Ordinary people - mathematicians among them - grasp when they take something to follow (deductively) from something ease. This is the backbone of our self-ascribed ability to reason. This book investigates the connection between that ordinary notion of consequence and the formal analogues developed by logicians. One crucial claim of the book is that, despite our apparent intuitive grasp of when something follows from something else, we have no introspective access to the rules by which we reason, nor to the scope and range of the domain, as it were, of our reasoning. The point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof.