1. Department of Biological Sciences, Stanford University, Stanford, California, 94305, USA
2. Center for the Study of Language and Information, Stanford University, Stanford, California, 94305, USA
3. Present addresses: Center for the Study of Language and Information, Ventura Hall, 200 Panama Street, Stanford, California 94305, USA (L.G.); eBay, Inc., 2145 Hamilton Avenue, San Jose, California 95125, USA (T.S.C.).

Correspondence to: Logan Grosenick^1,2,3 Correspondence and requests for materials should be addressed to L.G. (Email: logang@stanford.edu).

Transitive inference (TI) involves using known relationships to deduce unknown ones (for example, using A > B and B > C to infer A > C), and is thus essential to logical reasoning. First described as a developmental milestone in children¹, TI has since been reported in nonhuman primates^{2, 3, 4}, rats^{5, 6} and birds^{7, 8, 9, 10}. Still, how animals acquire and represent transitive relationships and why such abilities might have evolved remain open problems. Here we show that male fish (Astatotilapia burtoni) can successfully make inferences on a hierarchy implied by pairwise fights between rival males. These fish learned the implied hierarchy vicariously (as 'bystanders'), by watching fights between rivals arranged around them in separate tank units. Our findings show that fish use TI when trained on socially relevant stimuli, and that they can make such inferences by using indirect information alone. Further, these bystanders seem to have both spatial and featural representations related to rival abilities, which they can use to make correct inferences depending on what kind of information is available to them. Beyond extending TI to fish and experimentally demonstrating indirect TI learning in animals, these results indicate that a universal mechanism underlying TI is unlikely. Rather, animals probably use multiple domain-specific representations adapted to different social and ecological pressures that they encounter during the course of their natural lives.

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