非经典变体S4中的模态
Modalities in non-classical variations of $\mathsf{S4}$
AI总结：
该研究探讨S4的非经典类似逻辑CS4、IS4、GS4、GS4^c的模态数量，发现CS4有无限个{¬,◇}模态，IS4和GS4有无限个{¬,□,◇}模态，GS4^c的{¬,□,◇}模态数量有限。
AI中文摘要：
模态逻辑中的一个经典结果指出，S4有14种模态，即每个由否定词（¬）、必然算子（□）和可能算子（◇）组成的序列，都等价于这14种序列中的某一种。我们研究S4的非经典类似逻辑CS4、IS4、GS4和GS4^c的类似结果：首先，这些逻辑均有有限个{□,◇}和{¬,□}模态，但CS4有无限个{¬,◇}模态；其次，IS4和GS4有有限个{¬,◇}模态，却有无限个{¬,□,◇}模态；最后，GS4^c有有限个{¬,□,◇}模态。
英文摘要：
A classical result in modal logic states that $\mathsf{S4}$ has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues $\mathsf{CS4}$, $\mathsf{IS4}$, $\mathsf{GS4}$, and $\mathsf{GS4^c}$ of $\mathsf{S4}$. First, we show that, while all these logics have finitely many $\{\Box,\Diamond\}$- and $\{\neg,\Box\}$-modalities, the logic $\mathsf{CS4}$ has infinitely many $\{\neg,\Diamond\}$-modalities. Second, we show that $\mathsf{IS4}$ and $\mathsf{GS4}$ have finitely many $\{\neg,\Diamond\}$-modalities, but they have infinitely many $\{\neg,\Box,\Diamond\}$-modalities. At last, we show that $\mathsf{GS4^c}$ has finitely many $\{\neg,\Box,\Diamond\}$-modalities.
