Situation-Theoretic Account of Valid Reasoning with Venn Diagrams

Abstract

Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.