title: The triangle inequality constraint in similarity judgments.
abstract: Since Tversky's (1977) seminal investigation, the triangle inequality, along with symmetry and minimality, have had a central role in investigations of the fundamental constraints on human similarity judgments. The meaning of minimality and symmetry in similarity judgments has been straightforward, but this is not the case for the triangle inequality. Expressed in terms of dissimilarities, and assuming a simple, linear function between dissimilarities and distances, the triangle inequality constraint implies that human behaviour should be consistent with Dissimilarity (A,B) + Dissimilarity (B,C) ≥ Dissimilarity (A,C), where A, B, and C are any three stimuli. We show how we can translate this constraint into one for similarities, using Shepard's (1987) generalization law, and so derive the multiplicative triangle inequality for similarities, Sim(A,C)≥Sim(A,B)⋅Sim(B,C) where 0≤Sim(x,y)≤1. Can humans violate the multiplicative triangle inequality? An empirical demonstration shows that they can.
authors.affiliation: Department of Psychological Sciences, Vanderbilt University, USA. Electronic address: james.m.yearsley@vanderbilt.edu.
authors.forename: James M
authors.initials: JM
authors.lastname: Yearsley
authors.affiliation: Department of Psychology, City University London, UK.
authors.forename: Albert
authors.initials: A
authors.lastname: Barque-Duran
authors.affiliation: Department of Philosophy and Communication, University of Bologna, Italy.
authors.forename: Elisa
authors.initials: E
authors.lastname: Scerrati
authors.affiliation: Department of Psychology, City University London, UK.
authors.forename: James A
authors.initials: JA
authors.lastname: Hampton
authors.affiliation: Department of Psychology, City University London, UK. Electronic address: emmanuel.pothos.1@city.ac.uk.
authors.forename: Emmanuel M
authors.initials: EM
authors.lastname: Pothos
mesh_terms: D006801:Humans; D007600:Judgment; D008962:Models, Theoretical
keywords: Quantum theory; Shepard's generalization law; Similarity; Triangle inequality