  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 for
filtration closed under subformulas, when proving the finite model property
through filtration, neglecting that $\Box\varphi$ and $\Diamond\varphi$ 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.
