Abstract: Commonsense knowledge (CSK) supports a variety of AI applications, from visual understanding to chatbots. Prior works on acquiring CSK, such as ConceptNet, have compiled statements that associate concepts with properties that hold for most or some of their instances. Each concept and statement is treated in isolation from others, and the only quantitative measure (or ranking) is a confidence score that the statement is valid. This paper aims to overcome these limitations by introducing a multi-faceted model of CSK statements and methods for joint reasoning over sets of inter-related statements. Our model captures four different dimensions of CSK statements: plausibility, typicality, remarkability and salience, with scoring and ranking along each dimension. For example, hyenas drinking water is typical but not salient, whereas hyenas eating carcasses is salient. For reasoning and ranking, we develop a method with soft constraints, to couple the inference over concepts that are related in a taxonomic hierarchy. The reasoning is cast into an integer linear programming (ILP), and we leverage the theory of reduction costs of a relaxed LP to compute informative rankings. Our evaluation shows that we can consolidate existing CSK collections into much cleaner and more expressive knowledge.