Types and Tokens for Logic with Diagrams

It is well accepted that diagrams play a crucial role in human reasoning. But in mathematics, diagrams are most often only used for visualizations, but it is doubted that diagrams are rigor enough to play an essential role in a proof. This paper takes the opposite point of view: It is argued that rigor formal logic can carried out with diagrams. In order to do that, it is first analyzed which problems can occur in diagrammatic systems, and how a diagrammatic system has to be designed in order to get a rigor logic system. Particularly, it will turn out that a separation between diagrams as representations of structures and these structures themselves is needed, and the structures should be defined mathematically. The argumentation for this point of view will be embedded into a case study, namely the existential graphs of Peirce. In the second part of this paper, the theoretical considerations are practically carried out by providing mathematical definitions for the semantics and the calculus of existential Alpha graphs, and by proving mathematically that the calculus is sound and complete. 1 Motivation and Introduction The research field of diagrammatic reasoning investigates all forms of human reasoning and argumentation wherever diagrams are involved. This research area is constituted from multiple disciplines, including cognitive science and psychology as well as computer science, artificial intelligence, logic and mathematics. But it should not be overlooked that there has been until today a long-standing prejudice against non-symbolic representation in mathematics and logic. Without doubt diagrams are often used in mathematical reasoning, but usually only as illustrations or thought aids. Diagrams, many mathematicians say, are not rigorous enough to be used in a proof, or may even mislead us in a proof. This attitude is captured by the quotation below: [The diagram] is only a heuristic to prompt certain trains of inference; ... it is dispensable as a proof-theoretic device; indeed ... it has no proper place in a proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable area. Neil Tennant 1991, quotation adopted from [Ba93] Nonetheless, there exist some diagrammatic systems which were designed for mathematical reasoning. Well-known examples are Euler circles and Venn diagrams. More important to us, at the dawn of modern logic, two diagrammatic systems had been invented in order to formalize logic. The first system is Frege’s Begriffsschrift, where Frege tried to provide a formal universal language. The other one is of more relevance for this conference, as Sowa’s conceptual graphs are based on them: It is the systems of existential graphs (EGs) by Charles Sanders Peirce, which he used to study and describe logical argumentation. But none of these systems is used in contemporary mathematical logic. In contrast: For more than a century, linear symbolic representation systems (i.e., formal languages which are composed of signs which are a priori meaningless, and which are therefore manipulated by means of purely formal rules) have been the exclusive subject for formal logic. There are only a few logicians who have done research on formal, but non-symbolic logic. The most important ones are without doubt Barwise and Etchemendy. They say that there is no principle distinction between inference formalisms that use text and those that use diagrams. One can have rigorous, logically sound (and complete) formal systems based on diagrams. Barwise and Etchemendy 1994, quotation adopted from [Sh01] This paper advocates this view that rigor formal logic can carried out by means of manipulating diagrams. The argumentation for this point of view will be embedded into a case study, where the theoretical considerations are practically carried out to formalize the Alpha graphs of Peirce. Although the argumentation is carried out on EGs, it can be transferred to other diagrammatic systems (e.g., for conceptual graphs) as well. For those readers who are not familiar with EGs, Sec. 2 provides a short introduction into EGs. There are some authors who explored EGs, e.g. Zeman, Roberts or Shin. All these authors treated EGs as graphical entities. In Sec. 3, some of the problems which occur in this handling of EGs are analyzed. For this, the approach of Shin (see [Sh01]) will be used. It will turn out that informal definitions and a missing distinction between EGs and their graphical representations are the main problems. In fact, due to this problems, Shin’s (and other authors as well) elaboration of EGs is from a mathematicians point of view insufficient and cannot serve as diagrammatic approach to mathematical logic. I will argue that a separation between EGs as abstract structures and their graphical representations is appropriate, and that a mathematical definition for EGs is needed. A question which arises immediately is how the graphical representations should be defined and handled. In Secs. 4 and 6, two approaches to solve this question are presented. It will be shown that a mathematical formalization of the graphical representations may cause a new class of problems. In Sec. 5 we will discuss why mathematical logic does not have to cope with problems which arise in diagrammatic systems. It will turn out that the preciseness of mathematical logic is possible although the separation between formulas and their representation is usually not discussed. From this result we draw the conclusion that a mathematical formalization of the diagrammatic representations of EGs is not needed. In Sec. 6, the results of the preceding sections are brought together in order to describe my approach for a mathematical foundation of diagrams. In the remaining sections, the results of the theoretical discussion are applied to elaborate mathematically a complete description of the Alpha-part of EGs. 2 Existential Graphs In this paper, we consider the Alphaand Beta-part of existential graphs. Alpha is a system which corresponds to the propositional calculus of mathematical logic. Beta builds upon Alpha by introducing new symbols to Alpha, and it corresponds to first order predicate logic (FOPL), that is first order logic with predicate names, but without object names and without function names. We start with the description of Alpha. The EGs of Alpha consist only of predicate names of arity 0, which Peirce called medads, and of closed, double-point-free curves which are called cuts and used to negate the enclosed subgraph. Medads can