# Probabilistic models of vagueness

### Paul Egré, Steven Verheyen

### Language and Logic (Advanced)

### First week, from 17:00 to 18:30, room D

## Abstract

While many-valued logics are familiar to semanticists working on vagueness, the use of probabilities to represent vagueness is lesser known, although they are of common use in psychology and cognitive science more broadly. The goal of this course will be to introduce and compare four classes of probabilistic models that have been used in recent years to model categorization and to deal with vagueness more specifically:

(i) Probabilistic models based on conceptual spaces (Conceptual Space models).

(ii) Probabilistic models based on the notion of variable criteria (Item Response Theory models).

(iii) Probabilistic models based on a theory of noisy or imperfect discrimination (Gaussian models).

(iv) Probabilistic models based on Bayesian inference under uncertainty (Bayesian models).

The course will explain how such models can be related to more familiar models based on degrees of truth, and show specific applications, in particular to the modeling of borderline cases, of the sorites paradox, and of the informativity of vague sentences.

## Description

**Synopsis**

**Lecture 1:** *Vagueness and why probabilities can be useful.*

Main symptoms of vagueness. Centrality of borderline cases. Empirical motivations for degrees of truth. Fuzzy logic and supervaluationist conceptions of degrees.

**Lecture 2: ***The Conceptual spaces approach.*

Prototype theory. Conceptual spaces. The Douven & Decock derivation of graded membership. Gradable adjectives in the CS setting.

**Lecture 3: ***Item Response Theory Models.*

IRT fundamentals. Analyzing the relationship between typicality and graded membership. Degree vagueness vs. multidimensional vagueness.

**Lecture 4:** *A Gaussian model of categorization.*

Imprecision & Indeterminacy. The Central Limit Theorem. Gaussian categorization. Judgments about borderline cases. Relation with 3-valued logic (the strict-tolerant account).

**Lecture 5:** *Vague communication: the Bayesian perspective*

Representation vs. Communication. Metalinguistic vs. Descriptive updates. Bayes Theorem and applications. How can vague utterances be informative? Lassiter's approach.

**Goals & Pre-requisites: **Although the course is marked as "advanced", it will not be highly technical and probabilistic tools will be introduced as we go along (probabilities via proportions; the Central Limit Theorem; Bayes Theorem). The goal of the course is to present both classic and more recent work at the intersection of linguistics and cognitive science, and to discuss empirical studies on vagueness.

**Textbooks**: There is no textbook (yet) on this material, but we will highlight central references as we go along and this site will be updated daily.

**Selected References**

Black, M. (1937). Vagueness: An exercise in logical analysis. *Philosophy of Science, 4*:427–455.

Borel E. (1907). An economic paradox: the sophism of the heap of wheat and statistical truths. *Revue du Mois*. Engl. Translation in *Erkenntnis*, 79: 1080-1088, 2014.

Burnett, Heather. (2016). Gradability in Natural Language: Logical and Grammatical Foundations. *Oxford Studi**es in Semantics and Pragmatics 7.** *Oxford University Press.

Cobreros, P., Egré, P., Ripley D., & van Rooij R. (2012). Tolerant, Classical, Strict. *Th**e Journal of Philosophical Logic,* 41: 347-385

Decock L. and Douven I. (2014). What is graded membership? *Noûs, 48*: 653-682.

Douven I., Decock L., Dietz R., & Egré P. (2013). Vagueness: a conceptual spaces approach. *The Journal of Philosophical Logic,* 42: 137-160.

Douven I., Wenmackers S., Jraissati Y., & Decock L. (2016). Measuring graded membership: The case of color. *Cognitive Science*. doi: 10.1111/cogs.12359

Edgington D. (1997). Vagueness by degrees. In R. Keefe & P. Smith (Eds.), Vagueness: A reader, pp. 294–316, MIT Press.

Egré P. (2016). Vague judgment: A probabilistic account. *Synthese*, doi 10.1007/s11229-016-1092-2.

Egré P. (2011). Perceptual ambiguity and the sorites. In R. Nouwen et al. (Eds.), *Vagueness in communication. LNAI* (Vol. 6517, pp. 127–150). New York: Springer.

Egré P., & Barberousse A. (2014). Borel on the Heap. *Erkenntnis*, 79: 1043-1079.

Egré P., de Gardelle V., & Ripley D. (2013). Vagueness and order effects in color categorization. *Journal of Logic, Language and Information*, 22: 391-420.

Hampton J. A. (2007). Typicality, graded membership, and vagueness. *Cognitive Science, 31*: 355-384.

Kamp, H. and Partee, B. (1995). Prototype theory and compositionality. *Cognition, 57*:129–191.

Koriat A., & Sorka, H. (2015). The construction of categorization judgments: Using subjective confidence and response latency to test a distributed model. *Cognition, 134*: 21-38.

Lassiter D. (2011). Vagueness as probabilistic linguistic knowledge. In R. Nouwen et al. (Eds.), *Vagueness in communication. LNAI* (Vol. 6517, pp. 127–150). New York: Springer.

Lassiter D., & Goodman N. (2016). Adjectival vagueness in a Bayesian model of interpretation. *Synthese*. doi:10.1007/s11229-015-0786-1

McCloskey M. E., & Glucksberg S. (1978). Natural Categories: well defined or fuzzy sets ? *Memory & Cognition*, 6: 462-472.

Raffman, D. (1994). Vagueness without paradox. *Philosophical Review, *103: 41-74.

Smith, N. J. J. (2008). *Vagueness and degrees of truth*. Oxford: Oxford University Press.

Verheyen S., Hampton J. A., & Storms G. (2010). A probabilistic threshold model: Analyzing semantic categorization data with the Rasch model. *Acta Psychologica*, 135: 216-225.

Verheyen S., & Storms G. (2013). A mixture approach to vagueness and ambiguity. PLoS ONE, 8: e63507.

Verheyen S., & Storms G. (2018). Education as a source of vagueness in criteria and degree. In E. Castroviejo, L. McNally and G. Sassoon eds., Gradability, Scale Structure and Vagueness: Experimental Perspectives.

Verheyen S., Dewil S., & Egré P. (2016). Subjectivity in gradable adjectives: the case of “tall” and “heavy”. *Submitted for publication.*

Verheyen S. and Egré P. (2017). Typicality and Graded Membership in Dimensional Adjectives. Under review.

Qing C. and Franke M. (2014). Gradable adjectives, vagueness, and optimal language use: A speaker-oriented model. *Proceedings of SALT*, 24: 23–41,