Existential Graphs as Structural Reasoning

JAVIER LEGRIS

Lowell, MA

Congreso; The Charles S. Peirce International Centennial Congress.; 2014

The Charles S. Peirce Foundation

C. S. Peirce conceived his diagrammatic logic of the Existential Graphs (EGs) as the most natural formulation of logical deduction, according to his idea of ?exact logic?, and consequently they constituted a general methodology for the mathematical treatment of deduction based on notions from topology. Thus, the EGs open a fully new direction in symbolic logic. Peirce formulated different diagrammatic systems for the EGs that have been intensively studied and expanded in the last years. In this paper an attempt to build up a bridge between Peirce?s EGs and the current perspective in logic of Structural Reasoning is carried out. This perspective stems from the original ideas of Gerhard Gentzen in order to analyze deduction (Gentzen 1935). More specifically, the aim of this paper is to suggest that Peirce?s Existential Graphs can be properly understood as a kind of Structural Reasoning. Instead of using sequent style reasoning, EGs uses diagrams in order to formulate general properties of deduction and to define logical concepts. Instead of combinatorial analysis and recursion, EGs can be studied by topology. In this sense, the EGs can be interpreted as an account of logical structures. Here only the basic conception is outlined in an informal way, without making a full exposition of the technical details. The main claims of the paper can be summarized as follows: (1) Diagrams can be used in two different senses: (a) as an intuitive way to express the nature of deduction in terms of diagrams, (b) as a mathematical tool to characterize logical constants of different logic systems. (2) The theory for formulating logic systems evolves from a coherent philosophical perspective, so that there is a clear correspondence between logical theory (including a topological framework) and philosophical theory (the iconic nature of deduction). (3) The philosophical theory does not make deduction and logical concepts dependent on ordinary language. Furthermore, the ?scroll? of the EGs will be used to express more an implication structure than a conditional operator. It is to be noted that this idea is consistent with the ?productive ambiguity? of diagrammatic representation. Diagrams are open to different interpretations because of their purely structural nature. From the point of view of structural reasoning, logic begins with implication structures, so that implication is central to logic. The logical operators, as well as modal operators generally, can be characterized as such, by the way they interact with respect to implication (see v.g. Koslow 2005, p. 167). Anyway it must be noticed that Gentzen himself in his Investigations into Logical Deductions clarified his sequents in terms of conditional (see Gentzen 1935 secc. 2, engl. Translation p. 82).