 Peirce’s journey on formal deductive logic started with Boolean calculus and De Morgan’s logic of relatives. Boolean algebra created a path to generalize Aristotelian syllogism and De Morgan’s ambition to formalize relations opened a new territory to conquer. However, Peirce’s predicate logic is neither a mechanical expansion from the existing logics nor a simple combination of these two. A leap made by Peirce from his contemporary logic is qualitatively substantial enough to call Peirce a founder of modern deductive logic, as the entry explains. The first section explores Peirce’s development of predicate logic presented in his several well-known papers, by locating the root of Peirce’s introduction of quantifiers and bound variables. While formal details and notations of Peirce’s first-order logic can be overwhelming, one should not lose the sight of the bigger picture by paying attention to the main motivation behind Peirce’s enterprise for a new logic. Conquering new territory—relations—with new formal notation, Peirce’s adventure was launched into another dimension—a new mode of representation, that is, diagrammatic representation. This is the topic of the second section. While two systems of Peirce’s Existential Graphs (“EG” henceforth) are presented, the following perspective is at the background: Peirce’s EG were not invented just as random alternatives, logically equivalent to his own predicate logical notation, but a reflection of Peirce’s new approach to logic and formalization. As Peirce’s tireless attempts for predicate logic brought to us more powerful formal notation, Peirce’s search for better representation of relational states of affairs was pursued beyond his own symbolic system. Spatial, as opposed to linear, notation is still not familiar to some of us, and the second section introduces the basic notational aspects of Peirce’s EG, and discusses the fundamental differences between EG and symbolic systems. The third section about three-valued logic examines another new enterprise of Peirce, not in syntactic notation, but in semantic values. Peirce scholars propose various motivations behind Peirce’s three value semantics and those different views will be briefly discussed.