Abstract

This thesis challenges a general prejudice against visualization in the history of logic and mathematics, by providing a semantic analysis of two graphical representation systems--a traditional Venn diagram representation system and an extension of it. While Venn diagrams have been used to solve problems in set theory and to test the validity of syllogisms in logic, they have not been considered valid proofs but heuristic tools for finding valid formal proofs. ;I present Venn diagrams which have been used in logic as a formal system of representation equipped with its own syntax and semantics. I also show that Venn-I, with the rules of transformation that I specify, is sound and complete. The soundness of Venn-I refutes a worry that mathematicians and logicians have about graphical representation: Fallacies often have arisen from the misuse of diagrams in mathematical proofs, especially in geometry. However, the validity of the transformation rules assures us that the correct application of these rules will not lead to fallacies. ;I extend Venn-I so that this new system has a general way to convey disjunctive information. A new formation rule is introduced and new transformation rules are added. I call this extended system Venn-II. The semantics of Venn-II is formalized as an extension of the semantics of Venn-I. The soundness and completeness of Venn-II are also proven. I then show that Venn-II and a monadic first-order language are equivalent in their expressiveness. ;The analysis of Venn-I and Venn-II leads to interesting issues which have their analogues in other deductive systems. I conclude the thesis with a discussion of some fundamental differences between graphical systems and linguistic systems