Two case studies
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To illustrate how PDS bridges formal semantics and experimental data, we’ll examine two case studies that exemplify different aspects of the framework.
Case Study 1: Gradable Adjectives
Vague predicates provide an ideal starting point because effectively everyone agrees we need to incorporate into our theories of their meanings something representation that drives gradience. Expressions like tall, expensive, and old lack sharp boundaries—there’s no precise height at which someone becomes tall (Lakoff 1973; Sadock 1977; Lasersohn 1999; Krifka 2007; Solt 2015).
Formal semantic theories have long recognized this gradience. Degree-based approaches (Klein 1980; Bierwisch 1989; Kamp 1975; Chris Kennedy 1999; Christopher Kennedy and McNally 2005; Christopher Kennedy 2007; Barker 2002) analyze gradable adjectives as expressing relations to contextual thresholds:
- tall is true of \(x\) if \(\ct{height}(x) \geq d_\text{tall}\) (context)
The threshold \(d_\text{tall}\) varies with context—what counts as tall for a basketball player differs from tall for a child. But even within a fixed context, speakers show gradient judgments about borderline cases.
This makes gradable adjectives ideal for demonstrating how PDS works. They allow us to show how the framework maintains the compositional degree-based analysis from formal semantics while adding probability distributions over thresholds to capture gradient judgments and how it can support modeling how context shifts these distributions and link threshold distributions to slider scale responses. It also allows us to show how PDS is well-poisitioned to help us understand the additional complexity beyond this basic picture that recent experimental work reveals. Different adjective types show distinct patterns: relative adjectives like tall and wide show maximum gradience in their positive form, while absolute adjectives like clean and dry exhibit different threshold distributions altogether. Furthermore, the distinction between minimum and maximum standard adjectives reveals asymmetric patterns of imprecision that challenge simple threshold-based accounts. How do we account for these differences and relate them to judgment data?
Recent years have seen partial integration into computational models (Lassiter and Goodman 2013, 2017; Qing and Franke 2014; Kao et al. 2014; Bumford and Rett 2021). We’ll show that PDS allows us to synthesize and compare these different partial approaches.
Case Study 2: Factivity
While vagueness involves expected gradience, factivity presents a puzzle. Traditional theory treats factivity as discrete—predicates either trigger presuppositions or they don’t (Kiparsky and Kiparsky 1970; Karttunen 1971).1 Yet experimental data reveals pervasive gradience.
A predicate is factive if it triggers inferences about its complement that project through entailment-canceling operators. Love appears factive because Mo left is inferrable from the standard family of sentences in (1)–(3):
- Jo loves that Mo left.
- Jo doesn’t love that Mo left.
- Does Jo love that Mo left?
But when White and Rawlins (2018) (discussed above) and Degen and Tonhauser (2022) collected projection judgments at scale, they found continuous variation (Xue and Onea 2011; Smith and Hall 2011; Djärv and Bacovcin 2017 also observe similar patterns). Qualitatively, Degen and Tonhauser (2022) argue that there is no clear line separates factive from non-factive predicates. Mean projection ratings vary continuously from pretend (lowest) to be annoyed (highest).
This gradience poses a theoretical challenge (Simons 2007; Simons et al. 2010, 2017; Tonhauser, Beaver, and Degen 2018).
Kane, Gantt, and White (2022) later showed that this gradience is likely due largely to task effects and measurement noise. They demonstrate that when one applies a clustering model to these data that accounts for noise due to various factors, many of the standard subclasses of factives pop out.
To get a sense for how these clusters appear in the data, we can look back at this figure. Note that (i) there are clearly at least two separate “bumps” in the histogram; but (ii) the peak of the right bump is not at 1, as we might have expected. This is because some of these subclasses–e.g. the cognitive factives, which Karttunen (1971) observes to not always give rise factivity–appear to themselves be associated with non-necessary factive inferences.
In this case study, we’ll focus on understanding what gives rise to this gradience. We’ll take for granted that there are in fact discrete subclasses of factives, as Kane, Gantt, and White (2022) show, but that it remains an open question what kind of uncertainty drives the gradience internal to the subclasses. We’ll consider two hypotheses that PDS allows us to state precisely and test against the data collected by Degen and Tonhauser (2021), which uses the same experimental paradigm as Degen and Tonhauser (2022):
The Fundamental Discreteness Hypothesis: Factivity remains discrete; gradience reflects: - Multiple predicate senses (see Spector and Egré 2015) - Structural ambiguity affecting projection (Varlokosta 1994; Giannakidou 1998, 1999, 2009; Roussou 2010; Farudi 2007; Abrusán 2011; Kastner 2015; Ozyildiz 2017) - Contextual variation in whether or not complements are at-issue (Simons et al. 2017; Roberts and Simons 2024; Qing, Goodman, and Lassiter 2016)
The Fundamental Gradience Hypothesis: No discrete factivity property exists. Gradient patterns reflect different degrees to which predicates support complement truth inferences–e.g. by viewing at-issueness as itself fundamentally continuous (Tonhauser, Beaver, and Degen 2018).
PDS allows us to implement both hypotheses formally and test their predictions against fine-grained response distributions—not just means, but entire judgment patterns including multimodality that might indicate mixture distributions. We’ll show how this approach can be applied to judgment data aimed at capturing factivity using various experimental paradigms (Tonhauser 2016; Djärv and Bacovcin 2017; Djärv, Zehr, and Schwarz 2018; White and Rawlins 2018; White et al. 2018; White 2021; Degen and Tonhauser 2021, 2022; Jeong 2021; Kane, Gantt, and White 2022).
References
Footnotes
We’ll spend a lot of time on Day 4 saying exactly what we mean by discrete here. Karttunen (1971), of course, classically argues that there are predicates that sometimes trigger presuppositions and sometimes don’t. For our purposes, we’ll say that this behavior is discrete in the sense that it’s more like ambiguity than vagueness. That is, we’ll show that uncertainty around factivity displays the hallmarks of resolved uncertainty.↩︎