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.

"I will conclude by offering general praise for a work that really does deserve praise. Even more, it deserves to be read by those mathematicians and logicians who adhere to the general prejudice against diagrams. Shin has gone much farther than anyone in showing how a diagrammatic system can hold its own as a medium for reasoning....this book is clear and convincing....Shin's mastery and manipulation of her technical tools is always thorough and lucid. All in all, this is a very impressive, valuable piece of work." Modern Logic

"The book is refreshing in its clarity and modesty, and in its regular return to historial precedents. It will give welcome reassurance to all those logicians for whom pictures seem indisputably the simplest way for the mind to comprehend syllogistic reasoning." M. Schiff, Choice

"...The author's style is nice, very precise and approachable. The soundness and completeness of Venn systems provide a legitimate reason why logicians should be interested in nonlinguistic representation systems. Undoubtedly, this will be one of the rare inspiring sources motivating others to work in the field of visual representation systems." Branislav Boricic, Mathematical Reviews

"Shin has shown that sysytems of diagrams can be developed that precisley parallel axiomatic formal systems for monadic predicate logic....Shin correctly points out what diagram systems and verbal systems can both do...that realistic pictures...can't." Phillip L. Peterson, Philosophy in Review