Vagueness and Learning: A Type-Theoretic Approach Raquel Fernandez´ Staffan Larsson Institute for Logic, Language Department of Philosophy, Linguistics and Computation and Theory of Science University of Amsterdam University of Gothenburg [email protected] [email protected] Abstract expressions (Kelleher et al., 2005; Reiter et al., 2005; Portet et al., 2009) and, perhaps even more We present a formal account of the mean- strongly, on the field of robotics, where ground- ing of vague scalar adjectives such as ‘tall’ ing language on perceptual information is critical formulated in Type Theory with Records. to allow artificial agents to autonomously acquire Our approach makes precise how percep- and verify beliefs about the world (Siskind, 2001; tual information can be integrated into Steels, 2003; Roy, 2005; Skocaj et al., 2010). the meaning representation of these pred- Most of these approaches, however, do not build icates; how an agent evaluates whether an on theories of formal semantics for natural lan- entity counts as tall; and how the proposed guage. Here we choose to formalise our account semantics can be learned and dynamically in a theoretical framework known as Type Theory updated through experience. with Records (TTR), which has been shown to be suitable for formalising classic semantic aspects 1 Introduction such as intensionality, quantification, and nega- tion (Cooper, 2005a; Cooper, 2010; Cooper and Traditional semantic theories such as those de- Ginzburg, 2011) as well as less standard phenom- scribed in Partee (1989) and Blackburn and ena such as linguistic interaction (Ginzburg, 2012; Bos (2005) offer precise accounts of the truth- Purver et al., 2014), perception and action (Dob- conditional content of linguistic expressions, but nik et al., 2013), and semantic coordination and do not deal with the connection between meaning, learning (Larsson, 2009). In this paper we use perception and learning. One can argue, however, TTR to put forward an account of the semantics of that part of getting to know the meaning of lin- vague scalar predicates like ‘tall’ that makes pre- guistic expressions consists in learning to identify cise how perceptual information can be integrated the individuals or the situations that the expres- into their meaning representation; how an agent sions can describe. For many concrete words and evaluates whether an entity counts as tall; and how phrases, this identification relies on perceptual in- the proposed semantics for these expressions can formation. In this paper, we focus on characteris- be learned and dynamically updated through lan- ing the meaning of vague scalar adjectives such guage use. as ‘tall’, ‘dark’, or ‘heavy’. We propose a for- We start by giving a brief overview of TTR and mal account that brings together notions from tra- explaining how it can be used for classifying en- ditional formal semanticswith perceptual informa- tities as being of particular types integrating per- tion, which allows us to specify how a logic-based ceptual information. After that, in Section 3, we interpretation function is determined and modified describe the main properties of vague scalar pred- dynamically by experience. icates. Section 4 presents a probabilistic TTR for- The need to integrate language and percep- malisation of the meaning of ‘tall’, which captures tion has been emphasised by researchers work- its context-dependence and its vague character. In ing on the generation and resolution of referring Section 5, we then offer an account of how that This work is licensed under a Creative Commons Attribution meaning representation is acquired and updated 4.0 International Licence. Page numbers and proceedings footer are added by the organisers. Licence details: http: with experience. Finally, in Section 6 we discuss //creativecommons.org/licenses/by/4.0/ related work, before concluding in Section 7. 151 Proceedings of the Third Joint Conference on Lexical and Computational Semantics (*SEM 2014), pages 151–159, Dublin, Ireland, August 23-24 2014. 2 Meaning as Classification in TTR way of constructing ptypes where the arguments of a predicate are entities that have been intro- In this section we give a brief and hence inevitably duced before in the record type. A sample record partial introduction to Type Theory with Records. and record type are shown in (2). For more comprehensive introductions, we refer the reader to Cooper (2005b) and Cooper (2012). x = a x : Ind (2) c = prf(man(a)) : c : man(x) man man 2.1 Type Theory with Records: Main Notions crun = prf(run(a)) crun : run(x) As in any type theory, the most central notion in In (2), a is an entity of type individual and prf(P ) TTR is that of a judgement that an object a is is used as a placeholder for proofs of ptypes P . of type T , written as a : T . In TTR judgements In the record type above, the ptypes man(x) and are seen as fundamentally related to perception, in run(x) constructed from predicates are dependent the sense that perceiving inherently involves cate- on x (introduced earlier in the record type). gorising what we perceive. Some common basic types in TTR are Ind (the type of individuals) and 2.2 Perceptual Meaning + R (the type of positive real numbers). All basic Larsson (2013) proposes a system formalised in types are members of a special type Type. Given TTR where some perceptual aspects of meaning types T1 and T2, we can create the function type are represented using classifiers. For example, the T T 2 whose domain are objects of type T 1 → 1 meaning of ‘right’ (as in ‘to the right of ’) involves and whose range are objects of type T2. Types a two-input perceptron classifier κright(w, t, r), can also be constructed from predicates and ob- specified by a weight vector w and a threshold jects P (a1, . , an). Such types are called ptypes t, which takes as input a context r including an and correspond roughly to propositions in first or- object x and a position-sensor reading srpos. The der logic. In TTR, propositions are types of proofs, sensor reading consists of a vector containing two where proofs can be a variety of things, from situ- real numbers representing the space coordinates of ations to sensor readings (more on this below). x. The classifier classifies x as either being to the Next, we introduce records and record types. right on a plane or not.1 These are structured objects made up of pairs l, v h i x : Ind of labels and values that are displayed in a matrix: (3) if r : , then sr : RealVector (1) a. A record type: pos right(r.x) if (r.srpos w) > t `1 : T1 κ (w, t, r) = · right right(r.x) otherwise `2 : T2(`1) ¬ ... As output we get a record type containing either a `n : Tn(`1, `2, . , `n 1) ptype right(x) or its negation, right(x). Larsson − ¬ (2013) proposes that readings from sensors may `1 = a1 count as proofs of such ptypes. A classifier can `2 = a2 b. A record: r = ... be used for judging x as being of a particular type `n = an on the grounds of perceptual information. A per- ... ceptual proof for right(x) would thus include the Record r in (1b) is of the record type in (1a) if output from the position sensor that is directed to- and only if a1 : T1, a2 : T2(a1), . , and an : wards x. Here, this output would be the space co- Tn(a1, a2, . , an 1). Note that the record may − ordinates of x. contain more fields but would still be of type (1a) if the typing condition holds. Records and record 3 Vague Scalar Predicates types can be nested so that the value of a label is Scalar predicates such as ‘tall’, ‘long’ and ‘ex- itself a record (or record type). We can use paths pensive’, also called “relative gradable adjectives” within a record or record type to refer to specific (Kennedy, 2007), are interpreted with respect to a bits of structure: for instance, we can use r.`2 to refer to a2 in (1b). 1We are here assuming that we have a definition of dot product for TTR vectors a:RealVectorn and b:RealVectorn As can be seen in (1a), the labels `1, . `n in a n such that a b = Σi=1aibi = a1b1 + a2b2 + ... + anbn. We record type can be used elsewhere to refer to the also implicitly· assume that the weight vector and the sensor values associated with them. This is a common reading vector have the same dimensionality. 152 scale, i.e., a dimension such as height, length, or 2002; Kennedy and McNally, 2005; Kennedy, cost along which entities for which the relevant di- 2007; Solt, 2011; Lassiter, 2011). mension is applicable can be ordered. This makes We build on degree approaches but adopt a scalar predicates compatible with degree morphol- perception-based perspective and take a step fur- ogy, like comparative and superlative morphemes ther to formalise how the meaning of these pred- (‘taller than’, ‘the longest’) and intensifier mor- icates can be learned and constantly updated phemes such as ‘very’ or ‘quite’. In this pa- through language use. per, our focus is on the so-called positive form of these adjectives (e.g. ‘tall’ as opposed to ‘taller’ 4 A Perceptual Semantics for ‘Tall’ or ‘tallest’). To exemplify our approach, we will use the scalar A property that distinguishes the positive form predicate ‘tall’ throughout. from the comparative and the superlative forms is its context-dependance. To take a common exam- 4.1 Context-sensitivity ple: If Sue’s height is 180cm, she may be appro- We first focus on capturing the context- priately described as a tall woman, but probably dependence of relative scalar predicates.