Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic

TL;DR

This paper develops algebraic semantics for Continuous Propositional Logic by extending MV-algebras with a halving operator, establishing a correspondence with 2-divisible -groups, and providing an algebraic proof of weak completeness.

Contribution

It introduces continuous algebras as MV-algebras with a halving operator and extends Mundici's representation theory to this continuous setting.

Findings

Established a correspondence between continuous algebras and 2-divisible -groups.

Provided an algebraic proof of the weak completeness theorem for CPL.

Analyzed structural properties like ideals, quotients, and subdirect representations of continuous algebras.

Abstract

We present algebraic semantics for Continuous Propositional Logic, CPL, introduced by Itai Ben Yaacov, viewed as {\L}ukasiewicz propositional logic with a reversed truth-falsity orientation and enriched by a unary halving connective. We introduce continuous algebras as MV-algebras together with an unary operator $\kappa$ analogous to the halving operator introduced in CPL and analyze their core structural properties, including ideals, quotient constructions, and subdirect representations. We further establish a correspondence between continuous algebras and the class of 2-divisible $\ell u$-groups, extending Mundici's representation theory to the continuous setting. This correspondence leads to a purely algebraic proof of the weak completeness theorem for CPL.