The Role of Structural Reasoning in the Genesis of Graph Theory

History and Philosophy of Logic 40 (3):266-297 (2019)
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Abstract

The seminal book on graph theory by Dénes Kőnig, published in the year 1936, collected notions and results from precursory works from the mid to late nineteenth century by Hamilton, Cayley, Sylvester and others. More importantly, Kőnig himself contributed many of his own results that he had obtained in the more than twenty years that he had been working on this subject matter. What is noteworthy is the fact that the fundamentals of what he calls directed graphs are taken almost exhaustively from Paul Hertz' 1922 article on structural reasoning about sentences of the form a→b. This is not a fact that is well known in logical circles, even though Kőnig fully acknowledges this in his book. In view of the numerous trends in the recent decades to describe and explicate logical matters by means of graphs, the fact that it was Hertz' foundation of structural reasoning that informed basic notions of graph theory in the first place is highly significant. The main goal of this paper is to summarize Hertz' article and demonstrate how Kőnig integrates the notions and results presented therein in his book. This is followed by an exposition of how and when Hertz' results were reinvented in terms of graph theory. A critical discussion of the opinion expressed by both Hertz and Kőnig that the more general sentences of the form (a1,…,an)→b, introduced by Hertz in a companion article in 1923, cannot be interpreted by graphs concludes this paper.

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References found in this work

Untersuchungen über das logische Schließen. I.Gerhard Gentzen - 1935 - Mathematische Zeitschrift 35:176–210.
Substructural Logics.Peter Joseph Schroeder-Heister & Kosta Došen - 1993 - Oxford, England: Oxford University Press on Demand.
Gentzen's proof systems: byproducts in a work of genius.Jan von Plato - 2012 - Bulletin of Symbolic Logic 18 (3):313-367.
Substructural Logics.Peter Schroeder-Heister - 1996 - Erkenntnis 45 (1):115-118.

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