Diagrams are widely used in reasoning about problems in physics‎,‎ mathematics and logic‎,‎ but have traditionally been considered to be only heuristic tools and not valid elements of mathematical proofs‎.‎ This book challenges this prejudice against visualisation in the history of logic and mathematics and provides a formal foundation for work on natural reasoning in a visual mode‎.‎ The author presents Venn diagrams as a formal system of representation equipped with its own syntax and semantics and specifies rules of transformation that make this system sound and complete‎.‎ The system is then extended to the equivalent of a first‎-order monadic language‎.‎ The soundness of these diagrammatic systems refutes the contention that graphical representation is misleading in reasoning‎.‎ The validity of the transformation rules ensures that the correct application of the rules will not lead to fallacies‎.‎ The book concludes with a discussion of some fundamental differences between graphical systems and linguistic systems‎.‎ This groundbreaking work will have important influence on research in logic‎,‎ philosophy and knowledge representation‎.‎