Peirce’s Extended Euler Diagrams and the System Atl Based on Ladd-Franklin’s Exclusion Relations

We explore Charles Peirce’s innovative extension of Euler diagrams to incorporate negative terms through novel star-shaped curves, representing negation and dualizing the relation of inclusion to express exclusion. The study demonstrates the equivalence of Peirce’s extended Euler diagrams (PED) with the axiomatic system Atl, which is based on Christine Ladd-Franklin’s work on exclusion relations. The PED system is shown to be capable of expressing all fifteen valid syllogisms, thereby streamlining logical reasoning. We also explore the historical development of these diagrams and the implications of considering exclusion as a fundamental relation. By taking exclusion as primary, we connect Peirce’s advancements to Ladd-Franklin’s earlier work, highlighting the simplification of syllogistic reasoning and the historical evolution of Eulerian diagrammatic logic.