Existential Graphs as a Framework for Introducing Logical Constants and Logical Systems

JAVIER LEGRIS

Neubiberg bei MUENCHEN

Congreso; Colloquium Logicum 2014; 2014

Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG)

Charles S. Peirce conceived his diagrammatic logic of the Existential Graphs (EGs) as the best formulation of logical systems, according to his iconic conception of logic and his idea of ?exact logic?. 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 a new interpretation of Peirce?s EGs is introduced by applying them as a framework for defining logical constants and logical systems. This interpretation is coherent with the idea of the ?productive ambiguity? of diagrammatic representation implied in Peirce?s own conception. Diagrams are open to different interpretations because of their purely structural nature. Furthermore, this interpretation attempts to build up a bridge between Peirce?s EGs and the current perspective in logic of Structural Reasoning. According to this, EGs could be properly understood as a kind of Structural Reasoning, in the sense used by Peter Schroeder-Heister and originated in Gerhard Gentzen?s ideas. Instead of using sequent style systems, the EGs can be used in order to formulate general properties of deduction and to define logical concepts. Instead of combinatorial analysis and recursion (as in the case of sequent systems), EGs can be studied by topology. In this sense, the EGs can be interpreted as an account of logical structures. More specifically, the ?scroll? formulated by Peirce in early formulations of the EGs will be used to express  an implication structure. 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.