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We discuss four common mistakes in the teaching and textbooks of modal logic.The first one is missing the axiom$\Diamond\varphi\leftrightarrow\neg\Box\neg\varphi$, 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 forfiltration closed under subformulas, when proving the finite model propertythrough filtration, neglecting that $\Box\varphi$ and $\Diamond\varphi$ may beabbreviations of formulas. The third one is giving wrong definitions ofcanonical relations in minimal canonical models that are unmatched with theprimitive modal operators. The final one is misunderstanding the rule ofnecessitation, without knowing its distinction from the rule of modus ponens.To better understand the rule of necessitation, we summarize six ways ofdefining deductive consequence in modal logic: omitted definition, classicaldefinition, ternary definition, reduced definition, bounded definition, anddeflationary definition, and show that the last three definitions areequivalent to each other.

Some Common Mistakes in the Teaching and Textbooks of Modal Logic