Connecting the dots: hypergraphs to analyze and visualize the joint-contribution of premises and conclusions to the validity of arguments

Abstract

A detailed analysis of joint-contribution of premises and conclusions in classically valid sequents is presented in terms of hypergraphs. In (Saint-Germier, P., Verdée, P., & Villalonga, P. T. (2024). Relevant entailment and logical ground. Philosophical Studies (pp. 1–43). https://doi.org/10.1007/s11098-024-02101-1), this idea of joint-contribution is introduced and motivated as a method for characterizing four kinds of relevant validity, in the sense of selecting the relevantly valid sequents among the classically valid sequents. The account in (Saint-Germier, P., Verdée, P., & Villalonga, P. T. (2024). Relevant entailment and logical ground. Philosophical Studies (pp. 1–43). https://doi.org/10.1007/s11098-024-02101-1) is built on a calculus, called \(\textsf{GLK}^{\hbox {a}}\), which proves grounding claims for (enthymematically) valid sequents. In the present paper an adequate representation of \(\textsf{GLK}^{\hbox {a}}\) is given in terms of hypergraphs. The hypergraphs are a kind of diagrammatic proofs for Classical Propositional Logic, entirely based on the grounds of premises and conclusions. The hypergraphs and their visualization provide insights into the relations between premises and conclusions and into the way validity is produced by the binding of premises and conclusions via their partial grounds. They visualize the network of elements of the sequent that contribute to its logical validity. Non-contributing (i.e. irrelevant) premises and conclusions are then specified to be those that are disconnected from the network, however one constructs the graphs.