tives that adopts an ontologically promiscuous ap- inferential patterns depending on whether proach and thereby supports first order inference for they are intersective, privative, subsective all types of adjectives including extensional ones. or plain non subsective. Moreover, many Indeed, traditional semantic classifications of ad- of these classes are often described using jectives such as (?; ?; ?) subdivide adjectives second order constructs. In this paper, we into two classes namely extensional vs. intensional adopt Hobbs’s ontologically promiscuous adjectives, the latter grouping together adjectives approach and present a first order treatment which intuitively denote functions from properties of adjective semantics which opens the way to properties i.e., second order objects. for a sophisticated treatment of adjectival We present a compositional semantics for ad- inference. The approach was implemented jectives which both (i) defines a first order repre- and tested using first order automated rea- sentation and (ii) integrates interactions with other soners. sources of linguistic information such as lexical semantics and morpho-derivational relations. We then 1 Introduction show that the proposed semantics correctly predicts As has often been observed, not all of natural lan- the inferential patterns observed to hold of the var- guage meaning can be represented by first order ious adjective subclasses identified in the literature logic. There are expressions such as, most, former, (?; ?; ?; ?). I didn’t whose meaning intuitively involve higher- This paper is structured as follows. We start by order constructs. presenting a classification of adjectives which is mo- Nevertheless, as (?) and others have argued, se- tivated by the different inferential patterns observed. mantic representations for natural language need not We then propose a compositional semantics for each be higher-order in that ontological promiscuity can class and show that it correctly predicts their inferen- solve the problem. That is, by reifying all objects tial behaviour. We conclude with a brief discussion that can be predicated of, it is possible to retain a of related work and pointers for further research. semantic representation scheme for NL that is first- 2 Inferential patterns and adjective classes order. This observation is crucial for computational ap- In the literature (?; ?; ?; ?), adjectives are usually plications for two reasons. First, logics that goes be- divided into four main classes namely, intersective, yond first order are highly undecidable. Second and subsective, privative and plain non subsective de- more importantly, there is no off the shelf higher or- pending on whether or not the [Adj N]AP phrase entails the properties expressed by the noun and/or the by very) and most of them can only be used attribu- adjective. More specifically, each of the four classes tively (He is a former president but not The presi- is characterised as follows. dent is former). Semantically, they are usually taken to denote second order properties i.e., functions of Intersective adjectives. This class includes com- the type hhe,ti, he,tii. mon categorical (e.g., red, rectangular, French) and Intensional adjectives include denominal (or rela- tautological (e.g., real, present) adjectives. It is char- tional) adjectives (e.g polar bear, atomic scientist), acterised by the inferential patterns: manner (or adverbial) adjectives (e.g. a poor liar, a fast car), emotive (e.g. a poor man) and modals, i.e. [A N] |= N all adjectives which are related to adverbs, quanti- [A N] |= A fiers or determiners (e.g. a feeble excuse, the specific For instance, saying that there is a red table im- reason, a fake nose, etc.). plies both that there is something red and that there is a table. 3 Assigning FOL Representation to Intensional adjectives Subsective adjectives form an ontologically het- erogeneous class including for instance denominal We now show how adjectives can be assigned an ap- (e.g., gastronomical) and measure (e.g. big) adjec- propriate first order logic representation which ap- tives. They are characterised by the fact that the [Adj propriately reflects their inferential behaviour. N]AP phrase does not entail the Adj property: Following Hobbs, we adopt a promiscuous ontol- ogy and assume that for every predication that can [A N] |= N be made in natural language, there corresponds an [A N] 6|= A “eventuality”. As Hobbs has argued, this allows for higher order predications to remain first order in that a big mouse For instance, is a mouse but is not they become predications over (first order) eventual- big. Instead it is “big for a mouse”. In other words, ities. ’bigness’ cannot be directly inferred as e.g., a big Thus, in the domain there are entities which are mouse and a big elephant are big in very different either eventualities or individuals and relations be- ways. tween individuals. Moreover like Hobbs, we assume Privative adjectives denote adjectives such that a model to describe a platonic universe containing the [Adj N]AP phrase entails the negation of the N everything that can be spoken about whether or not property: these things exist in the real world. To express exis- [A N] |= ¬N tence in the real world, a special predicate (Exists) is introduced. For instance, the former king is not the king and a We use the following notation: fake weapon is not a weapon. • ei, for eventuality variables, Plain non subsective adjectives are adjectives which preclude any inference wrt to the N property: • xi, for individuals, [A N] |= (N ∨ ¬N) • Pi, for properties of individuals. [A N] 6|= A And the following types: Thus, if Peter is an alleged murderer, it is impos- sible to know whether or not he is a murderer. • e will denote the type of individuals, Now, the class of intensional adjectives groups together adjectives with a syntactic and semantic id- • ev the type of eventualities and iosyncratic behaviour. Syntactically, intensional adjectives are not gradable (e.g., cannot be modified • t a truth value. 3.1 The intuition As we shall shortly see, the additional lambda variable e is imposed by the treatment of adjective As shown in section ??, the semantics of [Adj N]AP semantics we propose and more specifically by the phrases has very different inferential properties de- necessity to sometimes distinguish between the indi- pending on the type of the adjective Adj. The differ- vidual described by the noun and the individual de- ences stem from three main points. scribed by the adjective. The variable P ol accounts The number of individuals introduced by the for the polarity of the noun, e.i. whether it occurs [Adj N]AP phrase. Thus, the red table evokes a with the negation or not. single individual x which is both red and a table We give here also the semantics assigned to the whilst the gastronomical book refers to a book x pronouns someone/something which will be used in which is about the gastronomy concept y. More gen- the derivations throughout this paper: erally, the variables predicated of by the noun and by the adjective can refer either to the same or to two (2) a. someone/something: λP ∃x.P (x) distinct individual(s). 3.3 The semantics of the copula The properties licensed by the adjective and the Following the proposal of Mantague, we assign a noun to contribute to the meaning of the [Adj unique representation for both the uses of the cop- N]AP phrase. Depending on the adjective type, ula in identity statements (e.g. John is Mary → the properties denoted by Adj and N will contribute john=mary) and in predicative assertions (e.g. John either directly or indirectly to the meaning of the is a man → man(john)): [Adj N]AP phrase. Thus in an intersective [Adj N]AP phrase, the meaning contributed by Adj and (3) a. be: λKλx.K(λy(x = y)) N are simply the properties they denote. By con- trast, the privative fake forces the negation of the N In the case of predicative assertions in which the property to be part of the Adj N meaning whilst the predicate is an adjective (e.g. John is brave), we subsective gastronomical induces a relation to the adjust the type of the argument of the copula in the morphoderivationally related noun concept (about following way: gastronomy) to be included in the the Adj N meaning. More generally, the properties that compose the (4) a. be Adj: be(Adj(λP olλeλx.true)) meaning of the Adj N phrase can be the denotation of Adj and/or N, the negation of N, its denotation in 3.4 The semantics of adjectives the past or some property derived from it. Given such a representation for nouns, we represent adjectives using the schema given in Figure ??. The existence in the real world of the entity de- Briefly, schema ?? captures the observations noted by the NP. In all cases the Adj N Phrase de- made in section (??) as follows. First it intro- notes a set of individuals but whilst in most cases the duces an existential quantification (in the platonic Adj N phrases is neutral with respect to the existence universe) over not one but two variables (ea and en) in the real world of these individuals, plain non sub- – depending on how the formula is instantiated (and sective (e.g., alleged murderer) explicitly questions in particular on the value of R1 and R2) these two it (an alleged murderer may or not exist in the real variables may or not denote the same object.

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