Starting with simple examples that demonstrate the relevance of cutting and pasting logic, the monograph develops a mathematical theory combining and decomposing logic, ranging from logic based on propositional and first for high-level logic and the logic of truth-based non-functional. It also provides examples of potential applications in fields such as security protocols arise, quantum computing, networking and argumentation theory, in addition to discussing the more classical applications such as software specification, knowledge representation, computational linguistics and modular automated reasoning. This monograph will appeal to researchers and graduate students in mathematical logic, computational theory and philosophy of logic that has no previous knowledge of the subject combine and decompose logic, but with knowledge of first-order logic.