Many description logics are decidable fragments of first-order logic (FOL), also known as first-order predicate calculus (FOPC), and many of two-variable logic or guarded logic, however, some description logics have more features than first-order logic.

First-Order Logic | Description Logic |
---|---|

A(x) |
A |

C(a) |
C(a), alternatively a : C |

A ≈ B |
A ≡ B |

¬C(x) |
¬C |

C(x) ^ D(x) |
C ⊓ D |

C(x) ∨ D(x) |
C ⊔ D |

∀x(C(x) → D(x)) |
C ⊑ D |

R(a,b) |
R(a,b), alternatively (a,b) : R |

∀x∀y(R(x,y) → S(x,y)) |
R ⊑ S |

∃y(R(x,y) ^ C(y)) |
∃R.C |

∀x∀y∀z(R(x,y) → R(y,z) → R(x,z)) |
R ◦ R ⊑ R |

Despite of the feasibility of direct translation between FOL and DL, guaranteeing complete and terminating reasoning requires a different transformation, such as the *structural transformation*. The structural transformation is based on a conjunction normal form, which replaces FOL subformulae with new predicates, for which it also provides definitions. A major advantage of the structural transformation is that it avoids the exponential size growth of the clauses.