Recent years have witnessed a renewed interest in the explanation of classifier systems in the field of explainable AI (XAI). The standard approach is based on propositional logic. We present a modal language which supports reasoning about binary input classifiers and their properties. We study a family of classifier models, axiomatize it as two proof systems regarding the cardinality of the language and show completeness of our axiomatics. Moreover, we show that the satisfiability checking problem for our modal language is NEXPTIME-complete in the infinite-variable case, while it becomes polynomial in the finite-variable case. We moreover identify an interesting NP fragment of our language in the infinite-variable case. We leverage the language to formalize counterfactual conditional as well as a variety of notions of explanation including abductive, contrastive and counterfactual explanations and biases. Finally, we present two extensions of our language: a dynamic extension by the notion of assignment enabling classifier change and an epistemic extension in which the classifier’s uncertainty about the actual input can be represented.