模态逻辑教材与教学中的几个常见错误
Some Common Mistakes in the Teaching and Textbooks of Modal Logic
摘要：
本文讨论了模态逻辑教学和教材中的四个常见错误。第一种是缺少公理$\Diamondφ\leftrightarrow\neg\Box\negφ$，当选择$\Diamond$作为原始模态算子时，误解了$\Box$和$\Diamond$是对称的。二是在用滤子证明有限模型性质时，忘记了使滤子公式集在子公式下封闭，忽略了$\Boxφ$和$\Diamondφ$可能是滤子公式的缩写。三是给出了与原始模态算子不匹配的极小典型模型中典型关系的错误定义。三是对必然性规则的误解，不了解必然性规则与肯定前件规则的区别。为了更好地理解必然性规则，本文总结了模态逻辑中演绎后件的六种定义方式：省略定义、经典定义、三元定义、约化定义、有界定义和紧缩定义，并证明了后三种定义是等价的。
英文摘要：
We discuss four common mistakes in the teaching and textbooks of modal logic. The first one is missing the axiom $\Diamondφ\leftrightarrow\neg\Box\negφ$, when choosing $\Diamond$ as the primitive modal operator, misunderstanding that $\Box$ and $\Diamond$ are symmetric. The second one is forgetting to make the set of formulas for filtration closed under subformulas, when proving the finite model property through filtration, neglecting that $\Boxφ$ and $\Diamondφ$ may be abbreviations of formulas. The third one is giving wrong definitions of canonical relations in minimal canonical models that are unmatched with the primitive modal operators. The final one is misunderstanding the rule of necessitation, without knowing its distinction from the rule of modus ponens. To better understand the rule of necessitation, we summarize six ways of defining deductive consequence in modal logic: omitted definition, classical definition, ternary definition, reduced definition, bounded definition, and deflationary definition, and show that the last three definitions are equivalent to each other.
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