Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic

Purbita Jana; Prateek

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arXiv · 2501.13114

Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic

Purbita Jana·Prateek

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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.

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BibTeX RIS

Purbita Jana, Prateek. 2025-01-18. Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic. https://arxiv.org/abs/2501.13114

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