Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.13114 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908726550593536 |
|---|---|
| author | Jana, Purbita Prateek |
| author_facet | Jana, Purbita Prateek |
| contents | We present algebraic semantics for Continuous Propositional Logic, CPL, introduced by Itai Ben Yaacov, viewed as Łukasiewicz 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 $κ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic Jana, Purbita Prateek Logic We present algebraic semantics for Continuous Propositional Logic, CPL, introduced by Itai Ben Yaacov, viewed as Łukasiewicz 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 $κ$ 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. |
| title | Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic |
| topic | Logic |
| url | https://arxiv.org/abs/2501.13114 |