Existential Graphs as an Efficient, Formal, Representation System for Logic

摘要

It has been believed that Peirce himself was more proud of his existential graphs than of his symbolic systems. The modern history of logic, however, gave an opposite verdict on these two different kinds of logical systems. How can we explain the discrepancy between Peirce's own evaluation and a commonly accepted view on this issue? The paper explores the rationale for Peirce's own preference from his thoughts on various issues—representation, formalization, diagrams, notations, abstractions, and so forth. At the same time, I would like to speculate why Peirce himself did not spell out his hunch for the superiority of existential graphs over symbolic systems. A representation system has its own semantic domain, and we aim to reason about a certain target domain by adopting a system. Hence, we would like to have a representation system which carries out this goal more accurately and more efficiently. It turns out, as I show, that there is a tension between accuracy and efficiency of a representation system. I claim that modern logicians and mathematicians, facing some disasters and anxiety around the turn of the 20th century, opted for accuracy, which is a correct thing to do. However, accuracy, unfortunately, has become the only thing logicians expect from a formal system. As a result, a narrow concept of formalization has been dominant since the dawn of post-modern logic. Presenting his theory of notation, I argue that Peirce's concept of formalization is much broader than the modern use of formalization and became the main source of his invention of non-symbolic systems. At the same time, his theory of abstraction provides us with theoretical grounds for the existence of a system where there needs to be no trade-off between accuracy and efficiency. If my picture is convincing, it should surprise us that Peirce never defended his existential graphs in this clear way. I will present a couple of speculations about Peirce's silence on the issue by relying both on a historical situation and on Peirce's own view about deductive systems.