Review by: Arnold, Ruedi (2004-10-19)

The two authors are well-known in the field of logic, reasoning, and education through their prior work. In this paper they address the fundamental question of how people reason and “do logic”. They talk about their experience as lecturers employing their educational software which uses suitable graphical representations.

The first autobiographic chapter is about the tradition of teaching elementary logic at Stanford University using computers. This started in the late 1960’s with the introduction of a program called Valid. They authors also describe Turing’s World, their software for visualizing Turing machines, and talk about its advantages. Tarski’s World is introduced which teaches symbolic logic and first order logic with geometric objects of various shapes and sizes.

Through experience with these programs, the authors became impressed by the cognitive power of graphical representations. They argue that a lot of what they call heterogeneous reasoning is happening while students solve exercises. Students naturally combine information from two kinds of representation: e. g. Tarki’s World diagrams and first-order sentences (called sentential sentences). The traditional theory of reasoning seems inadequate for this, as there seems to be a discrepancy between traditional logic theory and this empirical observation of heterogeneous reasoning. A practical solution to this is the authors latest software, Hyperproof, which allows the combination of visual and sentential information to solve many different types of reasoning problems.

The authors emphasize the power of diagrams and visual representations for reasoning. They hint at a new enriched theory of reasoning and claim that logic must be able to cope with a multitude of representations in rigorous ways. This paper claims that the traditional way logic and reasoning is thought about does not match well the way we think and reason naturally. We therefore need to extend and enrich the traditional notions of syntax, semantics, logical consequence, and proofs to allow new (visual) forms of representation.

This work illustrates how the authors came to realize and appreciate the power of suitable graphical representations. Computers and their sophisticated graphical capabilities are an important tool to support such an approach. This paper targets reasoning, one of the most fundamental concepts of all science. However, many of the authors’ insights about the power of suitable graphical representations hold for many other subjects and for computer-supported learning environments on these subjects too.