Integrating domain knowledge into deep neural networks is a promising way to improve generalization. Existing methods either encode prior knowledge in the loss function or apply post-processing modules, but both depend on identifying useful symbolic knowledge to integrate. Since such rules are often unavailable in real-world vision tasks, we propose a method for targeted knowledge discovery. We propose a Differentiable Knowledge Unit (DKU) that enables modulating the classifier logits, yielding refined class probabilities. The DKU uses implication rules to represent relationships between task classes and implicit concepts learned entirely from the main task supervision, without requiring concept labels. Concepts are identified by dedicated classifiers, whose probabilities are passed to DKU alongside the primary class probabilities. DKU computes a logic-based adjustment vector via fuzzy inference, which modulates the primary class logits to yield refined class probabilities. When concept classifiers represent concepts that do not support the logical rule structure, the resulting adjustments to the class probabilities do not directly minimize the supervision loss. Consequently, optimizing the supervision loss on these adjusted class probabilities implicitly trains the concept classifiers. We construct the rule base so that bidirectional logical relations connect concepts and classes. We enforce the concepts to be distinct from each other and with respect to the classes. This design enforces a clean supervision signal for concept learning. We evaluate our methods on the PASCAL-VOC, COCO, and MedMNIST datasets. We demonstrate improvement through our knowledge integration across these datasets. We conduct domain generalization and hard-sample ablation studies and find that our implicit knowledge discovery and integration outperforms the baseline.