A single mathematical operator can now derive all of propositional logic, modal logic, and the core rules of calculus.
April 26, 2026
Original Paper
Propagation Logic: A Single-Operator Foundation for Logic and Calculus
SSRN · 6439258
The Takeaway
This new formal system proves that the fundamental laws of logic and the rules of calculus are actually the same thing. They are just different applications of the same underlying propagation operator. For centuries, math and logic were treated as related but distinct languages with their own rules. This discovery unifies them into a single, cohesive foundation for all computation. It simplifies the way we build automated reasoning systems and formal proof checkers.
From the abstract
We present Propagation Logic (PL), a formal system in which the entirety of propositional, first-order, modal, and probabilistic logic-and the core operations of calculus-are recovered from a single operator: P/G → Q. Here P is a loaded pattern carrying an accumulated gradient demand, G is a gradient field supplied by a context, and Q is the reconfigured output. Logical connectives, quantifiers, inference rules, modalities, and probability are recovered by choosing gradient fields over a Boolean