Ranking-based Defeasible Reasoning for Restricted First-Order
                         Conditionals Applied to Description Logics
                         Alexander Hahn1 , Gabriele Kern-Isberner1 and Thomas Meyer2
                         1
                             Dept. of Computer Science, TU Dortmund University, Dortmund, Germany
                         2
                             University of Cape Town and CAIR, Cape Town, South Africa


                                             Abstract
                                             Nonmonotonic reasoning based on ordinal conditional functions (OCFs), often called ranking functions, and description logics are
                                             both well-established methodologies in knowledge representation and reasoning. However, nonmonotonic reasoning mainly focuses
                                             on propositional logic as a base logic, while description logics investigate fragments of first-order logic for efficient reasoning with
                                             terminological knowledge. In this paper, we investigate how OCFs can be employed to define inference relations induced from first-order
                                             conditional knowledge bases. The goal of this work is to present first steps towards an interpretation of defeasible subsumptions in
                                             description logics (DL) which is thoroughly based on conditionals and ranking functions. In the process, we adapt a recently proposed
                                             DL version of the KLM postulates, a popular framework for non-monotonic reasoning from propositional knowledge bases, for the use
                                             with conditional first-order logic. Moreover, we consider some additional recently proposed rationality postulates for a KLM approach
                                             based on (restricted) first-order logic. Concrete examples are provided for reasoning with strategic c-representations, a special type of
                                             ranking functions based on the underlying conditional structures of a knowledge base, yielding high-quality non-monotonic inferences
                                             without the need to specify external relations, e.g., expressing typicality among individuals.

                                             Keywords
                                             first-order logic, description logic, conditional reasoning, non-monotonic reasoning, ranking functions, c-representations



                         1. Introduction                                                                                                     DL version of these postulates, lifting the KLM approach to
                                                                                                                                             defeasible description logics.
                         Rules in the form of conditional statements “If A then (usu-                                                           The goal of this paper is to propose first steps towards
                         ally) B” (sometimes equipped with a quantitative degree)                                                            an interpretation of defeasible subsumptions in description
                         are basic to human reasoning and also to logics in Artificial                                                       logics (DL) which is thoroughly based on conditionals and
                         Intelligence, and have been explored in the area of non-                                                            ranking functions. To this end, we lift the notion of inductive
                         monotonic reasoning since the 80s of the past century. They                                                         inference operators defined in [7] to first-order conditional
                         can be formalized as conditionals (𝐵|𝐴), allowing for a non-                                                        knowledge bases. Additionally, we adapt the KLM postu-
                         classical, three-valued interpretation of conditional state-                                                        lates from [4], as well as additional rationality postulates for
                         ments. Semantics for knowledge bases consisting of condi-                                                           the KLM approach from [3] and show that they are fulfilled
                         tionals are provided by epistemic states, often equipped with                                                       by our approach. Moreover, we illustrate the application
                         total preorders on possible worlds. Using total preorders                                                           of ranking-based first-order conditional semantics to a DL
                         ensures a high quality of nonmonotonic reasoning in terms                                                           example well-known from the literature and compare it to
                         of broadly accepted axioms. Ordinal conditional functions                                                           concept-wise multipreference (cwm ) semantics from [5].
                         [1, 2], often called ranking functions, can be considered an                                                           The rest of this paper is organized as follows. In Section 2,
                         implementation of such epistemic states that assign to each                                                         the basics on first-order conditionals and defeasible 𝒜ℒ𝒞
                         possible world 𝜔 an implausibility rank 𝜅(𝜔) such that the                                                          are summarized. In Section 3, we describe inductive infer-
                         higher 𝜅(𝜔), the less plausible 𝜔 is, and with the normal-                                                          ence operators for first-order (conditional) knowledge bases.
                         ization constraint that there are worlds that are maximally                                                         In Section 4, postulates from [4] and [8] are adapted and
                         plausible.                                                                                                          evaluated for the use with first-order conditional logic. In
                            Similar to conditionals for propositional logic, statements                                                      Section 5, we compare the OCF-based semantics for first-
                         of the form “Usually, As are Bs” encoded as defeasible con-                                                         order knowledge bases with cwm -semantics [5] for defeasi-
                         cept inclusions 𝐴 ⊏ ∼ 𝐵, also called defeasible subsumptions,                                                       ble 𝒜ℒ𝒞 knowledge bases. In Section 6, we conclude this
                         are a natural extension for description logics (DLs) in or-                                                         paper with summarizing its main contributions and some
                         der to introduce conditional reasoning. Recently, different                                                         pointers for future work.
                         semantics for defeasible DL knowledge bases have been
                         proposed [3, 4, 5].
                            In order to compare and contrast different approaches to                                                         2. Preliminaries
                         non-monotonic reasoning, as well as to provide unifying
                         frameworks, postulates are necessary. A popular approach                                                            This section recalls some formal basics on conditional first-
                         for non-monotonic reasoning, preferential models, is char-                                                          order logic and defeasible description logics. For a more
                         acterized by the so-called KLM postulates [6]. However,                                                             thorough introduction to description logics, we recommend
                         preferential models have been mostly considered for propo-                                                          [9].
                         sitional logics. Recently, Britz et al. [4] have proposed a
                                                                                                                                             2.1. Conditionals in First-Order Logic
                          22nd International Workshop on Nonmonotonic Reasoning, November 2-4,
                          2024, Hanoi, Vietnam                                                                                               In this section, we recall relevant parts of the first-order
                          $ alexander.hahn@tu-dortmund.de (A. Hahn);                                                                         conditional logic introduced in [10]. We start with syntac-
                          gabriele.kern-isberner@tu-dortmund.de (G. Kern-Isberner);                                                          tical details. Let Σ = ⟨𝑃Σ , 𝑈Σ ⟩ be a first-order signature
                          tommie.meyer@uct.ac.za (T. Meyer)
                           0009-0008-6114-2594 (A. Hahn); 0000-0001-8689-5391
                                                                                                                                             consisting of a finite set of predicates 𝑃Σ and a finite set
                          (G. Kern-Isberner); 0000-0003-2204-6969 (T. Meyer)                                                                 of constant symbols 𝑈Σ but without function symbols of
                                     © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribu-
                                     tion 4.0 International (CC BY 4.0).
                                                                                                                                             arity > 0. An atom is a predicate of arity 𝑛 together with a

CEUR
                  ceur-ws.org
Workshop      ISSN 1613-0073
Proceedings
list of 𝑛 constants and/or variables. A literal is an atom or a        Definition 1. An ordinal conditional function (OCF) 𝜅 on
negated atom. Formulas are built on atoms using conjunc-               ΩΣ is a function 𝜅 : ΩΣ → N ∪ {∞} with 𝜅−1 (0) ̸= ∅.
tion (∧), disjunction (∨), negation (¬), material implication
(⇒), and quantification (∀, ∃). We abbreviate conjunctions               We can now make use of the possible world semantics
by juxtaposition and negations usually by overlining, e. g.            to assign degrees of disbelief also to formulas and (non-
𝐴𝐵 means 𝐴 ∧ 𝐵 and 𝐴 means ¬𝐴. The symbol ⊤ denotes                    quantified) conditionals. In the following, let 𝐴, 𝐵 ∈ ℒΣ
an arbitrary tautology, and ⊥ denotes an arbitrary contra-             denote closed formulas, and let 𝐴(𝑥        ⃗ ) ∈ ℒΣ denote
                                                                                                           ⃗ ), 𝐵(𝑥
diction. A ground formula contains no variables. In a closed           open formulas.
formula, all variables (if they occur) are bound by quanti-            Definition 2 (𝜅-ranks of closed formulas [10]). Let 𝜅 be
fiers, otherwise, the formula is open, and the variables that          an OCF. The 𝜅-ranks of closed formulas are defined (as in the
occur outside of the range of quantifiers are called free. If a        propositional case) via
formula 𝐴 contains free variables we also use the notation
𝐴(𝑥 ⃗ ) where ⃗𝑥 = (𝑥1 , . . . , 𝑥𝑛 ) contains all free variables in    𝜅(𝐴) = min 𝜅(𝜔)         and 𝜅(𝐵 | 𝐴) = 𝜅(𝐴𝐵) − 𝜅(𝐴).
                                                                                  𝜔|=𝐴
𝐴. If ⃗𝑐 is a vector of the same length as ⃗𝑥 then 𝐴(𝑐  ⃗) is meant
to denote the instantiation of 𝐴 with ⃗.     𝑐 A formula ∀𝑥  ⃗ 𝐴(𝑥⃗)      By convention, 𝜅(⊥) = ∞, because ranks are supposed
(∃𝑥⃗ 𝐴(𝑥⃗ )) is universal (existential) if 𝐴 involves no further       to reflect plausibility.
quantification. Let ℒΣ be the first-order language that al-               The ranks of first-order formulas are coherently based on
lows no nested quantification, i.e., all quantified formulas           the usage of OCFs for propositional formulas. These degrees
are either universal or existential formulas. ℒΣ contains              of beliefs are used to specify when a formula from (ℒΣ |ℒΣ )
both open and closed formulas.                                         is accepted by a ranking function 𝜅 (where acceptance is
   ℒΣ is extended by a conditional operator “|” to a condi-            denoted by |=). We will first consider the acceptance of
tional language (ℒΣ |ℒΣ ) containing first-order condition-            closed (conditional) formulas.
als (𝐵|𝐴) with 𝐴, 𝐵 ∈ ℒΣ . We write (𝐵(𝑥                     ⃗ )) to
                                                       ⃗ )|𝐴(𝑥
highlight free variables. Then we assume ⃗𝑥 to mention                 Definition 3 (Acceptance of closed formulas [10]). Let 𝜅
all free variables occurring in 𝐴 or 𝐵 where the positions             be an OCF. The acceptance relation between 𝜅 and closed
of the variables are suitably adapted. Note that 𝐴 and 𝐵               formulas from ℒΣ and (ℒΣ |ℒΣ ) is defined as follows:
usually will have free variables in common but may also                     • 𝜅 |= 𝐴 iff for all 𝜔 ∈ Ω with 𝜅(𝜔) = 0, it holds that
mention free variables which do not occur in the respec-                      𝜔 |= 𝐴.
tive other formula. E.g., the conditional (𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑦, 𝑥) ∧                 • 𝜅 |= (𝐵 | 𝐴) iff 𝜅(𝐴𝐵) < 𝜅(𝐴𝐵).
𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑥, 𝑧)|𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑥, 𝑦)) (if 𝑥 is a friend of 𝑦 then
usually 𝑦 is also a friend of 𝑥 and 𝑥 has also a(nother)                  Acceptance of a sentence by a ranking function is the
friend 𝑧) would be represented by (𝐵(𝑥, 𝑦, 𝑧)|𝐴(𝑥, 𝑦, 𝑧))              same as in the propositional case for ground sentences, and
with 𝐵(𝑥, 𝑦, 𝑧) = 𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑦, 𝑥) ∧ 𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑥, 𝑧) and                  interprets the classical quantifiers in a straightforward way.
𝐴(𝑥, 𝑦, 𝑧) = 𝐹 𝑟𝑖𝑒𝑛𝑑𝑠(𝑥, 𝑦). Note that conditionals cannot                The treatment of acceptance of open formulas is more
be nested, and that conditionals with tautological antecedent          intricate, as such formulas will be used to express default
are identified with the corresponding non-conditional state-           statements, like in “𝐴 is plausible”, or in “usually, if 𝐴 holds,
ment, i.e., (𝐴|⊤) ≡ 𝐴. Nevertheless, we distinguish be-                then 𝐵 also holds”.
tween such plausible statements (𝐴|⊤) ≡ 𝐴 and strict
                                                                       Definition 4 (𝜅-ranks of open formulas [10]). We define the
facts. Let ℒ𝑐Σ = ℒΣ ∪ (ℒΣ |ℒΣ ) be the language contain-
                                                                       ranks of open formulas and open conditionals by evaluating
ing both first-order formulas and conditionals as specified
                                                                       most plausible instances:
above.
   A first-order conditional knowledge base ℛ is a finite set                𝜅(𝐴(𝑥
                                                                                 ⃗ )) =      min     𝜅(𝐴(𝑎
                                                                                                         ⃗ ))
of conditional formulas. A first-order knowledge base 𝒦ℬ =                                 𝑎∈ℋ𝐴(𝑥
                                                                                           ⃗    ⃗)

⟨ℱ, ℛ⟩ consists of a first-order conditional knowledge base            𝜅(𝐵(𝑥
                                                                           ⃗ )|𝐴(𝑥
                                                                                 ⃗ )) =          min          𝜅(𝐴(𝑎    ⃗ )) − 𝜅(𝐴(𝑎
                                                                                                                  ⃗ )𝐵(𝑎          ⃗ )).
ℛ, together with a set ℱ of closed formulas from ℒΣ , called                               𝑎∈ℋ(𝐵(𝑥
                                                                                           ⃗     ⃗ )|𝐴(𝑥
                                                                                                       ⃗ ))

facts. The open formulas and conditionals in ℛ are meant                  Generalizing the notion of acceptance of a first-order for-
to represent defeasible plausible beliefs.                             mula or conditional is straightforward for closed formulas
   Regarding semantics, we base our first-order conditional            and conditionals. The basic idea here is that such (condi-
semantics on the Herbrand semantics. A possible world                  tional) open statements hold if there are individuals called
𝜔 is a subset of the Herbrand base ℋΣ , which contains all             representatives that provide most convincing instances of
ground atoms of the first-order signature Σ. Possible worlds           the respective conditional.
can be concisely represented as complete conjunctions or
minterms, i.e. conjunctions of literals where every atom               Definition 5 (representative [10]). Let 𝑟 = (𝐵(𝑥      ⃗ )|𝐴(𝑥 ⃗ ))
of ℋΣ appears either in positive or in negated form. The               be an open conditional. We call ⃗𝑎 ∈ ℋ𝑟 a weak representa-
set of all possible worlds is denoted by ΩΣ . For an open              tive of 𝑟 iff both of the following conditions are satisfied:
conditional 𝑟 = (𝐵(𝑥      ⃗ )|𝐴(𝑥⃗ )), the set ℋ𝑟 denotes the set
of all vectors from the Herbrand universe that appear in                           𝜅(𝐴(𝑎
                                                                                       ⃗ )𝐵(𝑎
                                                                                            ⃗ ))       =      𝜅(𝐴(𝑥
                                                                                                                  ⃗ )𝐵(𝑥
                                                                                                                       ⃗ ))          (1)
groundings of 𝑟, i.e.                                                              𝜅(𝐴(𝑎
                                                                                       ⃗ )𝐵(𝑎
                                                                                            ⃗ ))       <      𝜅(𝐴(𝑎
                                                                                                                  ⃗ )𝐵(𝑎
                                                                                                                       ⃗ ))          (2)

           ℋ(𝐵(𝑥⃗)|𝐴(𝑥⃗)) = {𝑎
                             ⃗ ∈ 𝑈Σ | |𝑎
                                       ⃗ | = |𝑥
                                              ⃗ |}.                    The set of weak representatives of 𝑟 is denoted by wRep(𝑟).
                                                                       Further, ⃗𝑎 ∈ wRep(𝑟) is a (strong) representative of 𝑟 iff
   Ordinal conditional functions [1], usually called rank-                                                    (︁         )︁
                                                                                                                  ⃗ )𝐵(𝑏
                                                                                                                       ⃗) .       (3)
                                                                                 (︀          )︀
ing functions, can be defined just as in the propositional                    𝜅 𝐴(𝑎  ⃗ )𝐵(𝑎
                                                                                          ⃗ ) = min 𝜅 𝐴(𝑏
                                                                                                   ⃗
                                                                                                   𝑏∈wRep(𝑟)
case. They associate degrees of (im)plausibility with possi-
ble worlds.                                                            The set of strong representatives of 𝑟 is denoted by Rep(𝑟).
   (Strong) Representatives of a conditional are character-      𝜅(𝐴(𝑏)𝐵(𝑏)) and 𝜅(𝐴(𝑎)𝐵(𝑎)) = ∞ = 𝜅(𝐴(𝑏)𝐵(𝑏)).
ized by being most general (1) and least exceptional (3). And    Moreover, we must have 𝜅 |= (𝐵(𝑥)|𝐴(𝑥)) and 𝜅 |=
of course, their instantiation should be accepted by 𝜅 (2).      (𝐵(𝑏)|𝐴(𝑏)). For the second closed conditional, this simply
Note that Rep(𝑟) ̸= ∅ iff wRep(𝑟) ̸= ∅. Now we can base          means 𝜅(𝐴(𝑏)𝐵(𝑏)) < 𝜅(𝐴(𝑏)𝐵(𝑏)), which clearly holds.
our definition of acceptance of open conditionals on the         For the open conditional (𝐵(𝑥)|𝐴(𝑥)), we must apply Def-
notion of representatives as follows.                            inition 6. In particular, we must consider representatives of
                                                                 this conditional. Since 𝜅(𝐴(𝑎)𝐵(𝑎)) = 0 = 𝜅(𝐴(𝑏)𝐵(𝑏)),
Definition 6 (acceptance of open conditionals [10]). Let 𝜅       we also have 𝜅(𝐴(𝑥)𝐵(𝑥)) = 0 = 𝜅(𝐴(𝑥)𝐵(𝑥)) by
be an OCF and 𝑟 = (𝐵(𝑥  ⃗ )|𝐴(𝑥⃗ )). Then 𝜅 |= 𝑟 iff Rep(𝑟) ̸=   Definition 4. Hence the more complicated option (B) of
∅ and either of the two following conditions holds.              Definition 6 applies, and we must also consider represen-
   (A) It holds that                                             tatives of (𝐵(𝑥)|𝐴(𝑥)). Natural candidates of representa-
                                                                 tives for (𝐵(𝑥)|𝐴(𝑥)) resp. (𝐵(𝑥)|𝐴(𝑥)) would be 𝑎 resp.
                 𝜅(𝐴(𝑥
                     ⃗ )𝐵(𝑥
                          ⃗ )) < 𝜅(𝐴(𝑥
                                     ⃗ )𝐵(𝑥
                                          ⃗ )).            (4)   𝑏, but let us go into more details here. For both, we have
                                                                 𝜅(𝐴(𝑎)𝐵(𝑎)) = ∞ = 𝜅(𝐴(𝑏)𝐵(𝑏)), so they are defi-
   (B) 𝜅(𝐴(𝑥 ⃗ )𝐵(𝑥⃗ ))    =     𝜅(𝐴(𝑥    ⃗ )) and either
                                     ⃗ )𝐵(𝑥                      nitely weak representatives of the respective conditional. How-
       Rep((𝐵(𝑥         ⃗ ))) = ∅, or for all 𝑎⃗1 ∈
                  ⃗ )|𝐴(𝑥                                        ever, for strong representatives, their rank of falsification
       Rep((𝐵(𝑥         ⃗ ))), 𝑎⃗2 ∈ Rep((𝐵(𝑥
                  ⃗ )|𝐴(𝑥                       ⃗ )|𝐴(𝑥
                                                      ⃗ )))      must also be minimal among all weak representatives, ac-
       it holds that                                             cording to (3). For 𝑎 and 𝑏, this rank is maximal due to
                                                                 𝜅(𝐴(𝑎)𝐵(𝑎)) = ∞ = 𝜅(𝐴(𝑏)𝐵(𝑏)), and at least prima fa-
               𝜅(𝐴(𝑎⃗1 )𝐵(𝑎⃗1 )) < 𝜅(𝐴(𝑎⃗2 )𝐵(𝑎⃗2 )).      (5)
                                                                 cie, it is well imaginable that there are other constants 𝑐 ∈ 𝑈Σ
   We also have to ensure that strict knowledge (facts) are      with lower, i.e., finite falsification ranks. So, at this point we
interpreted suitably by ranking functions. An OCF can            stop our investigations here and will come back later to this
enforce factual knowledge by setting the ranks of all worlds     example when we can use more detailed information about
which violate facts to infinity.                                 the structure of ranking models of 𝒦ℬ in the next section.

Definition 7 (enforcement of facts). Let 𝜅 be an OCF and let       Now we have set up the formal framework of our ranking-
𝐴 ∈ ℒΣ be a closed formula. Then we define that 𝜅 enforces       based approach that we need for reasoning from first-order
the fact 𝐴, denoted by 𝜅 ||− 𝐴, iff 𝜅(𝐴) = ∞.                    knowledge bases. Before dealing with inference from such
                                                                 knowledge bases in the next section, we briefly summarize
  Observe the difference between acceptance and enforce-         the syntactic basics of a description logics with defeasible
ment: while 𝜅 |= 𝐴 is the same as 𝜅 |= (𝐴|⊤) and                 subsumptions.
only means that 𝐴 has to hold in the 𝜅-minimal worlds,
𝜅 ||− 𝐴 means that 𝐴 holds in all feasible (i.e. finitely-
ranked) worlds. Nevertheless, enforcement is downward-           2.2. Defeasible 𝒜ℒ𝒞
compatible to plausible acceptance, as the next proposition      Let 𝑁𝐶 be a set of atomic concept names, 𝑁𝑅 be a set of
shows.                                                           role names and 𝑁𝐼 be a set of individual names. The set of
                                                                 𝒜ℒ𝒞-concepts is defined by the rule
Proposition 1. Let 𝜅 be an OCF and let 𝐴 ∈ ℒΣ be a closed
formula. 𝜅 ||− 𝐴 implies 𝜅 |= 𝐴.                                        𝐶 ::= 𝐴|⊤|⊥|¬𝐶|𝐶 ⊓ 𝐶|𝐶 ⊔ 𝐶|∃𝑟.𝐶|∀𝑟.𝐶 ,
Proof. If 𝜅 ||− 𝐴, then 𝜔 |= 𝐴 for all 𝜔 ∈ Ω such that
                                                                 where 𝐴 ∈ 𝑁𝐶 and 𝑟 ∈ 𝑁𝑅 .
𝜅(𝜔) < ∞. This implies particularly that 𝜔 |= 𝐴 for all
                                                                   An 𝒜ℒ𝒞-interpretation is a tuple 𝐼 = ⟨∆ℐ , ·ℐ ⟩, where
𝜔 ∈ Ω such that 𝜅(𝜔) = 0, i.e., 𝜅 |= 𝐴.
                                                                 ∆ is a domain and ·ℐ is an interpretation function which
                                                                  ℐ

  With the necessary notation for the treatment of both un-      maps 𝐴 ∈ 𝑁𝐶 , 𝑟 ∈ 𝑁𝑅 , 𝑎 ∈ 𝑁𝐼 to 𝐴ℐ ⊆ ∆ℐ , 𝑟ℐ ⊆
certain and factual knowledge in place, we are now ready to      ∆ℐ × ∆ℐ , 𝑎ℐ ∈ ∆ℐ , respectively. For complex concepts:
define the conditions for whether an OCF can be considered
                                                                            ⊤ℐ = ∆ℐ
a model of a first-order knowledge base.
                                                                            ⊥ℐ = ∅
Definition 8 ((ranking) model of a first-order KB [10]). Let
𝜅 be an OCF and let 𝒦ℬ = ⟨ℱ, ℛ⟩ be a first-order knowledge                ¬𝐶 ℐ = ∆ℐ ∖ 𝐶 ℐ
base. We say that 𝜅 is a (ranking) model of 𝒦ℬ if both of the       (𝐶 ⊓ 𝐷)ℐ = 𝐶 ℐ ∩ 𝐷ℐ
following conditions hold.
                                                                    (𝐶 ⊔ 𝐷)ℐ = 𝐶 ℐ ∪ 𝐷ℐ
     • 𝜅 ||− 𝐴 for every fact 𝐴 ∈ ℱ.
                                                                      (∃𝑟.𝐶)ℐ = {𝑥 ∈ ∆ℐ | ∃𝑦.(𝑥, 𝑦) ∈ 𝑟ℐ ∧ 𝑦 ∈ 𝐶 ℐ }
     • 𝜅 |= 𝑟 for every rule 𝑟 ∈ ℛ.
                                                                      (∀𝑟.𝐶)ℐ = {𝑥 ∈ ∆ℐ | ∀𝑦.(𝑥, 𝑦) ∈ 𝑟ℐ ⇒ 𝑦 ∈ 𝐶 ℐ }
  We illustrate first-order knowledge bases and their rank-
ing models in the following example.                             Classical (strict) subsumptions 𝐶 ⊑ 𝐷 (where 𝐶, 𝐷 are
Example 1. We consider a signature Σ = ⟨𝑃Σ , 𝑈Σ ⟩ con-           concepts) hold in an interpretation 𝐼 (short: 𝐼 |= 𝐶 ⊑ 𝐷)
sisting of two unary predicates 𝑃Σ = {𝐴, 𝐵} and at least         iff 𝐶 ℐ ⊆ 𝐷ℐ . Assertions of the form 𝐶(𝑎) or 𝑟(𝑎, 𝑏) (where
two constants {𝑎, 𝑏} ⊆ 𝑈Σ . Let the knowledge base 𝒦ℬ =          𝐶 is a concept, 𝑟 is a role and 𝑎, 𝑏 are individuals) hold if
⟨ℱ, ℛ⟩ be specified by ℱ = {𝐴(𝑎)𝐵(𝑎), 𝐴(𝑏)𝐵(𝑏)} and              𝑎 ∈ 𝐶 ℐ or (𝑎, 𝑏) ∈ 𝑟ℐ , respectively.
ℛ = {(𝐵(𝑥)|𝐴(𝑥)), (𝐵(𝑏)|𝐴(𝑏))}. Any model 𝜅 of 𝒦ℬ                    Beyond classical logics and similar to first-order condi-
must assign rank ∞ to all 𝜔 ̸|= ℱ, i.e., can have finite ranks   tionals, defeasible subsumptions 𝐶 ⊏∼ 𝐷 encode information
only for worlds 𝜔 satisfying 𝜔 |= 𝐴(𝑎)𝐵(𝑎)𝐴(𝑏)𝐵(𝑏). This         of the form “Usually, instances of 𝐶 are instances of 𝐷” or
implies, also by Proposition 1, that 𝜅(𝐴(𝑎)𝐵(𝑎)) = 0 =           “Typical 𝐶s are 𝐷s”.
   A defeasible (𝒜ℒ𝒞) knowledge base 𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩               3.2. Inductive FO-Inference Based on
consists of a TBox 𝒯 (containing strict subsumptions), a                c-Representations
DBox 𝒟 (containing defeasible subsumptions) and an ABox
𝒜 (containing assertions).                                         c-Representations, originally defined for the propositional
   A popular approach to provide semantics for defeasible          setting [11, 12], are a special kind of ranking models which
subsumptions (e.g. used in [4, 5]) is to introduce some order-     assign ranks to possible worlds in a regular way by adhering
ing over the domain elements ∆ℐ and require the minimal            to the conditional structures of knowledge bases. A (simpli-
instances of 𝐶 (with respect to said ordering) to be instances     fied) version of c-representations for first-order conditinal
of 𝐷 in order for 𝐶 ⊏                                              knowledge bases was proposed in [10].
                      ∼ 𝐷 to hold.
   In this paper, we understand defeasible subsumptions as         Definition 9 (c-Representation [10]). Let 𝒦ℬ = ⟨ℱ, ℛ⟩
open conditionals and interpret them via ranking functions.        with ℛ = {(𝐵1 (𝑥  ⃗ 1 )|𝐴1 (𝑥
                                                                                               ⃗ 1 )), . . . , (𝐵𝑛 (𝑥⃗ 𝑛 )|𝐴𝑛 (𝑥
                                                                                                                               ⃗ 𝑛 ))} be
                                                                   a first-order knowledge base. An OCF 𝜅 is a c-representation
3. Reasoning from First-Order                                      of 𝒦ℬ if 𝜅(𝜔) = ∞ for all 𝜔 ̸|= ℱ and 𝜅 |= 𝑟 for every
                                                                   𝑟 ∈ ℛ, and for all 𝜔 |= ℱ, 𝜅(𝜔) is of the form
   Knowledge Bases                                                                                   ∑︁
                                                                                   𝜅(𝜔) = 𝜅0 +                 𝑓𝑖 (𝜔)𝜂𝑖 ,              (7)
In this section, we consider inference relations induced by                                          1≤𝑖≤𝑛
first-order (FO) knowledge bases, similar to the ones con-
sidered for the propositional case in [7].                         where 𝑓𝑖 (𝜔) = #{𝑎               𝑟𝑖
                                                                                          ⃗ 𝑖 ∈ ℋ | 𝑟𝑖 = (𝐵𝑖 (𝑥     ⃗ 𝑖 ) | 𝐴𝑖 (𝑥
                                                                                                                                ⃗ 𝑖 )) ∈
                                                                   ℛ, 𝜔 |= 𝐴𝑖 (𝑎 ⃗ 𝑖 )𝐵𝑖 (𝑎
                                                                                          ⃗ 𝑖 )} is the number of possible grounding
3.1. Inductive FO-Inference Relations                              vectors that appear in falsifications of 𝑟𝑖 in 𝜔, and 𝜅0 , 𝜂𝑖 ∈ N
                                                                   with 𝜂𝑖 ≥ 0 are suitably chosen to ensure that 𝜅 is an OCF
Let 𝒦ℬ = ⟨ℱ, ℛ⟩ be a first-order knowledge base. We                and 𝜅 |= ℛ.
are interested in defeasible inferences that we can draw
from 𝒦ℬ, i.e., we consider (nonmonotonic) inferences of               The value 𝜅0 is a normalizing factor for ensuring that
the form 𝒦ℬ |∼ 𝜙 with 𝜙 ∈ ℒ𝑐Σ being a first-order formula          min𝜔∈Ω 𝜅(𝜔) = 0, and the values 𝜂𝑖 are called impact fac-
or conditional. More precisely, we study inductive inference       tors. Observe that the value of 𝜂𝑖 does not depend on the
relations |∼⊆ ℒ𝑐Σ × ℒ𝑐Σ similar to the ones presented in [7].      specific world 𝜔 under consideration, but on the other condi-
Two fundamental postulates for such inference relations pre-       tionals in ℛ, and can be determined via a set of inequalities
supposed in [7] are that formulas from the knowledge base          between the different 𝜂𝑖 . Therefore, c-representations exist
can be inferred (this is called Direct Inference (DI)), and that   iff this system of inequalities is solvable. This allows for
without conditionals in the knowledge base, conditionals           deriving sufficient conditions for the satisfiability of knowl-
can be inferred only trivially (Trivial Vacuity (TV)).             edge bases in terms of solutions of inequalities. However, in
                                                                   the first-order case, this system of inequalities is much more
(DI) 𝜙 ∈ ℱ ∪ ℛ implies ⟨ℱ, ℛ⟩ |∼ 𝜙.                                complex than in the propositional case because conditionals
                                                                   can be both verified and falsified by different constants in
(TV) ⟨ℱ, ∅⟩ |∼ (𝐵(𝑥 ⃗ )|𝐴(𝑥⃗ )) only if there is a constant
                                                                   the same world, and due to the interactions between facts
      vector ⃗𝑎 such that ℱ |= 𝐴(𝑎 ⃗ ) ⇒ 𝐵(𝑎⃗ ).
                                                                   and conditionals. Therefore, it is hardly possible to give a
   Note that (DI) is a bit basic concerning the treatments of      generic representation of these inequalities for first-order
facts. Actually, we would expect facts from ℱ to be enforced.      knowledge bases. We illustrate c-representations by contin-
We will see that our approach can guarantee this.                  uing our Example 1.
   One natural way to construct an inductive inference re-
                                                                   Example 2 (Example 1 cont’d). We consider the knowledge
lation is to choose a model 𝜅 for each knowledge base and
                                                                   base 𝒦ℬ = ⟨ℱ, ℛ⟩ with ℱ = {𝐴(𝑎)𝐵(𝑎), 𝐴(𝑏)𝐵(𝑏)} and
consider the inferences induced by 𝜅 via
                                                                   ℛ = {𝑟1 = (𝐵(𝑥)|𝐴(𝑥)), 𝑟2 = (𝐵(𝑏)|𝐴(𝑏))} from Exam-
                  ⟨ℱ, ℛ⟩ |≈𝜅 𝜙 iff 𝜅 |= 𝜙,                  (6)           A c-representation 𝜅 of 𝒦ℬ has the form 𝜅(𝜔) =
                                                                   ple 1. ∑︀
                                                                   𝜅0 +      1≤𝑖≤2 𝑓𝑖 (𝜔)𝜂𝑖 for 𝜔 |= 𝐴(𝑎)𝐵(𝑎)𝐴(𝑏)𝐵(𝑏),
where 𝜙 ∈ ℒ𝑐Σ , and ⟨ℱ, ℛ⟩ |≈𝜅 𝜙 means that 𝜙 can be               and 𝜅(𝜔) = ∞ for 𝜔 ̸|= 𝐴(𝑎)𝐵(𝑎)𝐴(𝑏)𝐵(𝑏). Since
inferred from ⟨ℱ, ℛ⟩ via its ranking model 𝜅.                      𝐴(𝑏)𝐵(𝑏) ∈ ℱ , conditional 𝑟2 cannot be falsified by finitely-
   However in general, it is not easy to decide on the ex-         ranked worlds, so the impact factor 𝜂2 is ineffective, and we
istence of models of a first-order knowledge base, i.e., on        just have
the satisfiability of such knowledge bases in our ranking-                           𝜅(𝜔) = 𝜅0 + 𝑓1 (𝜔)𝜂1                     (8)
based semantics, as we saw in Example 1. In particular,
                                                                   for 𝜔 |= 𝐴(𝑎)𝐵(𝑎)𝐴(𝑏)𝐵(𝑏). For any such 𝜔, 𝜅(𝜔) ≥
knowledge given by facts in ℱ may interact with plausi-
                                                                   𝜅0 + 𝜂1 because of the falsification of 𝑟1 by 𝑏, and if no
ble beliefs specified by conditionals in ℛ. For example, if
                                                                   other constant falsifies 𝑟1 , we obtain 𝜅(𝜔) = 𝜅0 + 𝜂1 as the
ℱ |= ∀𝑥      ⃗ ) ⇒ 𝐵(𝑥
        ⃗ .𝐴(𝑥         ⃗ ), the conditional (𝐵(𝑥       ⃗ )) ∈ ℛ
                                                 ⃗ )|𝐴(𝑥
                                                                   minimum rank which must be 0. This yields 𝜅0 = −𝜂1 .
cannot be accepted by a ranking function 𝜅. In this case,
                                                                      The impact factor 𝜂1 ≥ 0 has to be chosen in such a
the 𝒦ℬ = ⟨ℱ, ℛ⟩ would be not satisfiable.
                                                                   way that 𝑟1 is accepted by 𝜅. As for any model of 𝒦ℬ,
   In the next subsection, we recall a class of ranking models
                                                                   it holds that 0 = 𝜅(𝐴(𝑥)𝐵(𝑥)) = 𝜅(𝐴(𝑎)𝐵(𝑎)) <
of first-order knowledge bases that allow for more trans-
                                                                   𝜅(𝐴(𝑎)𝐵(𝑎)) = ∞ and 0 = 𝜅(𝐴(𝑥)𝐵(𝑥)) =
parent investigations into the satisfiability of first-order
                                                                   𝜅(𝐴(𝑏)𝐵(𝑏)) < 𝜅(𝐴(𝑏)𝐵(𝑏)) = ∞, so 𝑎 ∈ wRep(𝑟1 )
knowledge bases and usually provide a basis for quite well-
                                                                   and 𝑏 ∈ wRep((𝐵(𝑥)|𝐴(𝑥))), and Definition 6 (B) applies.
behaved inductive inference.
                                                                   Consider any constant 𝑐 ̸∈ {𝑎, 𝑏}. Since 𝜔 |= 𝐴(𝑐)𝐵(𝑐)
                                                                   can be chosen in such a way that 𝜔 |= 𝐴(𝑑) for any fur-
                                                                   ther constant 𝑑 ̸∈ {𝑎, 𝑏, 𝑐}, we obtain 𝜅(𝐴(𝑐)𝐵(𝑐)) =
𝜅(𝐴(𝑎)𝐵(𝑎)𝐴(𝑏)𝐵(𝑏)𝐴(𝑐)𝐵(𝑐)) = 0 = 𝜅(𝐴(𝑥)𝐵(𝑥)),                       Proof. Any c-representation 𝜅 of ⟨ℱ, ∅⟩ has the form (7)
and analogously, 𝜅(𝐴(𝑐)𝐵(𝑐)) = 𝜂1 .                                  for 𝜔 |= ℱ and satisfies 𝜅(𝜔) = ∞ for 𝜔 ̸|= ℱ. Since
   Consider the case 𝜂1 = 0. Then we would have                      there are no conditionals in the rule base, we simply have
𝜅(𝐴(𝑐)𝐵(𝑐)) = 0 = 𝜅(𝐴(𝑐)𝐵(𝑐)), so 𝑐 ̸∈ wRep(𝑟1 ) and                 𝜅(𝜔) = 𝜅0 for 𝜔 |= ℱ, hence the normalization constant
𝑐 ̸∈ wRep((𝐵(𝑥)|𝐴(𝑥))). Hence wRep(𝑟1 ) = {𝑎} and                    must satisfy 𝜅0 = 0. If 𝜅 |= (𝐵(𝑥            ⃗ )) holds, there
                                                                                                            ⃗ )|𝐴(𝑥
wRep((𝐵(𝑥)|𝐴(𝑥))) = {𝑏}, and therefore Rep(𝑟1 ) = {𝑎}                must be weak representative for (𝐵(𝑥          ⃗ )), hence there
                                                                                                             ⃗ )|𝐴(𝑥
and Rep((𝐵(𝑥)|𝐴(𝑥))) = {𝑏}. So finally, we have to check             must be a constant vector ⃗𝑎 such that 𝜅(𝐴(𝑎       ⃗ )𝐵(𝑎 ⃗ )) <
the last condition (5) from Definition 6 (B) for 𝑎 and 𝑏, and        𝜅(𝐴(𝑎⃗ )𝐵(𝑎⃗ )). This is possible only if 𝜅(𝐴(𝑎  ⃗ )𝐵(𝑎⃗ )) = 0
find that 𝜅(𝐴(𝑎)𝐵(𝑎)) = ∞ = 𝜅(𝐴(𝑏)𝐵(𝑏)), hence (5) is                and 𝜅(𝐴(𝑎 ⃗ )𝐵(𝑎 ⃗ )) = ∞, since 𝜅 has only these two ranks.
violated. Therefore, 𝜂1 = 0 cannot ensure the acceptance of          This implies 𝜅(𝜔) = ∞ for all 𝜔 |= 𝐴(𝑎        ⃗ )𝐵(𝑎⃗ ), i.e., for
𝑟1 .                                                                 all 𝜔 |= 𝐴(𝑎  ⃗ )𝐵(𝑎 ⃗ ), 𝜔 ̸|= ℱ. Via contraposition, ℱ |=
   On the other hand, for any (finite) 𝜂1 > 0 and for any            ¬(𝐴(𝑎      ⃗ )) ≡ 𝐴(𝑎
                                                                           ⃗ )𝐵(𝑎             ⃗ ) ⇒ 𝐵(𝑎⃗ ). This was to be shown.
constant 𝑐 ̸∈ {𝑎, 𝑏}, we then calculate 𝜅(𝐴(𝑥)𝐵(𝑥)) =
𝜅(𝐴(𝑐)𝐵(𝑐)) = 0 < 𝜂1 = 𝜅(𝐴(𝑐)𝐵(𝑐)). Hence each
such 𝑐 is a weak representative satisfying 𝜅(𝐴(𝑐)𝐵(𝑐)) =             3.3. c-Representations for Defeasible 𝒜ℒ𝒞
𝜂1 < ∞ = 𝜅(𝐴(𝑎)𝐵(𝑎)). So in this case, 𝑎 cannot be
a strong representative of 𝑟1 , and we obtain Rep(𝑟1 ) =             The basic idea of our approach is to understand defeasi-
𝑈Σ ∖ {𝑎, 𝑏}. Obviously, any 𝑐 ∈ 𝑈Σ ∖ {𝑏} cannot be a                 ble subsumptions as open first-order conditionals. This
(weak) representative of (𝐵(𝑥)|𝐴(𝑥)), and therefore we have          allows for considering defeasible 𝒜ℒ𝒞 knowledge bases
wRep((𝐵(𝑥)|𝐴(𝑥))) = Rep((𝐵(𝑥)|𝐴(𝑥))) = {𝑏}. Fi-                      as first-order (conditional) knowledge bases and make use
nally, since for any 𝑐 ∈ Rep(𝑟1 ), 𝜅(𝐴(𝑐)𝐵(𝑐)) = 𝜂1 <                of ranking functions to provide semantics for defeasible
∞ = 𝜅(𝐴(𝑏)𝐵(𝑏)), also (5) can be satisfied. Therefore, any           𝒜ℒ𝒞 knowledge bases. Even more, we are then able to
finite 𝜂1 > 0 in (8) yields a c-representation of 𝒦ℬ.                reason inductively from defeasible 𝒜ℒ𝒞 knowledge bases
                                                                     via c-representations. We will investigate both the general
   Nevertheless, if c-representations of a first-order knowl-        ranking-based semantics of defeasible 𝒜ℒ𝒞 reasoning and
edge base exist at all, then there are usually infinitely many       its more sophisticated version based on c-representations
of them. E.g., in Example 2 above, infinitely many 𝜂1 > 0            in the following to show the potential of this semantics for
define infinitely many c-representations. Therefore, some            description logics. Since this paper only takes first steps in
kind of selection procedure is needed in order to formal-            this direction, we want to focus on main techniques of our
ize which c-representations an inductive inference operator          approach to not burden the general line of thought with
should choose.                                                       too many technical details. Therefore, the following three
                                                                     prerequisites apply for the rest of this paper:
Definition 10 (selection strategy 𝜎). A selection strategy
(for c-representations) is a function 𝜎 assigning to each first-         1. Both components ℱ and ℛ of first-order conditional
order conditional knowledge base 𝒦ℬ = ⟨ℱ, ℛ⟩ an impact                      knowledge bases do not mention any constant. For
vector ⃗𝜂 ∈ N|ℛ|                                                            defeasible 𝒜ℒ𝒞 knowledge bases, this means that
                         𝜎 : 𝒦ℬ ↦→ ⃗𝜂                                       the ABox is empty.
such that the OCF obtained by using ⃗𝜂 as impacts in Defini-             2. The ranking-based semantics for open first-order
tion 9 is a c-representation of ℛ.                                          conditionals is restricted to option (A) of Definition 6,
                                                                            i.e., in the following, 𝜅 |= 𝑟 = (𝐵(𝑥            ⃗ )) iff
                                                                                                                       ⃗ )|𝐴(𝑥
  With the help of selection strategies, we are now able to                 Rep(𝑟) ̸= ∅ and
define inductive inference operators specifically for infer-
ences obtained from c-representations of a given knowledge                             𝜅(𝐴(𝑥
                                                                                           ⃗ )𝐵(𝑥
                                                                                                ⃗ )) < 𝜅(𝐴(𝑥    ⃗ )).
                                                                                                           ⃗ )𝐵(𝑥
base.
                                                                         3. Moreover, we also presuppose that there are
                   c-rep
Definition 11 (C𝜎 ). An inductive inference operator for                    “enough” constants available in 𝑈Σ to ensure that
c-representations with selection strategy 𝜎 is a function                   for every conditional there is some constant vector
                                                                            that can serve as a strong representative, and that
                   Cc-rep
                    𝜎     : 𝒦ℬ ↦→ 𝜅𝜎(𝒦ℬ)                                    non-acceptance of conditionals is not due to |𝑈Σ |
                                                                            being too small. E.g., one may assume that for every
where 𝜎 is a selection strategy for c -representations. As before,          conditional 𝑟 = (𝐵(𝑥  ⃗ )|𝐴(𝑥⃗ )) there exists a special
a corresponding inductive inference relation can be obtained                constant vector ⃗𝑎𝑟 with ⃗𝑎𝑟 ∈ ℋ𝑟 the components of
via Equation (6).                                                           which do not occur anywhere else in the knowledge
   It can easily be checked that the postulates (DI) and                    base.
(TV) are satisfied by all inference relations induced from           The first prerequisite is not uncommon for description logics
c-representations. (DI) is ensured by the fact that each c-          and is an intuitive justification for the second prerequisite.
representation is a model of the knowledge base, and (TV)            Although the ranking-based conditional semantics for first-
is immediate from Equation (7), as the following lemma               order knowledge bases from Section 2.1 is able very well
shows.                                                               to deal with information about individuals and even allows
                                                                     for having a defeasible ABox, as Examples 1 and 2 illustrate,
Lemma 1. Let ⟨ℱ, ∅⟩ be a first-order knowledge base, let
                                                                     these examples also show how intricate investigations can
𝜅 be a c-representation of ⟨ℱ, ∅⟩. Then for any conditional
                                                                     be when option (B) of Definition 6 must be applied. This
(𝐵(𝑥⃗ )|𝐴(𝑥⃗ )), 𝜅 |= (𝐵(𝑥
                         ⃗ )|𝐴(𝑥 ⃗ )) only if there is a constant
                                                                     option is typically relevant only in cases where knowledge
vector ⃗𝑎 such that ℱ |= 𝐴(𝑎 ⃗ ) ⇒ 𝐵(𝑎  ⃗ ).
                                                                     or beliefs about individuals are present. Since we focus on
                                                                     generic (conditional) beliefs in this paper, i.e., our knowledge
bases consist of quantified first-order sentences and open            (CM∃ ) If ∃𝑟.𝐴 ⊏
                                                                                     ∼ 𝐶 and ∃𝑟.𝐴 ∼ ∀𝑟.𝐵, then ∃𝑟.(𝐴⊓𝐵) ∼
                                                                                                  ⊏                     ⊏
conditionals representing defeasible subsumptions, we use                   𝐶.
only option (A) of Definition 6 in this paper.
   In fact, condition (4) is enough to ensure the acceptance          (CM∀ ) If ∀𝑟.𝐴 ⊏
                                                                                     ∼ 𝐶 and ∀𝑟.𝐴 ∼ ∀𝑟.𝐵, then ∀𝑟.(𝐴⊓𝐵) ∼
                                                                                                  ⊏                     ⊏
of a conditional, as the following proposition shows.                       𝐶.
                                                                      (RM∃ ) If ∃𝑟.𝐴 ⊏
                                                                                     ∼ 𝐶 and ∃𝑟.𝐴 ∼ ∀𝑟.¬𝐵, then ∃𝑟.(𝐴 ⊓
                                                                                                  ⊏
                                                                                                          ⧸︀
Proposition 2. Let 𝜅 be an OCF and let (𝐵(𝑥⃗ )|𝐴(𝑥
                                                 ⃗ )) be an
open conditional. If 𝜅(𝐴(𝑥 ⃗ )𝐵(𝑥
                                ⃗ )) < 𝜅(𝐴(𝑥
                                           ⃗ )𝐵(𝑥
                                                ⃗ )) holds,                 𝐵) ⊏∼ 𝐶.
then 𝜅 |= (𝐵(𝑥
             ⃗ )|𝐴(𝑥 ⃗ )).                                            (RM∀ ) If ∀𝑟.𝐴 ⊏
                                                                                     ∼ 𝐶 and ∀𝑟.𝐴 ∼ ∀𝑟.¬𝐵, then ∀𝑟.(𝐴 ⊓
                                                                                                  ⊏
                                                                                                          ⧸︀

                                                                            𝐵) ⊏∼ 𝐶.
Proof. We have to show that (4) ensures that the con-
ditional has strong representatives. Let ⃗𝑎 be such that                 We now present a version of the KLM-style postulates
𝜅(𝐴(𝑥 ⃗ )𝐵(𝑥 ⃗ )) = 𝜅(𝐴(𝑎   ⃗ )𝐵(𝑎 ⃗ )). Since 𝜅(𝐴(𝑥  ⃗ )𝐵(𝑥
                                                           ⃗ )) <     using first-order conditionals. Since concepts and roles in
𝜅(𝐴(𝑥 ⃗ )𝐵(𝑥 ⃗ )) ≤ 𝜅(𝐴(𝑎 ⃗ )𝐵(𝑎  ⃗ )), we have 𝜅(𝐴(𝑎  ⃗ )𝐵(𝑎
                                                            ⃗ )) <    description logics are unary and binary predicates, respec-
𝜅(𝐴(𝑎 ⃗ )𝐵(𝑎 ⃗ )). Therefore, ⃗𝑎 is at least a weak representative    tively, we use single variables 𝑥, 𝑦 instead of vectors ⃗𝑥 here
of (𝐵(𝑥 ⃗ )|𝐴(𝑥 ⃗ )), which means that Rep((𝐵(𝑥            ⃗ ))) ̸=
                                                     ⃗ )|𝐴(𝑥          in order to simplify notation. However, none of the proofs
∅. Because (A) holds by definition, it follows that 𝜅 |=              in this paper rely on the arity of the predicates. Moreover,
(𝐵(𝑥⃗ )|𝐴(𝑥 ⃗ )).                                                     in compliance with the prerequisites stated in the previous
                                                                      section, we can assume that there is at least one constant
    To motivate prerequisite (3), consider Example 2 again. If
                                                                      symbol, i.e. 𝑈Σ ̸= ∅.
𝑈Σ would consist only of the constants 𝑎 and 𝑏, conditional
𝑟1 could not be accepted for the only reason that neither 𝑎           (Ref) 𝜅 |= (𝐴(𝑥)|𝐴(𝑥)).
nor 𝑏 can be a strong representative for 𝑟1 (please see the
argumentation for case 𝜂1 = 0 in the example). At least a             (LLE) If 𝜅 ||− ∀𝑥.[𝐴(𝑥) ⇔ 𝐵(𝑥)] and 𝜅 |= (𝐶(𝑥)|𝐴(𝑥)),
third constant 𝑐 ̸∈ {𝑎, 𝑏} is needed to ensure the acceptance               then 𝜅 |= (𝐶(𝑥)|𝐵(𝑥)).
of 𝑟1 .
                                                                      (RW) If 𝜅 ||− ∀𝑥.[𝐵(𝑥) ⇒ 𝐶(𝑥)] and 𝜅 |= (𝐵(𝑥)|𝐴(𝑥)),
    However, even under all three prerequisites from above,
                                                                           then 𝜅 |= (𝐶(𝑥)|𝐴(𝑥)).
it is hard to make general statements about the consistency
of a first-order knowledge base, or the system of inequalities        (And) If 𝜅 |= (𝐵(𝑥)|𝐴(𝑥)), (𝐶(𝑥)|𝐴(𝑥)), then 𝜅 |=
that impact factors in c-representations have to solve. The                 (𝐵(𝑥) ∧ 𝐶(𝑥)|𝐴(𝑥)).
papers [13, 14] present a sufficient condition for the consis-
tency of a first-order knowledge base by lifting the concept          (Or) If 𝜅 |= (𝐶(𝑥)|𝐴(𝑥)), (𝐶(𝑥)|𝐵(𝑥)), then 𝜅 |=
of a tolerance partition (on which the propositional system                 (𝐶(𝑥)|𝐴(𝑥) ∨ 𝐵(𝑥)).
Z [15] is based) to the first-order case. However, it is still an
                                                                      (CM) If 𝜅 |= (𝐵(𝑥)|𝐴(𝑥)), (𝐶(𝑥)|𝐴(𝑥)), then 𝜅 |=
open question under which conditions c-representations for
                                                                           (𝐶(𝑥)|𝐴(𝑥) ∧ 𝐵(𝑥)).
a first-order knowledge base exist. Our conjecture here is
that they exist if the knowledge base is consistent, i.e., if it      (RM) If 𝜅 |= (𝐶(𝑥)|𝐴(𝑥)) and 𝜅 ̸|= (𝐵(𝑥)|𝐴(𝑥)), then
has a ranking model at all, just as in the propositional case.             𝜅 |= (𝐶(𝑥)|𝐴(𝑥) ∧ 𝐵(𝑥)).
We leave further investigations into this research question
for future work and focus on the quality of inductive rea-            The translation of the quantified postulates using first-order
soning based on c-representations for defeasible description          conditionals is given below.
logics in the following.
                                                                      (CM∃ ) If 𝜅 |= (𝐶(𝑥) | ∃𝑦.[𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦)]) and 𝜅 |=
                                                                            (∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐵(𝑦)] | ∃𝑦.[𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦)]), then
4. KLM-style Postulates and Beyond                                          𝜅 |= (𝐶(𝑥)|∃𝑦.𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦) ∧ 𝐵(𝑦)).

In [4], the well-known KLM postulates for non-monotonic               (CM∀ ) If 𝜅 |= (𝐶(𝑥) | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)]) and 𝜅 |=
reasoning were translated for use with defeasible description               (∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐵(𝑦)] | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)]),
logics, and also the postulate of rational monotonicity was                 then 𝜅 |= (𝐶(𝑥) | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ (𝐴(𝑦) ∧ 𝐵(𝑦))]).
considered. Let 𝐴, 𝐵, 𝐶 be concepts.                                  (RM∃ ) If 𝜅 |= (𝐶(𝑥) | ∃𝑦.[𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦)]) and 𝜅 ̸|=
(Ref) 𝐴 ⊏                                                                   (∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐵(𝑦)] | ∃𝑦.[𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦)]), then
        ∼ 𝐴.
                                                                            𝜅 |= (𝐶(𝑥) | ∃𝑦.[𝑅(𝑥, 𝑦) ∧ 𝐴(𝑦) ∧ 𝐵(𝑦)]).
(LLE) If 𝐴 ≡ 𝐵 and 𝐴 ⊏
                     ∼ 𝐶, then 𝐵 ∼ 𝐶.
                                 ⊏
                                                                      (RM∀ ) If 𝜅 |= (𝐶(𝑥) | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)]) and 𝜅 ̸|=
(RW) If 𝐴 ⊏
          ∼ 𝐵 and 𝐵 ⊑ 𝐶, then 𝐴 ∼ 𝐶.
                                ⊏                                           (∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐵(𝑦)] | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)]),
                                                                            then 𝜅 |= (𝐶(𝑥) | ∀𝑦.[𝑅(𝑥, 𝑦) ⇒ (𝐴(𝑦) ∧ 𝐵(𝑦))]).
(And) If 𝐴 ⊏
           ∼ 𝐵 and 𝐴 ∼ 𝐶, then 𝐴 ∼ (𝐵 ⊓ 𝐶).
                     ⊏           ⊏
                                                                      Proposition 3. All of the postulates given above hold for
          ∼ 𝐶 and 𝐵 ∼ 𝐶, then (𝐴 ⊔ 𝐵) ∼ 𝐶.
(Or) If 𝐴 ⊏         ⊏                 ⊏
                                                                      every OCF 𝜅.
(CM) If 𝐴 ⊏
          ∼ 𝐵 and 𝐴 ∼ 𝐶, then (𝐴 ⊓ 𝐵) ∼ 𝐶.
                    ⊏                 ⊏
                                                                      Proof. In the following proofs for the individual postulates,
(RM) If 𝐴 ⊏                                                           we implicitly use Proposition 2 and prove the acceptance
          ∼ 𝐶 and 𝐴 ∼ ¬𝐵, then (𝐴 ⊓ 𝐵) ∼ 𝐶.
                    ⊏                  ⊏
                           ⧸︀
                                                                      of desired conditionals by proving that their verification is
Moreover, in [4], quantified versions of (CM) and (RM)                more plausible than their falsification.
which are adapted to the specific form of DL concepts have               (Ref): This postulate is straightforward as 𝜅(𝐴(𝑥)) <
been presented.                                                       𝜅(⊥) by definition.
   (LLE): Let 𝐴(𝑥) be equivalent to 𝐵(𝑥) for all 𝑥 in all           Postulate (CLA) claims that each ranking model of a con-
feasible possible worlds, and let 𝜅 |= (𝐶(𝑥)|𝐴(𝑥)). Be-          ditional knowledge base respects all classical consequences
cause of the equivalence of 𝐴(𝑥) and 𝐵(𝑥), we have               of the facts. Postulate (SUB) reveals a compatibility between
𝜅(𝐴(𝑎)𝐶(𝑎)) = 𝜅(𝐵(𝑎)𝐶(𝑎)) and 𝜅(𝐴(𝑎)𝐶(𝑎)) =                      a conditional and its counterpart as material implication.
𝜅(𝐵(𝑎)𝐶(𝑎)) for every 𝑎. Therefore, if 𝑎 is a represen-          But note that this counterpart is only plausible.
tative of (𝐶(𝑥)|𝐴(𝑥)), it has to be a representative of             The next proposition shows that both these postulates
(𝐶(𝑥)|𝐵(𝑥)) as well. Hence, if condition (A) or (B) from         are also satisfied by our approach.
Definition 6 holds for (𝐶(𝑥)|𝐴(𝑥)), the respective condi-
tion has to hold for (𝐶(𝑥)|𝐵(𝑥)), too.                           Proposition 4. Let 𝜅 be an OCF. If 𝜅 is a model of ⟨ℱ, ℛ⟩
   (RW): We have 𝜅(𝐴(𝑥)𝐵(𝑥)) < 𝜅(𝐴(𝑥)𝐵(𝑥)) and                   then 𝜅 ||− 𝛼 for all 𝛼 ∈ ℒΣ with ℱ |= 𝛼. If 𝜅 is a model of
the fact ∀𝑥.[𝐵(𝑥) ⇒ 𝐶(𝑥)]. Therefore, we have                    ⟨∅, {(𝐵(𝑥      ⃗ ))}⟩ then 𝜅 |= (𝐴(𝑥
                                                                          ⃗ )|𝐴(𝑥                   ⃗ ) ⇒ 𝐵(𝑥⃗ )|⊤).
𝜅(𝐴(𝑥)𝐶(𝑥)) ≤ 𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)) = 𝜅(𝐴(𝑥)𝐵(𝑥)) and
                                                                 Proof. Let 𝜅 be a model of ⟨ℱ, ℛ⟩, let 𝛼 ∈ ℒΣ with
𝜅(𝐴(𝑥)𝐶(𝑥)) = 𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)) ≥ 𝜅(𝐴(𝑥)𝐵(𝑥)).
                                                                 ℱ |= 𝛼. Then 𝜅(𝜔) = ∞ for all 𝜔 ̸|= ℱ and hence
Hence, 𝜅(𝐴(𝑥)𝐶(𝑥)) < 𝜅(𝐴(𝑥)𝐶(𝑥)).
                                                                 also for all 𝜔 ̸|= 𝐴. Therefore, 𝜅 ||− 𝛼. If 𝜅 is a
   (And): Because of 𝜅(𝐴(𝑥)𝐵(𝑥)) < 𝜅(𝐴(𝑥)𝐵(𝑥))
                                                                 model of ⟨∅, {(𝐵(𝑥        ⃗ ))}⟩ then 𝜅 |= (𝐵(𝑥
                                                                                     ⃗ )|𝐴(𝑥                       ⃗ )|𝐴(𝑥⃗ )),
and 𝜅(𝐴(𝑥)𝐶(𝑥)) < 𝜅(𝐴(𝑥)𝐶(𝑥)), the minimal worlds
                                                                 i.e., 𝜅(𝐴(𝑥
                                                                           ⃗ )𝐵(𝑥
                                                                                ⃗ )) < 𝜅(𝐴(𝑥        ⃗ )). Since 𝜅(𝐴(𝑥
                                                                                               ⃗ )𝐵(𝑥                  ⃗) ⇒
𝜔 in 𝜅 with 𝜔 |= 𝐴(𝑎) for some 𝑎 have to sat-
                                                                 𝐵(𝑥 ⃗ )) = 𝜅(¬𝐴(𝑥 ⃗ ) ∨ 𝐵(𝑥⃗ )) ≤ 𝜅(𝐴(𝑥      ⃗ )), the state-
                                                                                                         ⃗ )𝐵(𝑥
isfy both 𝐵(𝑎) and 𝐶(𝑎) as well.            Therefore, we
                                                                 ment follows.
can conclude that 𝜅(𝐴(𝑥)𝐵(𝑥)) = 𝜅(𝐴(𝑥)𝐶(𝑥)) =
𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)). It follows that 𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)) <                  The next postulate deals with obviously irrelevant vari-
min{𝐴(𝑥)𝐵(𝑥), 𝐴(𝑥)𝐶(𝑥)} = 𝜅(𝐴(𝑥)(𝐵(𝑥) ∨ 𝐶(𝑥))).                  ables in an open conditional, i.e., variables that do not occur
   (Or): It holds that 𝜅(𝐴(𝑥)𝐶(𝑥) ∨ 𝐵(𝑥)𝐶(𝑥)) =                  in both the antecedent and the consequent of the conditional.
min{𝜅(𝐴(𝑥)𝐶(𝑥)), 𝜅(𝐵(𝑥)𝐶(𝑥))}                              <     It adapts the postulate (IRR) from [8].
min{𝜅(𝐴(𝑥)𝐶(𝑥)), 𝜅(𝐵(𝑥)𝐶(𝑥))} = 𝜅(𝐴(𝑥)𝐶(𝑥) ∨
𝐵(𝑥)𝐶(𝑥)).                                                       (IRR) Let ⃗𝑥, ⃗𝑦 , ⃗𝑧 mention variables from pairwise dis-
   (CM): Because of 𝜅(𝐴(𝑥)𝐵(𝑥)) < 𝜅(𝐴(𝑥)𝐵(𝑥)) and                       joint sets. Then ⟨∅, {(𝐵(𝑥                     ⃗ , ⃗𝑧 ))}⟩ |∼
                                                                                                            ⃗ , ⃗𝑦 )|𝐴(𝑥
𝜅(𝐴(𝑥)𝐶(𝑥)) < 𝜅(𝐴(𝑥)𝐶(𝑥)), the minimal worlds 𝜔                        (𝐵(𝑥 ⃗ , ⃗𝑏)|𝐴(𝑥⃗ , ⃗𝑎)) for all proper groundings ⃗𝑎, ⃗𝑏 of
in 𝜅 with 𝜔 |= 𝐴(𝑎) for some 𝑎 have to satisfy both                    ⃗𝑧 resp. ⃗𝑦 in 𝐴 resp. 𝐵.
𝐵(𝑎) and 𝐶(𝑎) as well. Therefore, we can conclude that
𝜅(𝐴(𝑥)𝐵(𝑥)) = 𝜅(𝐴(𝑥)𝐶(𝑥)) = 𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)).                       This postulate does not hold in general for our ranking
It follows that 𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)) < 𝜅(𝐴(𝑥)𝐶(𝑥)) ≤                  semantics but we can show that it holds for ranking models
𝜅(𝐴(𝑥)𝐵(𝑥)𝐶(𝑥)).                                                 which are c-representations.
   (CM∃ ): Let 𝜔 be a minimal world in 𝜅 such that 𝑥, 𝑦 ex-
ist with 𝜔 |= 𝑅(𝑥, 𝑦)𝐴(𝑦). Since (A) holds, for every 𝜔 ′        Proposition 5. Let ⃗𝑥, ⃗𝑦 , ⃗𝑧 mention variables from pairwise
with 𝜅(𝜔 ′ ) = 𝜅(𝜔) we have 𝜔 ′ |= ∀𝑥.∀𝑦.[𝑅(𝑥, 𝑦)𝐴(𝑦) ⇒          disjoint sets, and let 𝒦ℬ = ⟨∅, {(𝐵(𝑥       ⃗ , ⃗𝑦 )|𝐴(𝑥
                                                                                                                        ⃗ , ⃗𝑧 ))}⟩. Let
𝐶(𝑥)𝐵(𝑦)], and for every 𝜔 ′′ with 𝜅(𝜔 ′′ ) < 𝜅(𝜔)               𝜅 = 𝜅𝜎(𝒦ℬ) be a strategic c-representation of 𝒦ℬ. Then
we have 𝜔 ′′ |= ∀𝑥.∀𝑦.𝑅(𝑥, 𝑦) ∨ 𝐴(𝑦). Therefore,                 𝜅 |= (𝐵(𝑥   ⃗ , ⃗𝑏)|𝐴(𝑥
                                                                                       ⃗ , ⃗𝑎)) for all proper groundings ⃗𝑎, ⃗𝑏 of ⃗𝑧
𝜅(𝜔) = 𝜅(𝐶(𝑥) ∧ ∃𝑦.[𝑅(𝑥, 𝑦)𝐴(𝑦)𝐵(𝑦)]) < 𝜅(𝐶(𝑥) ∧                 resp. ⃗𝑦 in 𝐴 resp. 𝐵.
∃𝑦.[𝑅(𝑥, 𝑦)𝐴(𝑦)𝐵(𝑦)]).
   (CM∀ ): Let 𝜔 be a minimal world in 𝜅 such that 𝑥 ex-         Proof. Let ⃗𝑥, ⃗𝑦 , ⃗𝑧 mention variables from pairwise disjoint
ists with 𝜔 |= ∀𝑦.𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦). Since (A) holds, for           sets, and let 𝒦ℬ = ⟨∅, {(𝐵(𝑥                   ⃗ , ⃗𝑧 ))}⟩. Each c-
                                                                                                     ⃗ , ⃗𝑦 )|𝐴(𝑥
every 𝜔 ′ with 𝜅(𝜔 ′ ) = 𝜅(𝜔) it holds for all 𝑥 that 𝜔 ′ |=     representation 𝜅 of 𝒦ℬ has the form
∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)] implies 𝜔 ′ |= 𝐶(𝑥)∧∀𝑦.[𝑅(𝑥, 𝑦) ⇒
                                                                                        𝜅(𝜔) = 𝜅0 + 𝑓1 (𝜔)𝜂1 ,
𝐵(𝑦)]. And for every 𝜔 ′′ with 𝜅(𝜔 ′′ ) < 𝜅(𝜔) we have
𝜔 ′′ |= ∀𝑥.∃𝑦.𝑅(𝑥, 𝑦)𝐴(𝑦). Therefore, 𝜅(𝜔) = 𝜅(𝐶(𝑥) ∧            where 𝑓1 (𝜔) = #{(𝑎         ⃗ , ⃗𝑏, ⃗)|(𝑎
                                                                                                     𝑐 ⃗ , ⃗𝑏, ⃗)
                                                                                                                𝑐 are proper groundings
∀𝑦.[𝑅(𝑥, 𝑦) ⇒ 𝐴(𝑦)𝐵(𝑦)]) < 𝜅(𝐶(𝑥) ∧ ∀𝑦.[𝑅(𝑥, 𝑦) ⇒                of ⃗𝑧 , ⃗𝑦 , ⃗𝑥 in (𝐵(𝑥
                                                                                       ⃗ , ⃗𝑦 )|𝐴(𝑥  ⃗ , ⃗𝑧 ))} and 𝜅0 , 𝜂1 ∈ N with 𝜂1
𝐴(𝑦)𝐵(𝑦)]).                                                      suitably chosen to ensure that 𝜅 |= (𝐵(𝑥                       ⃗ , ⃗𝑧 )). This
                                                                                                                     ⃗ , ⃗𝑦 )|𝐴(𝑥
   For the case (A), (RM), (RM∃ ), and (RM∀ ) are implied by     latter condition enforces that
(CM), (CM∃ ), and (CM∀ ), respectively.

  In [8], the authors present an approach to defeasible rea-            min 𝜅(𝐴(𝑐       ⃗, ⃗𝑏) < min 𝜅(𝐴(𝑐
                                                                                ⃗, ⃗𝑎)𝐵(𝑐                        ⃗, ⃗𝑏).
                                                                                                         ⃗, ⃗𝑎)𝐵(𝑐
                                                                        ⃗ ⃗ ,𝑐
                                                                        𝑎,𝑏  ⃗                           ⃗ ⃗ ,𝑐
                                                                                                         𝑎,𝑏  ⃗
soning for a restricted first-order logic which they evaluate
according to postulates that are inspired by rational closure    The left hand side here is 0 (all instantiations verifying the
[16]. Beyond the KLM-postulates the satisfaction of which        conditional), and the right hand side here is 𝜂1 (just one
we proved above, they also propose further postulates. E.g.,     falsification of the conditional), so we obtain 𝜂1 > 0 from
the postulate (INCL) in that paper corresponds to our (DI). In   that.
the following, we adapt and extend three of those properties        Now, if we take any proper groundings ⃗𝑎, ⃗𝑏 of ⃗𝑧 resp. ⃗𝑦
that deal with relations to classical logic and irrelevance to   in 𝐴 resp. 𝐵 and check whether
the framework here. First, we consider relations to classical
logic resp. implication:                                                min 𝜅(𝐴(𝑐
                                                                          ⃗
                                                                          𝑐
                                                                                ⃗, ⃗𝑧 )𝐵(𝑐
                                                                                         ⃗, ⃗𝑦 ) < min 𝜅(𝐴(𝑐
                                                                                                           ⃗, ⃗𝑧 )𝐵(𝑐
                                                                                                           ⃗
                                                                                                           𝑐
                                                                                                                    ⃗, ⃗𝑦 ),

(CLA) Let 𝒦ℬ = ⟨ℱ, ℛ⟩ be a first-order conditional               we find again that the left hand side is 0 and the right hand
      knowledge base, and let 𝜅 be a model of 𝒦ℬ. If             side is 𝜂1 . Since 𝜂1 > 0 must hold, we conclude that 𝜅 |=
      ℱ |= 𝛼 ∈ ℒΣ , then 𝜅 ||− 𝛼.
                                                                 (𝐵(𝑥⃗ , ⃗𝑏)|𝐴(𝑥
                                                                               ⃗ , ⃗𝑎)) for all proper groundings ⃗𝑎, ⃗𝑏 of ⃗𝑧 resp. ⃗𝑦
(SUB) ⟨∅, {(𝐵(𝑥     ⃗ ))}⟩ |∼ (𝐴(𝑥
              ⃗ )|𝐴(𝑥            ⃗ ) ⇒ 𝐵(𝑥
                                         ⃗ )|⊤).                 in 𝐴 resp. 𝐵.
5. Application of OCF-based                                        A cwm -model ℳ satisfies a defeasible subsumption 𝐶 ⊏∼
                                                                 𝐷 iff the <ℳ -minimal instances of 𝐶 are instances of 𝐷:
   Reasoning to a DL Knowledge
                                                                                              ℳ   ℐ     ℐ
   Base                                                                          ∼ 𝐷 iff min(< , 𝐶 ) ⊆ 𝐷 ,
                                                                          ℳ |= 𝐶 ⊏

The goal of this section is to provide an example for how        where min(<, 𝑆) = {𝑠 ∈ 𝑆 | ∄𝑠′ ∈ 𝑆 : 𝑠′ < 𝑠} as usual.
a DL knowledge base can be translated into a first-order           Now we move towards defining 𝑐𝑤m -entailment from
knowledge base, so that OCF-based inductive reasoning can        defeasible knowledge bases. Let 𝒮𝒦ℬ be the set that contains
be applied. Further, we point out some commonalities and         𝐶 and ¬𝐶 for all concepts 𝐶 that occur in a knowledge base
differences between the OCF-based semantics and the cwm -        𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩. We say that {𝐷1 , . . . , 𝐷𝑚 } ⊆ 𝒮𝒦ℬ is
semantics introduced by Giordano and Theseider Dupré in          consistent with 𝒦ℬ if
[5] which we consider first.
                                                                               𝒯 ̸|= (𝐷1 ⊓ · · · ⊓ 𝐷𝑚 ) ⊑ ⊥ ,

5.1. Concept-Wise Multipreference                                i.e. if the intersection of 𝐷1 to 𝐷𝑚 does not have to be
     Semantics                                                   empty.

In [5], a concept-wise multipreference (cwm ) semantics for      Definition 14 (canonical interpretation [5]). A cwm -model
ranked defeasible knowledge bases was presented, which           ℳ = ⟨∆ℐ , ·ℐ , <ℳ ⟩ is canonical for a knowledge base
makes use of a typicality operator T on concepts used for        𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩ if ⟨∆ℐ , ·ℐ ⟩ satisfies 𝒯 , and for any set of
the construction of typicality inclusions of the form T(𝐶) ⊑     concepts {𝐷1 , . . . , 𝐷𝑚 } ⊆ 𝒮𝒦ℬ consistent with 𝒦ℬ, there
𝐷 (where 𝐶, 𝐷 are concepts). We provide an equivalent            exists 𝑥 ∈ (𝐷1 ⊓ · · · ⊓ 𝐷𝑚 )ℐ .
definition using the notation 𝐶 ⊏   ∼ 𝐷 here. Note that our
definition here is a simplified version of the one given in         In other words, an interpretation is canonical if there is
[5], because we only consider non-ranked knowledge bases         at least one domain element 𝑥 ∈ ∆ℐ in every intersection
in this paper.                                                   of concepts that occur in 𝒦ℬ.
   In order to define a preference relation over individuals,    Definition 15 (T-compliant interpretation [5]). A cwm -
the DBox 𝒟 is partitioned based on the left-hand side of the     model ℳ = ⟨∆ℐ , ·ℐ , <ℳ ⟩ is T-compliant for a knowledge
defeasible inclusions. Let 𝒞 = {𝐶 | (𝐶 ⊏     ∼ 𝐷) ∈ 𝒟}. For      base 𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩ if ⟨∆ℐ , ·ℐ ⟩ satisfies 𝒯 and for all 𝐶 ∈
each concept 𝐶 ∈ 𝒞, let 𝒟𝐶 be the set that contains all de-      𝒞 with 𝐶 ℐ ̸= ∅, there exists 𝑥 ∈ 𝐶 ℐ such that 𝒟𝐶
                                                                                                                  ℐ
                                                                                                                    (𝑥) = 𝒟𝐶 .
feasible inclusions (𝐶 ⊏∼ 𝐷) ∈ 𝒟, and for an interpretation
ℐ = ⟨∆ℐ , ·ℐ ⟩, let 𝒟𝐶
                     ℐ
                       (𝑥) be the set of defeasible inclusions     The definition above means that for all non-empty con-
from 𝒟𝐶 which are not violated by 𝑥, i.e.                        cepts 𝐶, there is at least one instance of 𝐶 which does not
                                                                 violate any defeasible subsumptions in with 𝐶 on the left-
      ℐ                                   ℐ
     𝒟𝐶 (𝑥) = {(𝐶 ⊏
                  ∼ 𝐷) ∈ 𝒟𝐶 | 𝑥 ∈ (¬𝐶 ⊔ 𝐷) }.                    hand side.

Based on the amount of non-violated defeasible subsump-          Definition 16 (cwm -entailment [5]). A defeasible subsump-
                                                                 tion 𝑑 = 𝐶 ⊏∼ 𝐷 is cw -entailed by a knowledge base 𝒦ℬ
                                                                                        m
tions, for each concept 𝐶 ∈ 𝒞 a preference relation ≤𝐶 is
defined via                                                      (short: 𝒦ℬ |≈cwm 𝑑) if all canonical and T-compliant cwm -
                                                                 models of 𝒦ℬ satisfy 𝑑.
                           ℐ          ℐ
              𝑥 ≤𝐶 𝑦 iff |𝒟𝐶 (𝑥)| ≥ |𝒟𝐶 (𝑦)|.              (9)
                                                                    It was proven in [5] that cwm -entailment fulfills the prop-
  Before we can define cwm -models, we need one more             erties (Ref), (LLE), (And), (Or), and (CM).
definition: If a concept 𝐶 is a (potentially) strict subset of
another concept 𝐷, the subset 𝐶 can be viewed as more            5.2. Translation of a DL Knowledge Base
specific then 𝐷.
                                                                 In order to allow for a comparison between the approach
Definition 12 (specificity of concepts [5]). Given a defea-      of [5] and the OCF-based semantics in the next part of this
sible knowledge base 𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩ and two concepts            section, we now give an example for how a defeasible knowl-
𝐶, 𝐷 ∈ 𝒞, we call 𝐶 more specific than 𝐷 (short: 𝐶 ≻ 𝐷)          edge base can be transformed into a first-order knowledge
iff 𝒯 |= 𝐶 ⊑ 𝐷 and 𝒯 ̸|= 𝐷 ⊑ 𝐶.                                  base.
  In cwm -models of defeasible knowledge bases, the prefer-      Example 3. We consider the following example DL knowl-
ence relations for the specific concepts defined in Equation 9   edge base, which is very similar to the running example pre-
are combined into a global preference relation based on the      sented in [5].
concepts’ specificity.
                                                                      𝒯 = {Employee ⊑ Adult, PhdStudent ⊑ Student,
Definition 13 (cwm -model [5]). A cwm -model of a defea-                    (∃has_funding.⊤ ⊓ ¬Funded) ⊑ ⊥},
sible knowledge base 𝒦ℬ = ⟨𝒯 , 𝒟, 𝒜⟩ is a tuple ℳ =
⟨∆ℐ , ·ℐ , <ℳ ⟩, where ∆ℐ ̸= ∅, ⟨∆ℐ , ·ℐ ⟩ is an 𝒜ℒ𝒞-              𝒟Employee = { 𝑑1 : Employee ⊏
                                                                                               ∼ ¬Young,
interpretation satisfying 𝒯 and 𝒜, and <ℳ is an ordering                         𝑑2 : Employee ⊏
                                                                                               ∼ ∃has_boss.Employee },
over ∆ℐ such that 𝑥 <ℳ 𝑦 iff                                        𝒟Student = { 𝑑3 : Student ⊏
                                                                                              ∼ ∃has_classes.⊤,
    1. 𝑥 <𝐶 𝑦 for some 𝐶 ∈ 𝒞, and                                                𝑑4 : Student ⊏
                                                                                              ∼ Young,
    2. for all 𝐶 ∈ 𝒞: 𝑥 ≤𝐶 𝑦, or there exists 𝐶 ′ such that                      𝑑5 : Student ⊏
                                                                                              ∼ ¬Funded },
       𝐶 ′ ≻ 𝐶 and 𝑥 <𝐶 ′ 𝑦.
                                                                 𝒟PhdStudent = { 𝑑6 : PhdStudent ⊏
                                                                                                 ∼ ∃has_funding.Money,
                                                                                 𝑑7 : PhdStudent ⊏
                                                                                                 ∼ Bright }.
As description logics are fragments of first-order logic, the            In the following, we present an example which shows that
knowledge base above can easily be translated into a first-           the desirable properties of cwm -semantics w.r.t. inheritance
order knowledge base. We start by translating the strict sub-         of properties are fulfilled by the OCF-based semantics as
sumptions in the TBox as facts.                                       well. A full axiomatization of “proper inheritance of proper-
                                                                      ties” is out of the scope of this paper, and will be addressed
    ℱ = {∀𝑥.[Employee(𝑥) ⇒ Adult(𝑥)],                                 in future work.
            ∀𝑥.[PhdStudent(𝑥) ⇒ Student(𝑥)],
                                                                      Example 4. We consider again the knowledge base from
            ∀𝑥.[∃𝑦.has_funding(𝑥, 𝑦) ⇒ Funded(𝑥)]} .
                                                                      Example 3. In [5], several possible queries and desirable re-
The defeasible subsumptions can be translated as open con-            sults are mentioned. Formulated as queries for the first-order
ditionals. From now on, all predicates are shortened to their         knowledge base defined above, they read as follows.
initial letters.                                                           1. ⟨ℱ, ℛ⟩ |≈𝜅 ∃𝑦.hb(𝑥, 𝑦) | 𝑆(𝑥) ∧ 𝐸(𝑥) ?
                                                                                          (︀                             )︀

        ℛ = { 𝑟1 : (¬𝑌 (𝑥) | 𝐸(𝑥)),                                           ˓→ should be(︀yes                       (inheritance)
                                                                           2. ⟨ℱ, ℛ⟩ |≈𝜅 ∃𝑦.hc(𝑥, 𝑦) | 𝑆(𝑥) ∧ 𝐸(𝑥) ?
                                                                                                                         )︀
                  𝑟2 : (∃𝑦.[hb(𝑥, 𝑦) ∧ 𝐸(𝑦)] | 𝐸(𝑥)),
                                                                              ˓→ should be(︀yes                    )︀ (inheritance)
                  𝑟3 : (∃𝑦.hc(𝑥, 𝑦) | 𝑆(𝑥)),
                                                                           3. ⟨ℱ, ℛ⟩ |≈𝜅 ¬𝐹 (𝑥) | 𝑆(𝑥) ∧ 𝐸(𝑥) ?
                  𝑟4 : (𝑌 (𝑥) | 𝑆(𝑥)),                                        ˓→ should be(︀yes                    )︀ (inheritance)
                  𝑟5 : (¬𝐹 (𝑥) | 𝑆(𝑥)),                                    4. ⟨ℱ, ℛ⟩ |≈𝜅 𝑌 (𝑥) | 𝑆(𝑥) ∧ 𝐸(𝑥) ?
                  𝑟6 : (∃𝑦.[hf (𝑥, 𝑦) ∧ 𝑀 (𝑦)] | 𝑃 (𝑥)),                      ˓→ should be(︀no                               (conflict)
                                                                           5. ⟨ℱ, ℛ⟩ |≈𝜅 ¬𝑌 (𝑥) | 𝑆(𝑥) ∧ 𝐸(𝑥) ?
                                                                                                                   )︀
                  𝑟7 : (𝐵(𝑥) | 𝑃 (𝑥)) }
                                                                              ˓→ should be(︀no                          )︀ (conflict)
   We demonstrate below how the inequality for the acceptance              6. ⟨ℱ, ℛ⟩ |≈𝜅 ¬𝑌 (𝑥) | 𝑆(𝑥) ∧ Italian(𝑥) ?
of 𝑟6 by a c-representation 𝜅 can be computed. In order to                    ˓→ should be(︀no                        (irrelevance)
                                                                           7. ⟨ℱ, ℛ⟩ |≈𝜅 ¬𝐹 (𝑥) | 𝑃 (𝑥) ?
                                                                                                           )︀
keep formulas compact and readable, we indicate by a dot
                     ̇ (𝑎)) that the literal may be either positive
over a literal (e.g. 𝐴                                                        ˓→ should be no                               (override)
or negative (𝐴(𝑎) or ¬𝐴(𝑎)) and an underscore serves as a
                                                                      When the OCF 𝜅 used for answering the queries above is a
wildcard that may be filled by all suitable constants 𝑐 ∈ 𝑈Σ .
                                                                      (minimal) c-representation, all of the queries are answered
For roles, i.e., binary predicates, the constants 𝑏𝑎𝑖 are the ones
                                                                      correctly. As an example, we will compute the answer for
used together with a constant 𝑎 in order to form a candidate
                                                                      query 7 below. The conditional mentioned in the query is
for a strong representative for the rule 𝑟𝑖 .
                                                                      accepted by 𝜅 iff 𝜅(𝐹 (𝑥)𝑃 (𝑥)) < 𝜅(𝐹 (𝑥)𝑃 (𝑥). Hence, we
            (4)                                                       need to compute these two ranks.
𝜅 |= 𝑟6    ⇔      𝜅(𝑃 (𝑥)hf (𝑥, 𝑦)𝑀 (𝑦))
                   < 𝜅(𝑃 (𝑥) ∧ ∀𝑦.hf (𝑥, 𝑦)𝑀 (𝑦))                     𝜅(𝑃 (𝑥)𝐹 (𝑥))      =     min 𝜅(𝑃 (𝑎)𝐹 (𝑎))
                                                                                              𝑎∈𝑈Σ
           ⇔       min 𝜅(𝑃 (𝑎)hf (𝑎, 𝑏𝑎6 )𝑀 (𝑏𝑎6 ))                                                 (︀
                  𝑎∈𝑈Σ                                                                   =     min 𝜅 𝑃 (𝑎)𝐹 (𝑎)𝑆(𝑎)hf (𝑎, _)
                                                                                              𝑎∈𝑈Σ
                                      ⋀︁
                   < min 𝜅(𝑃 (𝑎)            hf (𝑎, 𝑏)𝑀 (𝑏))                                        𝐸(𝑎))hḃ (𝑎, _)hc(𝑎, 𝑏𝑎3 )𝑌 (𝑎)
                      𝑎∈𝑈Σ
                                     𝑏∈𝑈Σ
                                                                                                        ̇ (𝑎)𝑀̇ (𝑎)
                                                                                                                    )︀
                        (︁                                                                         𝐵(𝑎)𝐴
           ⇔       min 𝜅 𝑃 (𝑎)hf (𝑎, 𝑏𝑎6 )𝑀 (𝑏𝑎6 )                                       =    𝜂6
                  𝑎∈𝑈Σ

                         𝐹̇ (𝑎)𝑆(𝑎)𝐸(𝑎)hc(𝑎, 𝑏𝑎3 )𝑌 (𝑎)               𝜅(𝑃 (𝑥)𝐹 (𝑥))      =     min 𝜅(𝑃 (𝑎)𝐹 (𝑎))
                                                                                              𝑎∈𝑈Σ
                                                   )︁
                          ̇ (𝑎)𝐵(𝑎)𝑀      ̇ (𝑎, _)
                                    ̇ (𝑎)hb                                                    min 𝜅 𝑃 (𝑎)𝐹 (𝑎)𝑆(𝑎)hf (𝑎, 𝑏𝑎6 )
                                                                                                    (︀
                         𝐴                                                               =
                                                                                              𝑎∈𝑈Σ
                          (︁      ⋀︁ (︀
                                                                                                   𝑀 (𝑏𝑎6 )𝐸(𝑎))hḃ (𝑎, _)hc(𝑎, 𝑏𝑎3 )
                                                         )︀
                   < min 𝜅 𝑃 (𝑎)       hf (𝑎, 𝑏) ∨ 𝑀 (𝑏)
                      𝑎∈𝑈Σ
                                                                                                              ̇ (𝑎)𝑀 ̇ (𝑎)
                                     𝑏∈𝑈Σ
                                                                                                                          )︀
                                                                                                   𝑌 (𝑎)𝐵(𝑎)𝐴
                               𝐹̇ (𝑎)𝑆(𝑎)𝐸(𝑎)hc(𝑎, 𝑏𝑎3 )𝑌 (𝑎)                            =    𝜂5
                                                         )︁
                               𝐴̇ (𝑎)𝐵(𝑎)𝑀      ̇ (𝑎, _)
                                          ̇ (𝑎)hb
                                                                      The computation above shows that the conditional
           ⇔      𝜂5 < 𝜂6                                             (𝐹 (𝑥)|𝑃 (𝑥)) is accepted by 𝜅 if 𝜂6 < 𝜂5 . However, we know
                                                                      from Example 3 that 𝜂5 < 𝜂6 . Hence, the answer to the query
The other inequalities can be computed in a similar way. The
                                                                      is no.
resulting system of inequalities can be solved, i.e. the knowl-
edge base ⟨ℱ, ℛ⟩ is consistent for the OCF-based semantics               The example above shows that the OCF-based semantics,
as well.                                                              just like cwm -semantics, allows subclasses to appropriately
                                                                      inherit and override information specified for their respec-
5.3. Inheritance of Properties and                                    tive superclass.
                                                                         Observe another interesting feature of c-representations
     Conflicting Information
                                                                      that comes to light in the example above: We did not have
One particular advantage of the cwm -semantics is that it sup-        to compute the ranks for all possible worlds or even just the
ports sub-concepts to both inherit and override properties            values of the 𝜂𝑖 , but could answer the query based on the
defined for their parent-concepts, depending on whether               underlying conditional structures of the knowledge base,
there is a conflict of information. This is not the case for e.g.     captured by the inequalities between the 𝜂𝑖 .
Rational Closure [4], which suffers from the well-known                  There are some key differences between our approach and
drowning problem.                                                     the semantics proposed for defeasible description logics in
the literature. While most of these approaches are based on         More advanced properties like syntax splitting [7] could
an ordering over individuals and uses some notion of typical      be considered for the first-order case as well. Our results
individuals for specific concepts, our approach uses an order-    concerning the property (IRR) dealing with splitting of vari-
ing over possible worlds and makes use of representatives         ables can be considered as first steps in this direction.
for conditionals. Even if canonical models look very similar
to orderings over possible worlds, only considering typi-
cality between domain elements (even on a concept-level)          References
when constructing the global ordering < seems to be less
                                                                   [1] W. Spohn, Ordinal conditional functions: A dynamic
restrictive than considering representatives for conditionals,
                                                                       theory of epistemic states, in: W. L. Harper, B. Skyrms
as it allows knowledge bases to have models that would be
                                                                       (Eds.), Causation in Decision, Belief Change, and
considered inconsistent under our semantics. However, this
                                                                       Statistics, Springer Netherlands, 1988, pp. 105–134.
breaks (DI), as the following example shows.
                                                                       doi:10.1007/978-94-009-2865-7_6.
Example 5. Consider the following knowledge base.                  [2] W. Spohn, The Laws of Belief - Ranking Theory and Its
                                                                       Philosophical Applications, Oxford University Press,
                   𝒯 = {𝐴 ⊑ 𝐵}                                         Oxford, 2014.
                 𝒟𝐴 = {𝐴 ⊏
                         ∼ ¬𝐶}                                     [3] G. Casini, T. Meyer, K. Moodley, R. Nortjé, Relevant
                 𝒟𝐵 = {𝐵 ⊏      ⊏
                         ∼ 𝐴, 𝐵 ∼ 𝐶}
                                                                       closure: A new form of defeasible reasoning for de-
                                                                       scription logics, in: E. Fermé, J. Leite (Eds.), Log-
A canonical and T-compliant model ℳ for this knowledge                 ics in Artificial Intelligence - 14th European Confer-
base is given by the orderings below. The individuals are              ence, JELIA 2014, Funchal, Madeira, Portugal, Septem-
named after the concepts that they are interpreted in, with            ber 24-26, 2014. Proceedings, volume 8761 of Lecture
overlines indicating negation, i.e. for the individual 𝑎𝑏𝑐 we          Notes in Computer Science, Springer, 2014, pp. 92–106.
have 𝑎𝑏𝑐 ∈ (¬𝐴 ⊓ ¬𝐵 ⊓ 𝐶)ℐ .                                            doi:10.1007/978-3-319-11558-0_7.
              𝑎𝑏𝑐, 𝑎𝑏𝑐, 𝑎𝑏𝑐, 𝑎𝑏𝑐, 𝑎𝑏𝑐 <𝐴 𝑎𝑏𝑐                       [4] K. Britz, G. Casini, T. Meyer, K. Moodley, U. Sattler,
                                                                       I. Varzinczak, Principles of KLM-style defeasible de-
            𝑎𝑏𝑐, 𝑎𝑏𝑐, 𝑎𝑏𝑐 <𝐵 𝑎𝑏𝑐, 𝑎𝑏𝑐 <𝐵 𝑎𝑏𝑐                           scription logics, ACM Transactions on Computational
         𝑎𝑏𝑐, 𝑎𝑏𝑐 <ℳ 𝑎𝑏𝑐, 𝑎𝑏𝑐 <ℳ 𝑎𝑏𝑐 <ℳ 𝑎𝑏𝑐                            Logic 22 (2020) 1–46. doi:10.1145/3420258.
                                                                   [5] L. Giordano, D. Theseider Dupré, An ASP ap-
We have min< (𝐵 ℐ ) = {𝑎𝑏𝑐, 𝑎𝑏𝑐}, i.e. min< (𝐵 ℐ ) ⊈ 𝐴ℐ                proach for reasoning in a concept-aware multipref-
and min< (𝐵 ℐ ) ⊈ 𝐶 ℐ . Therefore, 𝒦ℬ ̸|≈cwm 𝐵 ⊏
                                               ∼ 𝐴 and                 erential lightweight DL, Theory and Practice of
𝒦ℬ ̸|≈cwm 𝐵 ⊏
            ∼ 𝐶.                                                       Logic Programming 20 (2020) 751–766. doi:10.1017/
                                                                       s1471068420000381.
                                                                   [6] S. Kraus, D. Lehmann, M. Magidor, Nonmonotonic
6. Conclusions                                                         reasoning, preferential models and cumulative logics,
                                                                       Artificial Intelligence 44 (1990) 167–207. doi:10.1016/
In this paper, we have presented an approach for first-order
                                                                       0004-3702(90)90101-5.
conditional logic from [10], and added a definition of in-
                                                                   [7] G. Kern-Isberner, C. Beierle, G. Brewka, Syntax split-
ductive inference operators for first-order knowledge bases.
                                                                       ting = relevance + independence: New postulates for
Moreover, we have shown that an inductive inference op-
                                                                       nonmonotonic reasoning from conditional belief bases,
erator based on strategic c-representations fulfills the DL-
                                                                       in: Proceedings of the Seventeenth International Con-
version of the KLM postulates defined in [4], as well as
                                                                       ference on Principles of Knowledge Representation
additional postulates from [8]. Additionally, we have shown
                                                                       and Reasoning, KR-2020, International Joint Confer-
how to apply our approach to defeasible DL knowledge
                                                                       ences on Artificial Intelligence Organization, 2020.
bases, while pointing out some commonalities and differ-
                                                                       doi:10.24963/kr.2020/56.
ences with cwm -semantics for defeasible description logics
                                                                   [8] G. Casini, T. Meyer, G. Paterson-Jones, I. Varzinczak,
[5].
                                                                       KLM-style defeasibility for restricted first-order logic,
   The work done in this paper lays the foundation for future
                                                                       in: Lecture Notes in Computer Science, Springer In-
research on the capabilities of OCF-based semantics for
                                                                       ternational Publishing, 2022, pp. 81–94. doi:10.1007/
first-order conditional knowledge bases and, in particular,
                                                                       978-3-031-21541-4_6.
for more in-depth comparisons between c-representation-
                                                                   [9] F. Baader, I. Horrocks, C. Lutz, U. Sattler, An Introduc-
based inductive inference operators and different entailment
                                                                       tion to Description Logic, Cambridge University Press,
relations proposed for defeasible DL knowledge bases like
                                                                       2017. doi:10.1017/9781139025355.
rational entailment [17, 4], relevant entailment [3], and cwm -
                                                                  [10] G. Kern-Isberner, M. Thimm, A ranking semantics for
entailment [5]. There is some recent work on connections
                                                                       first-order conditionals, in: L. De Raedt, C. Bessiere,
between defeasible DL semantics and OCF-based semantics
                                                                       D. Dubois, P. Doherty, P. Frasconi, F. Heintz, P. J. F.
[18], albeit only using propositional conditional logic.
                                                                       Lucas (Eds.), ECAI 2012 - 20th European Conference
   Additionally, most work done so far on first-order condi-
                                                                       on Artificial Intelligence. Including Prestigious Appli-
tional knowledge bases and defeasible DL knowledge bases
                                                                       cations of Artificial Intelligence (PAIS-2012) System
focus on the general case where no facts or no ABox, respec-
                                                                       Demonstrations Track, Montpellier, France, August
tively, are present. Our approach is basically also capable
                                                                       27-31 , 2012, volume 242 of Frontiers in Artificial Intel-
of dealing with information from an ABox, but for this, we
                                                                       ligence and Applications, IOS Press, 2012, pp. 456–461.
must also include option (B) from Definition 6 into our con-
                                                                       doi:10.3233/978-1-61499-098-7-456.
siderations. We will work this out in future work. Moreover,
                                                                  [11] G. Kern-Isberner, Conditionals in Nonmonotonic Rea-
also more postulates describing how different approaches
                                                                       soning and Belief Revision, volume 2087 of Lecture
deal specifically with facts are needed.
     Notes in Computer Science, Springer, 2001. doi:10.
     1007/3-540-44600-1.
[12] G. Kern-Isberner, A thorough axiomatization of a
     principle of conditional preservation in belief revision,
     Annals of Mathematics and Artificial Intelligence 40
     (2004) 127–164. doi:10.1023/a:1026110129951.
[13] C. Beierle, T. Falke, S. Kutsch, G. Kern-Isberner,
     Minimal tolerance pairs for system Z-like rank-
     ing functions for first-order conditional knowledge
     bases, in: Z. Markov, I. Russell (Eds.), Proceedings
     of the Twenty-Ninth International Florida Artificial
     Intelligence Research Society Conference, FLAIRS
     2016, AAAI Press, Menlo Park, CA, 2016, pp. 626–
     631. URL: http://www.aaai.org/ocs/index.php/FLAIRS/
     FLAIRS16/paper/view/12868.
[14] C. Beierle, T. Falke, S. Kutsch, G. Kern-Isberner, System
     ZFO: Default reasoning with system Z-like ranking
     functions for unary first-order conditional knowledge
     bases, International Journal of Approximate Reason-
     ing 90 (2017) 120–143.
[15] J. Pearl, System Z: A natural ordering of defaults with
     tractable applications to nonmonotonic reasoning, in:
     Proc. of the 3rd conf. on Theor. asp. of reasoning about
     knowledge, TARK ’90, Morgan Kaufmann Publishers
     Inc., San Francisco, CA, USA, 1990, pp. 121–135.
[16] D. Lehmann, M. Magidor, What does a conditional
     knowledge base entail?, Artificial Intelligence 55 (1992)
     1–60. doi:10.1016/0004-3702(92)90041-u.
[17] L. Giordano, V. Gliozzi, N. Olivetti, G. L. Pozzato,
     Semantic characterization of rational closure: From
     propositional logic to description logics, Artificial In-
     telligence 226 (2015) 1–33. doi:10.1016/J.ARTINT.
     2015.05.001.
[18] J. Haldimann, C. Beierle, Characterizing multipref-
     erence closure with system W, in: F. D. de Saint-
     Cyr, M. Öztürk-Escoffier, N. Potyka (Eds.), Scalable
     Uncertainty Management - 15th International Confer-
     ence, SUM 2022, Paris, France, October 17-19, 2022,
     Proceedings, volume 13562 of Lecture Notes in Com-
     puter Science, Springer, 2022, pp. 79–91. doi:10.1007/
     978-3-031-18843-5\_6.