Peirce on Diagrammatic Reasoning and Semeiotic

Charles Sanders Peirce (1839–1914) is one of the “grounding fathers” of mathematical logic, having developed all of the key formal results of modern logic. He did it firstly (from 1860 on) in the algebraic tradition of mathematical logic stemming from Boole, combining it with the logic of relations, explicitly developed by Augustus De Morgan. From this, Peirce obtained a system that included quantifiers—a term he seems to have invented—and relative predicates. Developing his own system of relative terms, Peirce started from Boole’s system, trying to apply it to De Morgan’s logic of relations. Indeed, Peirce’s aim is to include the logic of relations into the calculus of algebra using his own system of algebraic signs. On the one hand, Peirce’s algebraic notation will be presented, specially: (a) relative terms as iconic representations of logical relations; (b) Peirce’s quantifiers and the passage from a linear notation to a diagrammatic one. On the other hand, Peirce’s graphical notation will be presented, specially: (a) his Alpha and Beta systems, which are fully compatible with what is nowadays called first-order logic, (b) and his unfinished Gamma system, designed for second-order logic and modal logic.