Meaning and Use of Gradable Adjectives: Formal Modeling Meets Empirical Data

Meaning and Use of Gradable Adjectives: Formal Modeling Meets Empirical Data Ciyang Qing (qciyang@gmail.com) Michael Franke (m.franke@uva.nl) Institute for Logic, Language and Computation, University of Amsterdam Science Park, Kruislaan 107, 1098 XG Amsterdam, The Netherlands Abstract pragmatic applicability, and it is this component of the model that we test empirically. We fit our model to data that was gathered by Solt and Gotzner (2012) (for a quite different pur- pose) and test model predictions against data from our own replication of their experiment. Despite its simplicity, our optimality-based model explains the data astonishingly well. The next section first introduces our speaker-based proba- bilistic model. Thereafter we detail Solt and Gotzner’s (2012) experiment and our replication. We discuss the model’s fit to the data and conclude with a critical reflection. The meaning of gradable adjectives is highly context- dependent, and is notoriously difficult to capture precisely. Re- cent work in theoretical linguistics suggests that the way we use gradable adjectives can be explained in terms of optimal language use. To test this hypothesis we formulate a prob- abilistic speaker model that combines ideas from Bayesian approaches to pragmatic reasoning as social cognition with broader optimality considerations, as suggested by evolution- ary linguistics. We demonstrate that, despite its simplicity, the model explains empirical data on the applicability of adjectives in context astonishingly well. Keywords: gradable adjectives; context-dependence; natural language production/generation; Bayesian cognitive modeling Optimal Use of Gradable Adjectives Introduction The meaning and use of gradable adjectives like tall, big, dark or full is elusive in manifold ways and continues to inspire research in not only linguistics and psychology, but also ma- chine learning and other related fields. It is notoriously diffi- cult to pin down the meaning and use of gradable adjectives mainly because of their context-dependence and vagueness. Whether a sentence like “John is tall” is felt to be true or pragmatically appropriate depends on the context in which the sentence is used, in particular on a contrasting set of indi- viduals against which John is to be compared (in terms of his height). The goal of the work presented here is to shed light on this immediate context-dependence of gradable adjectives. Building on previous work in the same direction (Barner & Snedeker, 2008; Schmidt, Goodman, Barner, & Tenen- baum, 2009; Syrett, Kennedy, & Lidz, 2010), we investi- gate the dependence of intuitive judgements of applicability of gradable adjectives on statistical properties of a visually presented comparison class. The key novelty of the present proposal is a fully predictive probabilistic model that formal- izes the idea that the applicability of gradable adjectives is de- termined by pragmatic reasoning about optimal language use in context. The model we present enriches previous Bayesian approaches to pragmatic reasoning in terms of social cogni- tion (e.g. Frank & Goodman, 2012; Goodman & Stuhlm¨uller, 2013; Lassiter & Goodman, 2013) with ideas borrowed from evolutionary game theory (Potts, 2008; Franke, 2012). More concretely, the model presented here is superficially similar, but conceptually distinct from the interpretation- based model of Lassiter and Goodman (2013). In contrast to the latter, the model we present here makes straightforward predictions about both comprehension and production. The key assumption in our modeling is that the speaker’s prag- matic reasoning is anchored in considerations of global opti- mality of the conventional usage conditions. We argue that it is the production side that is responsible for judgements of One of the most fundamental reasons of using gradable ad- jectives is to ensure communicative success when trying to describe a referent, e.g., such as to pick it out, or learn about its properties. To make this intuition more concrete, we adopt a degree-based approach to the semantics of gradable adjec- tives (Kennedy & McNally, 2005; Kennedy, 2007). Degree- based approaches hold that a sentence of the form “x is A” is true iff the degree d A (x) to which object x has property A exceeds a contextually given standard of comparison θ, i.e., d A (x) ≥ θ. However, exactly how θ is conventionally derived from the context is left open. Following the general tenet of evolutionary linguistics that language conventions are shaped to achieve optimal communicative success, we propose to fill this gap by defining which values of θ are optimal in a given context. The motivating idea behind our production model is then that speakers employ a standard of comparison θ with a probability proportional to the communicative efficiency of using θ as a general convention. Specifically, we measure communicative efficiency of θ in terms of the extent to which the utterance “x is A” would help resolve the (possibly implicit and hypothetical) Question un- der Discussion “how A is x? ” against the background of a contextually given comparison class of objects with varying levels of A-ness. We use p to denote the common knowl- edge about prior probability distribution of A-degrees in the comparison class. For example, if the conversation is about basketball players’ heights, then p is the prior distribution of the height of a basketball player from one’s common world knowledge. (Here we only consider discrete degree scales but it is easy to adapt the model to continuous density functions.) When the conventional standard of comparison θ is fixed, a literal listener ρ 0 , upon hearing the utterance “x is A,” learns from its semantic truth that the actual degree d A (x) ≥ θ. Thus his new belief about d A (x), denoted as ρ 0 (d A (x) | A; θ), is the conditional probability p(d A (x) | d A (x) ≥ θ). If on the other hand the speaker says nothing, the literal listener has no additional information and thus his belief stays the same: