Decomposing Analogy: A Logic Characterization

Analogical proportions, i.e., relational assertions of the form “a is to b as c is to d” are fundamental for analogical reasoning, a cognitively motivated form of reasoning that has several important applications in AI, including problem solving and learning. This paper contributes to the logic characterization of analogical reasoning by establishing a link between two perspectives. Analogical proportions can either be viewed atomically as a quaternary relation or as the implicit relation induced by applying analogical comparison a : b to an element d in order to identify a suitable element c. For linking these views we tackle a general question: Given a quaternary relation between four elements a, b, c, d, how can it be decomposed into a representation \(a:b\,{::}\,c:d\) with “ \({:}\,\!{:}\) ” denoting a binary relation and “:” denoting a binary function—possibly under additional constraints on  :  and \({:}\,\!{:}\) ? In particular we show that for a whole class of analogical proportions such a decomposition is possible with “:” denoting the same function in all of the analogical proportions of the class.