Neural networks are state-of-the-art classification approaches but are generally difficult to interpret. This issue can be partly alleviated by constructing a precise decision process within the neural network. In this work, a network architecture, denoted as Classification-By-Components network (CBC), is proposed. It is restricted to follow an intuitive reasoning based decision process inspired by Biederman's recognition-by-components theory from cognitive psychology. The network is trained to learn and detect generic components that characterize objects. In parallel, a class-wise reasoning strategy based on these components is learned to solve the classification problem. In contrast to other work on reasoning, we propose three different types of reasoning: positive, negative, and indefinite. These three types together form a probability space to provide a probabilistic classifier. The decomposition of objects into generic components combined with the probabilistic reasoning provides by design a clear interpretation of the classification decision process. The evaluation of the approach on MNIST shows that CBCs are viable classifiers. Additionally, we demonstrate that the inherent interpretability offers a profound understanding of the classification behavior such that we can explain the success of an adversarial attack. The method's scalability is successfully tested using the ImageNet dataset.