Dialectical logic: the Process Calculus

TL;DR

Dialectical logic extends traditional logic by emphasizing processes and flows, integrating sequential, parallel, and internal/external aspects, and evolving from intuitionistic to classical forms through double negation closure.

Contribution

This paper introduces dialectical logic as a dynamic framework that unifies process-oriented notions with logical systems, expanding standard logic to include types, nonsymmetry, and process flows.

Findings

Reveals the internal/external aspects of dialectical logic.

Shows how dialectical logic generalizes linear and process logics.

Demonstrates classical dialectical logic via double negation closure.

Abstract

Dialectical logic is the logic of dialectical processes. The goal of dialectical logic is to reveal the dynamical notions inherent in logical computational systems. The fundamental notions of proposition and truth-value in standard logic are subsumed by the notions of process and flow in dialectical logic. Standard logic motivates the core sequential aspect of dialectical logic. Horn-clause logic requires types and nonsymmetry and also motivates the parallel aspect of dialectical logic. The process logics of Milner and Hoare reveal the internal/external aspects of dialectical logic. The sequential internal aspect of dialectical logic should be viewed as a typed or distributed version of Girard's linear logic with nonsymmetric tensor. The simplest version of dialectical logic is inherently intuitionistic. However, by following Glivenko's approach in standard logic using double negation closure, we can define a classical version of dialectical logic.