□ is the modal operator for necessity.

Alethic modal logic is concerned with the concepts of truth and necessity, while deontic modal logic is concerned with the concepts of obligation and permission.

The formula for the modal logic axiom K is □(φ → ψ) → (□φ → □ψ).

From □φ, we can infer φ because necessity implies truth.

Modal logic can be used to analyze the concept of space, such as the relationship between physical space and logical space.

S5 is a modal logic system that is used to analyze the concept of space because it includes the axiom of universal accessibility, which states that every possible world is accessible from every other possible world.

The formula for the modal logic axiom T is □φ → φ.

T is a modal logic system that is used to analyze the concept of time because it includes the axiom of transitivity, which states that if one possible world is accessible from another possible world, and that other possible world is accessible from a third possible world, then the first possible world is accessible from the third possible world.

The formula for the modal logic axiom B is ¬□φ → φ.

K is a modal logic system that is used to analyze the concept of knowledge because it includes the axiom of knowledge, which states that if an agent knows a proposition, then that proposition is true.

The formula for the modal logic axiom 4 is □(φ → ψ) → (□φ → □ψ).

B is a modal logic system that is used to analyze the concept of belief because it includes the axiom of belief, which states that if an agent believes a proposition, then that proposition is possible.

The formula for the modal logic axiom 5 is □φ → φ.

S5 is a modal logic system that is used to analyze the concept of obligation because it includes the axiom of universal obligation, which states that every proposition is obligatory.

The formula for the modal logic axiom D is □(φ → ψ) → (□φ → □ψ).