Relation Between Description Logics and Modal Logic

Description logics are related to, but developed independently from, modal logic (ML). Many description logics are syntactic variants of modal logic, so many logical representations expressed in modal logic correspond to description logic formalisms:

Examples for Modal Logic to Description Logic Translation
Modal logic	Description Logics
M, x ⊨ ¬ϕ iff M, x ⊭ ϕ (¬ϕ)^M = { x \| x ∉ ϕ^M }	(¬C)^I = Δ^I \ C^I
M, x ⊨ ϕ₁ ∧ ϕ₂ iff M, x ⊨ ϕ₁ and M, x ⊨ ϕ₂ (ϕ₁ ∧ ϕ₂)^M = ϕ₁^M ∩ ϕ₂^M	(C ⊓ D)^I = C^I ∩ D^I
M, x ⊨ ϕ₁ ∨ ϕ₂ iff M, x ⊨ ϕ₁ and M, x ⊨ ϕ₂ (ϕ₁ ∨ ϕ₂)^M = ϕ₁^M ∪ ϕ₂^M	(C ⊔ D)^I = C^I ∪ D^I
M, x ⊨ ⋄ϕ iff ∃y((x,y) ∈ R and M, y ⊨ ϕ) (⋄ϕ)^M = { x \| ∃y((x,y) ∈ R and y ∈ ϕ^M)}	(∃R.C)^I = {x ∈ Δ^I \| ∃y.(x, y) ∈ R^I ^ y ∈ C^I}
M, x ⊨ □ϕ iff ∀y((x,y) ∈ R → M, y ⊨ ϕ) (□ϕ)^M = { x \| ∀y((x,y) ∈ R → y ∈ ϕ^M)}	(∀R.C)^I = {x ∈ Δ^I \| ∀y.(x, y) ∈ R^I → y ∈ C^I}