## Franklin's Notes

### Axiom schemas in modal logic

Here are some examples of axiom schemas that are valid in modal logic :

• T refers to the axiom scheme $\square\mathrm A\rightarrow \mathrm A$

• 5 refers to the axiom scheme $\diamondsuit\mathrm A\rightarrow \square\diamondsuit\mathrm A$

• K refers to the axiom scheme $\square(\mathrm A\rightarrow\mathrm B)\rightarrow (\square\mathrm A\rightarrow\square\mathrm B)$

• Df♢ refers to the axiom scheme $\diamondsuit \mathrm A\leftrightarrow \neg\square\neg\mathrm A$

which can be interpreted informally as follows:

• If $\mathrm A$ is necessary, then it is true.

• If $\mathrm A$ is possible, then it is necessarily possible.

• If it is necessary that $\mathrm B$ be true whenever $\mathrm A$ is true, and if $\mathrm A$ is necessary, then $\mathrm B$ is necessary.

• $\mathrm A$ is possible if and only if it is not necessarily false.