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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.13537 |
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Table of Contents:
- In this paper, we discuss models of the common knowledge logic. The common knowledge logic is a multi-modal logic that includes the modal operators $\mathsf{K}_{i}$ ($i\in\mathcal{I}$, where $\mathcal{I}$ is a finite set of agents) and $\mathsf{C}$ in the language. The intended meanings of $\mathsf{K}_{i}ϕ$ ($i\in\mathcal{I}$) and $\mathsf{C}ϕ$ are ''the agent $i$ knows $ϕ$'' ($i\in\mathcal{I}$) and ''$ϕ$ is common knowledge among $\mathcal{I}$'', respectively. Semantically, this can be expressed as follows: $\mathsf{C}ϕ$ is true if and only if all of $ϕ$, $\mathsf{E}ϕ$, $\mathsf{E}^{2}ϕ$, $\mathsf{E}^{3}ϕ,\ldots$ are true, where $\mathsf{E}ϕ=\bigwedge_{i\in\mathcal{I}}\mathsf{K}_{i}ϕ$. A Kripke frame that satisfies the condition is $\langle W,R_{\mathsf{K}_{i}} (i\in\mathcal{I}), R_{\mathsf{C}}\rangle$, where $R_{\mathsf{C}}$ is the reflexive and transitive closure of $R_{\mathsf{E}}=\bigcup_{i\in\mathcal{I}}R_{\mathsf{K}_{i}}$. We refer to such Kripke frames as CKL-frames. An algebra that satisfies the condition is a modal algebra with modal operators $\mathrm{K}_{i}$ ($i\in\mathcal{I}$) and $\mathrm{C}$, which satisfies that $\mathrm{C}x\leq x$, $\mathrm{C} x\leq\mathrm{E}\mathrm{C} x$, and $\mathrm{C} x$ is the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\inω\}$, where $\mathrm{E} x=\bigwedge_{i\in\mathcal{I}} \mathrm{K}_{i} x$. We refer to such modal algebras as CKL-algebras. In this paper, we show that the class of CKL-frames is modally definable, whereas the class of CKL-algebras is not. That is, the class of CKL-algebras does not form a variety, and there exists a modal algebra in which the common knowledge logic is valid, but $\mathrm{C}x$ is not the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\inω\}$.