Yearsley James M, Barque-Duran Albert, Scerrati Elisa, Hampton James A, Pothos Emmanuel M
Department of Psychological Sciences, Vanderbilt University, USA.
Department of Psychology, City University London, UK.
Prog Biophys Mol Biol. 2017 Nov;130(Pt A):26-32. doi: 10.1016/j.pbiomolbio.2017.03.005. Epub 2017 Mar 28.
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.
自特沃斯基（1977年）的开创性研究以来，三角不等式与对称性和最小性一起，在对人类相似性判断的基本限制的研究中发挥了核心作用。在相似性判断中，最小性和对称性的含义很直接，但三角不等式并非如此。用差异来表示，并假设差异与距离之间存在简单的线性函数，三角不等式约束意味着人类行为应符合差异（A，B）+差异（B，C）≥差异（A，C），其中A、B和C是任意三个刺激。我们展示了如何使用谢泼德（1987年）的泛化定律将这个约束转化为相似性的约束，从而推导出相似性的乘法三角不等式，即相似性（A，C）≥相似性（A，B）·相似性（B，C），其中0≤相似性（x，y）≤1。人类会违反乘法三角不等式吗？一项实证证明表明他们会。
