Binary classifiers are traditionally studied by propositional logic (PL). PL can only represent them as white boxes, under the assumption that the underlying Boolean function is fully known. Binary classifiers used in practical applications and trained by machine learning are however opaque. They are usually described as black boxes. In this paper, we provide a product modal logic called PLC (Product modal Logic for binary input Classifier) in which the notion of "black box" is interpreted as the uncertainty over a set of classifiers. We give results about axiomatics and complexity of satisfiability checking for our logic. Moreover, we present a dynamic extension in which the process of acquiring new information about the actual classifier can be represented.