Boolean proportions

Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, commonsense reasoning, learning, language acquisition, and story telling. This paper studies analogical proportions between booleans of the form `$a$ is to $b$ what $c$ is to $d$' called boolean proportions. Technically, we instantiate the abstract algebraic framework of analogical proportions recently introduced by the author in the boolean domain consisting of the booleans 0 and 1 together with boolean functions. It turns out that our notion of boolean proportions has a simple logical characterization which entails appealing mathematical properties. In a broader sense, this paper is a further step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.