Reconstituting beta graphs into an efficacious system (Q1306167)

In this paper the author discusses Charles S. Peirce's Beta system of Existential Graphs, which is a sound and complete deductive system equivalent to first-order systems. The Beta system has three kinds of primitive vocabulary: line, cut and predicate symbols. The author examines the reasons why logicians have been so doubtful of the practical use of this graphical system. Inference rules of a natural deduction system are based on how the formula in question is composed, and accordingly render the system more efficacious. However, as noted by the author, in the case of diagrams a building history of a diagram is not helpful in finding the correct rule and hence does not provide us with an efficient way to state the rules of inference. Peirce's Beta system contains only five rules and keeps a certain symmetry: erasure versus insertion, iteration versus deiteration, while the fifth rule is the rule of the double cut. The author argues that these rules do not take full advantage of visual distinctions in graphs and that reading off visual features as much as possible and implementing them in the rules of inference are the main type of efficacy a diagrammatic system can have over symbolic systems. The naturalness of a graphical system must stem from visual intuitiveness. The author re-presents Peirce's Beta inference rules with more specific symmetries built around the visual features of this system. This reconstruction is much easier to apply than Peirce's original rules. While the system of natural deduction is natural in the sense that the inference rules are formulated around each connective and quantifier, the reconstruction of Peirce's system is natural in the sense that it takes full advantage of the visual features of the diagrammatic system.