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Algebraic Semantics for Interpretability Logics
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DOI：10.1007/s10849-026-09460-4.png)
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Abstract
En 中文
In this paper, we define algebraic semantics for interpretability logics, which are a family of logics which extend modal provability logic GL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{GL}$$\end{document} and which are aimed to formalize the notion of relative interpretability between arithmetical theories. The standard Kripke-like semantics for these logics, called Veltman semantics, lacks completeness for some extensions of the basic system IL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{IL}$$\end{document}. We define the notion of interpretability algebras and we show that Veltman semantics is just a special case of this semantics by showing that each Veltman frame corresponds to a particular interpretability algebra. We also show that the basic system IL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{IL}$$\end{document} is complete with respect to the class of all interpretability algebras defined in this paper. Moreover, every extension of IL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{IL}$$\end{document} is sound and complete with respect to an appropriate class of interpretability algebras.
Keywords:
Modal logic
Interpretability logic
Veltman semantics
Algebraic semantics
Boolean algebras
Modal algebras
Completeness
Journal
J
IF:
0.6
Papers:
29
Citations:
0