DAML+OIL, RDFS, and OWL support a family of knowledge representation languages called *description logics* (DL). A description logic can model concepts, roles and individuals, and their relationships. A basic description logic is AL, the Attributive Language, which supports atomic negation, concept intersection, universal restrictions, and limited existential quantification. An extension of AL is the Attributive Language with Complements, the description logic abbreviated as ALC. ALC supports terminological knowledge representations (TBox axioms) using subclass relationships (⊑) and equivalence (≡), conjunction (⊓), disjunction (⊔), negation (¬), property restrictions (∀, ∃), tautology (⊤), and contradiction (⊥). ALC can also represent assertional knowledge (ABox axioms), such as individual assertions (e.g., Jet Li is an actor) and property assertions (e.g., the running time of Zambezia is 83 minutes). The naming convention of description logics is to indicate additional constructors by appending a corresponding letter, as shown in the table below.

Symbol | Constructs | Example |
---|---|---|

AL | Atomic negation, concept intersection, universal restrictions, limited existential quantification | copyrightHolder ≡ Organization ⊔ Person |

C | Complex class expressions by combining mathematical operators such as subclass relationships, equivalence, conjunction, disjunction, negation, property restrictions, tautology, and contradiction | cartoon ≡ animatedMovie ⊓ ¬(liveAction ⊔ computerAnimation) |

S | ALC with transitive roles | partOf ◦ partOf ⊑ partOf |

E | Full existential quantification | ∃hasDiscRelease.Blu-Ray |

F | Functional properties, a special case of uniqueness quantification | ⊤⊑⩽1officialWebsite.⊤ |

U | Concept union | broadcastChannel ≡ webcastChannel ⊓ televisionChannel |

H | Role hierarchy/subproperties | remakeOf ⊑ basedOn |

R | Complex role inclusion, reflexivity and irreflexivity, role disjointness | supervisorOf ◦ castMemberOf ⊑ directorOf |

O | Enumerated classes of object value restrictions (nominals) | AnimationStudios ≡ { DreamWorks, Walt Disney Animation Studios, Pixar } |

I | Inverse properties | directedBy ≈ directorOf^{–} |

N | Unqualified cardinality restrictions | TVSeries ≡ ≥2hasEpisode.LiveAction |

Q | Qualified cardinality restrictions | Actor ⊑= 1hasBirthplace.⊤ |

V | Variable nominal classes (nominal schemas) | ∃hasParent.{z} ⊓ ∃hasParent.∃married.{z} ⊑ C |

^{(D)} |
Data type properties, data values, or data types | runningTime(Zambezia, 83) |

Note that supersets of ALC defining transitive roles denote the underlying constructs with S. Some DL languages have overlapping constructs, as, for example, union and full existential quantification can be expressed using negation, therefore U and E are never used together in DL names, and C is used instead.

By extending ALC and transitivity roles (i.e., S) with role hierarchies (H), inverse roles (I), functional properties (F), and datatypes (D), we get the SHIF^{(D)} description logic, which roughly corresponds to OWL Lite. By adding nominals (O) and cardinality restrictions (N) to SHIF^{(D)}, we get SHOIN^{(D)}, the description logic underlying OWL DL. After industrial applications highlighted several key features missing from SHOIN^{(D)} to model complex knowledge domains, SHOIN^{(D)} has been extended with complex role inclusion axioms, reflexive and irreflexive roles, asymmetric roles, disjoint roles, the universal role, self-constructs, negated role assertions, and qualified number restrictions, leading to SROIQ^{(D)}, one of the most expressive description logic whose decidability is proven, and which largely corresponds to OWL 2 DL. Furthermore, SROIQ^{(D)} supports not only TBox and ABox axioms, but also so-called Role Boxes (RBox) to collect all statements related to roles and the interdependencies between roles. Each RBox consists of a role hierarchy (including generalized role inclusion axioms) and a set of role assertions.

## Relation to Other Logics

Many description logics are decidable fragments of *first-order logic* (FOL), also known as first-order predicate calculus (FOPC), and are fully-fledged logics with formal semantics. Description logics are closely related to propositional *modal logic*, which extends classical propositional and predicate logic to include operators expressing modality in the form of necessity, possibility, impossibility, and related notions, and guarded fragments, which generalize modal quantification through finding relative patterns of quantification.