The starting points of my talk are two of the most important subsystems of classical logic arising in the foundations of mathematics and foundations of physics: intuitionistic logic (Heyting 1930) and orthologic (Birkhoff and von Neumann 1936). In a Fitch-style formulation of natural deduction, intuitionistic logic and orthologic can be obtained from a more basic logic, defined using only the introduction and elimination rules for the logical connectives, by the addition of rules of Reiteration and Reductio ad Absurdum, respectively. In this talk (based on https://arxiv.org/abs/2207.06993), I will characterize this “fundamental” logic both proof-theoretically and semantically. The semantic characterization is based on representation theorems for complete lattices with additional operations (e.g., negation, implication) using directed graphs.