Early developments of linear logic can be traced back to Barr's *-autonomous categories and Lambek's bilinear logic. We show here that C. S. Peirce's early work on the logic of relations should be placed within this tradition. Peirce understood linear distributivity and the linear negation laws, understood linear implication in the form of residuation, and emphasized the dialogical nature of the linear connectives. Much of this can be found in Peirce's early algebraic work on the study of relations going back as early as the 1880s. Peirce eventually went on to develop a diagrammatic calculus – what he called the Existential Graphs – he thought better suited for the purpose. We go on to show graphs corresponding to these notions and confirm that many of Peirce's later studies in the graphs employ these concepts. The result is a dramatic revision of our understanding of the Existential Graphs, as well as Peirce's place in the logic tradition.
Nathan Haydon (Wed,) studied this question.