=Paper=
{{Paper
|id=Vol-2708/foust6
|storemode=property
|title=Free Description Logic for Ontologists
|pdfUrl=https://ceur-ws.org/Vol-2708/foust6.pdf
|volume=Vol-2708
|authors=Fabian Neuhaus,Oliver Kutz,Guendalina Righetti
|dblpUrl=https://dblp.org/rec/conf/jowo/NeuhausKR20
}}
==Free Description Logic for Ontologists==
https://ceur-ws.org/Vol-2708/foust6.pdf
Free Description Logic for Ontologists
Fabian NEUHAUS a,b,1 and Oliver KUTZ a and Guendalina RIGHETTI a
a Conceptual and Cognitive Modelling Research Group (CORE),
KRDB Research Centre for Knowledge and Data, Faculty of Computer Science, Free
University of Bozen-Bolzano, Italy
b Otto-von-Guericke University of Magdeburg, Magdeburg, Germany
Abstract. Free Logic deviates from classical first-order logic in that singular terms
need not to refer, or not refer to existing entities, i.e. those in the scope of the usual
first-order quantifiers. We thus face the basic questions whether sentences referring
to such non-existing objects can be true, or rather denied a truth value. Moreover,
the ontologist needs to decide whether she wants to endorse the existence of non-
existing, or ‘fictional’, objects in the meta-theory, or rather deny reference. We here
explore the various possible answers to these questions in the paradigm of dual-
domain semantics and analyse the choices in the context of basic Description Logic
languages. We finally sketch a treatment of definite descriptions under the different
choices.
Keywords. Free Logic, Existential Commitment, Description Logic, Fictional
Objects, Definite Descriptions
1. Introduction
According to conceptualists, ontologies are about concepts, including possibly fictional
concepts. Rather obviously, fictions often use names that fail to refer to anything exist-
ing: this is the case, for instance, of the name ‘The Hulk’ within a statement such as
‘The Hulk is green’. From a modelling point of view, this may not cause problems, or
inconsistencies, if we assume the point of view of the story. But the situation may be-
come more difficult if we need to distinguish between what is true in the story and what
is true in the real world, e.g. compare ‘The Hulk is a person and is green’ with ‘The
Hulk is stronger than Muhammad Ali’. If we take conceptualism seriously, ontologies
and knowledge bases may then include singular terms that refer to something that ex-
ists (e.g., ‘Pope Francis’) singular terms that do not refer to anything in the domain of
quantification (e.g., ‘The Hulk’ or ‘Santa Claus’) and singular terms of which we are
uncertain whether they refer to something (e.g., ‘Homer’).
The representation of scientific theories and empirical knowledge also requires
the consideration of non-existing or possibly non-existing entities. As Dumontier and
Hoehndorf pointed out in [1], this is particularly evident within scientific domains, where
hypothesis and theories often involve entities that are unknown to exist, or not observ-
able, or even idealisations that hardly exist, but that are still fundamental in the develop-
1 Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0).
ment and success of the theory. As argued by the authors, one example is the Higgs bo-
son, which was predicted by the Standard Model of particle physics, even if, up to 2012,
no instances of the predicted entity had been found. A representation of the Standard
Model in 2011 would have been incomplete without the Higgs boson, in spite of the fact
that its existence was hypothetical at this time. In contrast, the non-existence of the ideal
gas is not open for debate, since the ideal gas is a gas that consists of particles that lack
spatial extension (point particles) and that do not attract or repel each other. While the
ideal gas does not exist, it is subject to the ideal gas law P V “ nRT , which describes the
relationship between pressure, volume, and temperature of a given amount of an ideal
gas. For many purposes real gases (like hydrogen and oxygen) may be treated as the
ideal gas, and, thus, the ideal gas law is frequently used for calculations in Chemistry.
For this reason, a representation of chemical knowledge would be incomplete without
the representation of the ideal gas.
Fictions, scientific hypotheses, and idealisations are not the only origin of non-
referring terms. A more mundane source are errors during the modelling process. For
example, if one adds an IRI as name for an individual to an OWL ontology, and later re-
alises that this was a mistake, there is no elegant way to deal with the situation. If one just
deletes the IRI from the ontology, then references to the ontology that use the IRI may
break. However, if one leaves the IRI in the ontology (possibly annotated as “obsolete”),
then from a logical point of view a corresponding entity does exists in the universe of
discourse and may play a role for automatic reasoning (see e.g. [2]).
These three examples illustrate that there is need for allowing non-referring singular
terms in a knowledge representation language. Classical first-order logic presupposes
that all individual constants refer to something in the universe of discourse. Description
Logics (DLs) are one of the most important formalism for knowledge representation
and provide the logical foundation of the most widely used ontology language, OWL.
Most modern DLs are designed as fragments of classical first-order logic. From this
FOL heritage DLs inherit the presupposition that names always refer to something in the
universe of discourse, and that this universe is always non-empty.
In this paper, we develop the first steps towards a general framework to handle non-
referring singular terms in the area of Description Logic. Free Logic is the branch of logic
that studies logical systems free of existential presuppositions, in particular the presup-
position that singular terms denote something in the domain of quantification. Although
an important subject in its own right [3, 4], it is of particular importance for instance
in the context of counterpart theory or more generally the combination of modality and
quantification [5, 6] where existence of an object in a possible world is a central concept.
For instance, objects may go in and out of existence along the flow of time.2
In particular, we discuss here three different philosophical intuitions about the truth-
values of sentences that include non-referring singular terms. Positive free logics allow
atomic sentences with non-referring singular terms to be true. This is a natural position
if one intends to represent information about fictional entities or idealisations; e.g., in
atomic sentence like Santa has a beard or The ideal gas is a gas. In negative free logics,
on the other hand, the truth of an atomic sentence requires that all arguments refer to
something in the domain of quantification; hence, atomic sentences about Santa and the
ideal gas are always false. A negative free logic is the natural choice if one considers non-
2 Specifically, [5] used a free quantificational logic in order to provide a generic completeness proof for
different theories of cross-world identity.
referring singular terms as errors. Finally, according to gapped free logics which we also
call Fregean (and which sometimes are called ‘neutral’ [4]), a semantic precondition for
the evaluation of the truth-value of a sentence is that all singular terms refer to something.
Thus, sentences that fail that condition lack a truth-value; there is a truth-value gap. As
noted in [7, 8], this option is closest in spirit to the classical Fregean position, according
to which a sentence only has a truth-value if all singular terms that occur in the sentence
have a referent [9].
We present three different semantics for ALCO without existential presuppositions,
which reflect the three different philosophical choices (positive, negative, Fregean) and
discuss their implications. The different choices are implemented based on a dual domain
semantics, where we distinguish between an inner domain, which represents the standard
domain of discourse, and an outer domain, where individual names are interpreted that
do not denote anything that exists. The dual domain semantics may, arguably, be seen as
a less natural fit for negative or Fregean free logic than some of its alternatives (see Sec-
tion 2). However, it enables us to compare the implications of the different philosophical
stances within one framework.
The rest of the paper is organised as follows: In the next section we provide an
introduction to Free Logic as extensions of classical FOL. In Section 3 we introduce three
alternative semantics for a free description logic representing the different philosophical
stances on the truth-value of sentences with non-referring terms. Section 4 sketches how
these semantics may be applied to an extension of the language with definite descriptions.
2. Free Logic: The basic landscape
We summarise some of the basic semantic distinctions that can be made. More detail can
be found in [4, 3, 10].
Classical Logic requires that each singular term (i.e., a constant or functional ex-
pression) refers to an object in the domain of interpretation. Consequently, Dxpx “ tq is
a logical truth in classical logic. Following Quine’s famous maxim “To be is to be the
value of a variable” [11], the existential quantifier is usually read as there exists. Hence,
classical FOL seems to entail that t exists, regardless of the choice of “t”, including
“Santa Claus” and “ 10 ”. Thus, the existential presupposition of classical FOL concerning
singular terms leads to unintended ontological commitments. Another existential presup-
position of classical FOL is the assumption that something exists and, thus, the universe
of discourse is not empty.
Free Logic rejects the first and, typically, both of these existential presuppositions.3
In particular, Free Logic allows to handle formulas which contain singular terms which
do not refer to anything in the domain of quantification; in these cases we speak of empty
terms, following [4], or of non-referring terms. In order to do this, existence is treated as
a predicate of objects/individuals: an existence predicate E! is introduced in the language
in order to explicitly declare whether a term (or the object it denotes) exists in the domain
or not. It is usually defined as follows: E!t “def Dxpx “ tq. Quantification in free logic
must then range over all and only those objects that satisfy the E! predicate.
3 Sometimes ‘free logic’ is just associated with the rejection of the existential presupposition of singular
terms, while logical systems that allow an empty domain of quantification are called ‘inclusive’. In this paper
we will consider only free logics that are also inclusive in this sense.
Free Logic is an extension of classical FOL in the sense that it preserves the se-
mantics of classical FOL for sentences that contain no non-referring singular terms. One
way to categorise the different approaches to the semantics of free logic is according
to their philosophical position about the truth-value of sentences with non-referring sin-
gular terms. This leads to the distinction between free logic (NFL), positive free logic
(PFL), and Fregean free logic (FFL).
Let us imagine we want to formalise statements about Pegasus. Since Pegasus does
not exist, there is no Pegasus in the domain of quantification of our ontology. In other
words, the singular term “Pegasus” is empty (i.e., does not refer to anything).
PFL claims that some atomic sentences involving non-referring terms are true. Can-
didates are sentences about fictions like “Pegasus is a horse” and “Pegasus is the off-
spring of Poseidon” or about idealisations “The ideal gas is a gas”. In PFL these sen-
tences may be true without requiring the existence of Pegasus or the ideal gas. The equa-
tion pegasus “ pegasus is typically considered to be logically true in PFL.
In contrast, according to NFL, every atomic formula involving “Pegasus” (or any
other non-referring term) is false, including the equation pegasus “ pegasus. There-
fore, for any atomic sentence P n pt1 , . . . , tn q and any of its arguments ti , the sentence
P n pt1 , . . . , tn q Ñ E!ti is a logical truth in NFL. Since the negation in NFL is classical, it
follows that P n pt1 , . . . , tn q is true, if some ti is empty.
Lastly, FFL endorses the Fregean position that the reference of all singular terms
is a precondition for assigning truth-values to atomic sentences. Hence in FFL atomic
formulas involving non-referring terms are truth-valueless, or, as it is sometime said,
have a truth-value gap. Thus, any atomic formula involving Pegasus lacks a truth-value.
The three philosophical stances regarding truth-values (or lack thereof) of sentences
involving non-referring terms may be realised by different semantic approaches. The
choice of a suitable semantics for a given philosophical position is, arguably, one of the
most interesting and important questions in this context [12].
A dual domain semantics distinguishes between two domains, the inner domain and
the outer domain. The inner domain is the domain of existing entities and, thus, the
domain of quantification. The outer domain is used to interpret empty terms like “Santa
Claus”. In the following, we will distinguish between the reference of a singular term and
its denotation. In a dual domain semantics all singular terms denote some entity, but only
some of them refer to something; namely the entities that denote something in the inner
domain. This distinction is particularly useful for positive free logics, since it enables, for
example, to explain why “Santa Claus is Father Christmas” is true, while “Santa Claus
is the Easter Bunny” is false. While all these terms are without reference, “Santa Claus”
and “Father Christmas” denote the same fictional entity, while “Easter Bunny” denotes a
different one.
Single domain semantics avoid the addition of additional entities. Instead, their in-
terpretation function is partial and, thus, the denotation of non-referring terms is unde-
fined. For this reason, the interpretation function on its own does not determine the truth-
value of all atomic sentences. One possible way to ensure bivalence for all sentences is
to adopt a convention that completes the interpretation function. In NFL the choice is to
assign all atomic sentences with a non-referring term the truth-value False. However, it is
important to note that any function that assigns truth-values to atoms with non-referring
terms serves the same purpose: restoring a total assignment of truth values. An alterna-
tive approach is to relax the principle of bivalence and leave the truth-value of sentences
with non-referring terms undefined. Given this strategy, the partial interpretation function
results in a partial valuation function of sentences, because there are truth-value gaps.
Hence, this choice aligns naturally with FFL.
A lack of truth-value for some atomic formulas yields the question how to han-
dle complex formulae and compositionality. For instance, given two formulas φ and ψ,
where φ is truth-valueless while ψ is true, what should the truth value of φ_ψ be? In [8]
Lehmann proposes an approach where truth-value gaps are ‘infectious’ in the sense that
a complex sentence lacks a truth-value as soon as it contains some non-referring singular
term. A different kind of approach uses supervaluations to ‘close’ some of the truth-value
gaps [13]. For this purpose, one extends a given interpretation I by assigning arbitrary
truth-values to the atomic formulas with non-referring terms. A formula is supertrue (su-
perfalse) in I iff it is true (false) under all such extensions of I. E.g., if t does not exists
in the interpretation I, then P ptq _ P ptq would lack a truth-value in I according to the
Fregean semantics, but it would evaluate to true under the supervaluational semantics.
2.1. Definite descriptions
One of the main applications of Free Logic is the treatment of definite descriptions. Def-
inite descriptions are expressions which consist of a combination of the definite arti-
cle ‘the’ and a noun phrase, such as ‘The main application of Free Logic’ or ‘The man
wearing a blue t-shirt’.
In analytical philosophy, the treatment of definite descriptions is a long-established
subject. Consider the following statement: ‘The present king of France is bald’. Accord-
ing to Russell [14], the logical form of this sentence consists of two assertions: (i) there
is one and only one entity who is a king of France and (ii) this entity is bald. Follow-
ing this intuition, according to Russell definite descriptions should not be represented
in FOL as kind of singular terms, but as part of a formula that represents the whole
sentence: DxpFKingpxq ^ @pyqpFKingpyq Ñ x “ yq ^ Baldpxqq. It follows that in cases
where a definitive description fails to apply to a unique individual (e.g., because there
is no present King of France), the sentence containing that definite description is false.
Note that according to Russell’s analysis, definite descriptions are not treated as singular
terms, but instead they are evaluated contextually as part of a complex first-order for-
mula. Given the goal of representing definite description in DLs, Russell’s approach is
then not feasible, because DLs lack the quantificational apparatus.
In contrast to Russell, Frege treats definite descriptions as a type of singular term.
A sentence which contains a definite description (or, more generally, a singular term)
with no reference (as in the case of ‘The present king of France’) would in turn have no
reference, and would then lack a truth value, being neither true nor false [9].
Since Free Logic studies the logical behaviour of non-referring singular terms, the
study of definite descriptions in the context of free logic suggests itself. In free logic
systems definite descriptions are represented as singular terms. Given some formula φ
containing the free variable x, the singular term xφ represents the definite description
ι
The x such that φ. E.g., our running example would be represented as Baldp x FKingpxqq.
ι
Typically, free logics with definite descriptions obey Lambert’s law [4]:
@ypy “ xφ Ø @xpφ Ø x “ yqq,
ι x free in φ
Depending on their philosophical stance on non-referring singular terms, free logics
differ with respect to their treatment of non-referring definite descriptions. For example,
in NPL atomic formulas involving non-referring definite description are false, leading to
a position similar to the one of Russell.
3. Free Description Logics
In the following we will present free-logical variants of description logic, based seman-
tically on the dual-domain approach.
3.1. The Description Logic ALCO
We use description logics (DLs) as well-known examples of languages for describing
concepts. We briefly introduce the well-known DL ALCO, i.e. ALC enriched with the
nominal construct; for a full introduction to DL, see [15]. The syntax of ALCO concepts
is based on three pairwise disjoint sets, the set of concept names NC , the set of role names
NR , and the set NI for individual names. The set of ALCO singular terms is identical
with NI .4 The set of ALCO concepts is generated by the grammar
C ::“ J | K | A | C | C [ C | C \ C | @R.C | DR.C | ttu
where A P NC , R P NR , t is a singular term.
A TBox is a finite set of concept inclusions (GCIs) of the form C Ď D where C and
D are concepts. It is used to store terminological knowledge regarding the relationships
between concepts. An ABox is a finite set of assertions, i.e., formulas of the following
form:5
t1 “ t2 t1 ‰ t2 Cpt1 q „Cpt1 q t1 Rt2 „pt1 Rt2 q
where t1 , t2 are ALCO singular terms, R P NR , and C is a ALCO concept. A knowledge
base K is a set of inclusions and assertions. Note that the negated assertions use a differ-
ent negation symbol in order to distinguish the negation of sentences from the negation
of concepts. E.g., the concept A is the negation of the concept A, which may be used in
the positive assertion Apbq (in the sense that we affirm the predication of A) or in the
negative assertion „ Apbq (the statement Apbq is false). In classical ALCO, „ Apbq
is equivalent to Apbq, but as we will see below, this is not the case in free logic. These
two types of negation are well known in philosophical logic [16].
The semantics of ALCO is defined through interpretations I “ p∆I , ¨I q, where ∆I is
a non-empty domain, and ¨I is a function mapping every individual name to an element of
∆I , each concept name to a subset of the domain, and each role name to a binary relation
on the domain. The semantics of complex concepts is as follows: JI “ ∆I , KI “ ∅, and
4 We will extend the grammar of ALCO by adding definite descriptions as singular terms below.
5 Note that these assertions are syntactic sugar in the context of ALCO, but will be important in the free
version of the logic. in the case of ALC on the other hand, these assertions are in fact not expressible within
the language. To see the reduction in ALCO, consider for instance the subsumption tau Ď DR.tbu, which
holds in a model iff paRbq, etc.
ttuI “ tt I u, for any singular term t. the Boolean operations on concepts are given by the
usual set-theoretic intersection (conjunction [), union (disjunction \), and complement
(negation ) of the extension of concepts. The extensions of universal and existential
restrictions are defined as
p@R.CqI “ td P ∆I | for all e P ∆I : pd, eq P RI implies e P C I u, and
pDR.CqI “ td P ∆I | exists e P ∆I : pd, eq P RI and e P C I u
The satisfaction relationship |ù between interpretation and formulas is defined as
follows:
I |ù C Ď D iff C I Ď D I I |ù Cpt1 q iff t1I P C I
I |ù t1 Rt2 iff pt1I , t2I q P RI I |ù t1 “ t2 iff t1I “ t2I
I |ù „φ iff I * φ
The interpretation I is a model of the TBox T (of the knowledge base K) if it satisfies
all the elements of T (K, respectively). Given two concepts C and D, we say that C is
subsumed by D w.r.t. the TBox T (C ĎT D) if C I Ď D I for every model I of T . We
write C ”T D when C ĎT D and D ĎT C. We say that T (resp. K) is consistent if there
exists an interpretation that is a model of T (resp. K). A concept C is satisfiable wrt. T
(K) if there exists an interpretation I that is a model of T (K) for which C I is not empty.
A knowledge base K entails an inclusion or an assertion φ, K |ù φ, if φ is satisfied in
every model of K.
3.2. Dual Domain Semantics
An advantage of the dual-domain semantics that we will introduce below is that it al-
lows to handle not only the case of non-existent objects having non-trivial properties
(i.e. positive free logic), but equally well the frameworks of negative, inclusive, or neu-
tral, i.e. Fregean semantics. It can therefore serve as a semantic foundation for all these
paradigms. Informally, we distinguish between an inner domain and an outer domain.
The inner domain represents the universe of discourse, i.e., the set of entities that quan-
tifiers range over. The outer domain, extending the inner one, is utilised to analyse the
semantics of terms that do not refer to anything that exists, see Fig. 1.
In the following we define the basic dual-domain semantics for the language of
ALCO f . The basic intuition is that concepts, in general, have extensions within the
global, outer domain of reference, which includes fictional objects. However, explicit
quantification in the language is restricted to the inner domain of existing objects.
The free logics we will consider only disagree in how they evaluate the truth of
statements containing non-referring terms. The notions of dual-domain interpretation and
of concept extensions are independent of this.
Definition 1 (Dual-domain interpretations) A dual-domain interpretation, or DDI, is
a triple I “ p∆I , ΩI , ¨I q, where ∆I Ď ΩI is a (possibly empty) inner domain, ΩI a non-
empty outer domain, and ¨I is a function mapping every individual name a to an element
of ΩI , each concept name to a subset of ΩI , and each role name to a binary relation on
ΩI ˆ ΩI .
We clarify the terminology regarding terms:
Definition 2 (Denotation–Reference-Emptiness) Let I be a DDI and let t be an indi-
vidual name. We call an element d P Ω the denotation of t if Iptq “ d. If d P ∆, we
call the individual name t referring, and otherwise (if d P Ωz∆) non-referring (or also
empty).
The extension of complex concepts is defined independently of the assignment of
truth-values to statements, see Definition 3. This definition is inspired by the standard
dual-domain semantics of free first-order logic: concepts denote subsets of the outer
domain, J plays the role of the existence predicate E!, and the quantifiers are restricted
to quantify over the inner domain. However, the Boolean operations are defined as usual
in DL.
Definition 3 (Extension of concepts) Given a DDI I, concepts A and B, and a name a
the extension of complex concepts is defined inductively as follows:
pKqI “ ∅ pJqI “ ∆I ptauqI “ taI u
p AqI “ ΩzAI pA [ BqI “ AI X BI pA \ BqI “ AI Y BI
p@R.CqI “ td P ΩI | for all e P ∆I : pd, eq P RI implies e P C I u
pDR.CqI “ td P ΩI | exists e P ∆I : pd, eq P RI and e P C I u
We next define three different semantics to interpret ABox and TBox statements
in the language of ALCO. To this end, we define three (partial) valuation functions,
g
VI` , VI´ , VI that assign truth values True or False to ALCO statements, but which are
sometimes undefined. Note that we are therefore still operating within a bivalent se-
mantics which however allows for truth-value gaps. Moreover, we write VI˚ for any of
g
VI` , VI´ , VI , in case a definition is uniform across all three semantics, as in the case of
defining the subsumption.
Definition 4 (TBox Statements / GCI)
VI˚ pA Ď Bq “ True ðñ for all e P ∆I : if e P AI then e P BI ;
otherwise VI˚ pA Ď Bq “ False
Since GCIs represent quantified statements of the form All X are Y , the semantics
of GCIs is defined as quantification over the inner domain, i.e., the existing entities.
Note that VI˚ is a total function on the set of TBox statements. That is, under all three
semantics there are no truth value gaps on TBox axioms.
The positive semantics for ABox assertions is just like the classical one. Moreover, it
should be clear that the semantics collapses to the standard semantics under the assump-
tion p JqI “ ∅ “ KI , which, however, is not expressible with the above definition of
GCI since it only quantifies over the existing objects.
`
Definition 5 (Positive semantics of ALCO f )
VI` pCptqq “ True iff t1I P C I otherwise VI` pCptqq “ False
VI` pRpt1 , t2 qq “ True iff pt1I , t2I q P RI otherwise VI` pRpt1 , t2 qq “ False
VI` pt1 “ t2 q “ True iff t1I “ t2I otherwise VI` pt1 “ t2 q “ False
VI˚ p„φq “ True iff VI˚ pφq “ False VI˚ p„φq “ False iff VI˚ pφq “ True
`
In ALCO f the valuation function is therefore a total and bivalent function to
tTrue, Falseu on the above ABox statements. The semantics for the negative assertions
` ´ g
is classical and is the same for ALCO f , ALCO f , and ALCO f . Next, in the negative
semantics, atomic ABox assertions can only be true of existing things, and are false
otherwise:
´
Definition 6 (Negative Semantics of ALCO f )
VI´ pCptqq “ True iff t I P C I and t I P ∆ otherwise VI´ pCptqq “ False
VI´ pRpt1 , t2 qq “ True iff pt1I , t2I q P RI and t1I , t2I P ∆ otherwise VI´ pRpt1 , t2 qq “ False
VI´ pt1 “ t2 q “ True iff t1I “ t2I and t1I , t2I P ∆ otherwise VI´ pt1 “ t2 q “ False
´
In ALCO f , the valuation function VI´ is also total. The gapped semantics defined next
corresponds to the Fregean semantics discussed earlier and does include truth value gaps,
g
i.e. VI is partial.
g
Definition 7 (Gapped Semantics of ALCO f )
g
VI pCptqq “ True iff t I P C I and t I P ∆
g g
VI pCptqq “ False iff t I R C I and t I P ∆ otherwise VI pCptqq is undefined
g
VI pRpt1 , t2 qq “ True iff pt1I , t2I q P RI and t1I , t2I P ∆
g g
VI pRpt1 , t2 qq “ False iff pt1I , t2I q R RI and t1I , t2I P ∆ otherwise VI pRpt1 , t2 qq is undefined
VIn pt1 “ t2 q “ True iff t1I “ t2I and t1I , t2I P ∆
g g
VI pt1 “ t2 q “ False iff t1I ‰ t2I and t1I , t2I P ∆ otherwise VI pt1 “ t2 q is undefined
For all three semantics the notions of model, satisfiability, entailment etc. are defined
in the standard way (see Section 3.1 on classical ALCO).
3.3. Some basic properties of free ALCO without definite descriptions
In free DL based on dual domains, J plays the role of the existence predicate. Hence,
Jpbq may be read as ‘b exists’. What is a trivial truth in the classical version of ALCO
(Jpbq is always true and its negation always false) turns into a statement that illustrates
Table 1. The behaviour of some basic ALCO assertions under the different semantics. ‘—’ represents that the
situation cannot happen. Ó is used for an undefined truth-value.
` ´ g
Condition on I ALCO ALCO f ALCO f ALCO f
t I P ∆zt I R ∆ I |ù Jptq True \— True \False True \False True \Ó
t I P ∆zt I R ∆ I |ù Jptq False \— False \True False \False False \Ó
t I P ∆zt I R ∆ I |ù „Jptq False \— False \True False z True False z Ó
for all I p JqI “ pKqI yes no no no
|ù J Ď K yes yes yes yes
|ù Dr.C ” @r. C yes yes yes yes
Cpaq, C Ď D |ù Dpaq yes no yes yes
Cpaq |ù Jpaq yes no yes yes
Cpaq |ù Jpaq yes no yes yes
„Cpaq |ù Jpaq yes no no yes
Jpaq |ù DR.tau Ď K yes yes yes yes
Rpa, bq |ù pDR.tbuqpaq yes no yes yes
|ù a “ a yes yes no no
Jpaq |ù a “ a yes yes yes yes
the basic semantic distinctions between positive, negative, and Fregean paradigms, as
shown in Table 1.
Since the existing entities (the inner domain) are a subset of the entities that are
considered in an interpretation (the outer domain), note that if ∆I Ă ΩI (a proper subset),
then KI ‰ p JqI . Indeed, one can consider p JqI as the set of fictitious (= non-existing)
`
entities in I. Thus, in positive free ALCO f we may represent ‘Santa has a Beard and
does not exists’ as J [ Beardpsantaq. In spite of the fact that KI ‰ p JqI (for some
`
interpretations I), J Ď K is still logically true in ALCO f , because the semantics of
Ď quantifies only over existing entities; i.e., the entities in the inner domain ∆. (We will
discuss this point in more detail below.)
Since the Boolean operators , [, \ are all defined with respect to the outer domain
ΩI , they behave classically (e.g., p CqI “ C I , p pC [ DqqI “ p C \ DqI , pC \
I
Cq “ Ω ). I
A characteristic feature of classical logic as well as the classical version of descrip-
tion logics such as ALCO is the duality and interdefinability of the existential and uni-
versal quantifiers. It is therefore instructive to notice that the dualities remain valid also
in our four basic scenarios. In order to better understand the behaviour of a dual domain
semantics, we will discuss below a few examples and use them to illustrate the different
semantics.
`
tApaq, Bpaq, A Ď Bu is satisfiable in ALCO f . Because subsumption is defined
only over the elements of the inner domain, but it specifies nothing about the element of
ΩI z∆I . Conversely, the negation is defined over the outer domain, and behaves classi-
cally. So, for instance, if all the elements of the set p BqI lie in ∆, and the elements of
the set pAqI lie both in ∆I and in ΩI z∆I , A Ď B could be true while A shares some
elements with B in ΩI z∆I . To better visualise the situations, see the Fig. 1.
`
Analogously, we have that Cpaq, C Ď D * Dpaq in ALCO f . The subsumption
requires that the element of ∆ that are in pCq must be elements of the set pDqI . But it
I I
`
could be the case that aI P ΩI z∆I . In this case Dpaq would be true in ALCO f . On the
`
Figure 1. tApaq, Bpaq, A Ď Bu is satisfiable in ALCO f .
other hand, if we assume the existence of a we have a different result: Cpaq, Jpaq, C Ď
`
D |ù Dpaq in ALCO f . Here, since we assume Jpaq, aI must belong to ∆, and, according
to the definition of the subsumption, we get the result Dpaq. Please notice that here we
are using the J exactly as the predicate E! is normally used in Free Logic.
` ´ g
In contrast to ALCO f , in ALCO f and ALCO f the truth of Cpaq requires a to
be in ∆. Thus in these logics, Cpaq |ù Jpaq. Consequently, Cpaq, C Ď D |ù Dpaq in
´ g
ALCO f and ALCO f .
f´
In ALCO the two negations differ in their behaviour. In Cpaq the complex con-
´
cept C is asserted of the individual a. Hence, in ALCO f Cpaq may only be true,
if a exists. In contrast, „Cpaq is the negation of Cpaq. Thus, if Cpaq is false (possibly
´
because a does not exist), „Cpaq is true.6 Hence, in ALCO f Cpaq entails Jpaq, but
„Cpaq does not. Note that Cpaq |ù „Cpaq. E.g., Alan is unhappy entails It is not the
case that Alan is happy.
Since the quantifiers are quantifying over the inner domain ∆, it follows that
pDR.tauqI “ ∅, if aI R ∆. Thus, Jpaq |ù DR.tau ” K. For the same reason Rpa, bq *
`
pDR.tbuqpaq in ALCO f .
´ g
Since a may not exist, a “ a is not a logical truth in ALCO f and ALCO f , unless
Jpaq is assumed.
4. Notes on Definite Descriptions in DLs
This section sketches some of the difficulties encountered when adding the idea of defi-
ι
nite description to description logics. ALCO extends ALCO syntactically by adding the
basic machinery for definite descriptions as follows. For any given (complex) concept C
ι
of ALCO, we introduce C as a ALCO definite description. Intuitively, we want C to
ι ι
pick out, in accordance with Lambert’s Law, the unique object that is a C, if it exists.
ι ι
The set of ALCO singular terms is then the union of NI with the set of ALCO
definite descriptions.
6 A similar observation was made by Lambert in [17].
Table 2. Some examples for definite descriptions in free ALCO. Here, C, D are (possibly distinct) arbitrary
concepts, and the ddp is assumed for all interpretations.
` ´ g
ALCO ,f ALCO ,f ALCO ,f
ι ι ι
Dp Cq, Cpaq |ù a “ C
ι ι yes yes yes
Dp Cq |ù Jp Cq
ι ι no yes yes
„Dp Cq |ù Jp Cq
ι ι no no yes
Jp Cq |ù Cp Cq
ι ι yes yes yes
Kp Cq |ù Cp Cq
ι ι yes no no
ι
If φ is a ALCO statement, let Dpφq be the set of definite descriptions C that ι
occur in φ. An interpretation I meets the definite descriptions presupposition (ddp) of a
statement φ iff for all C P Dpφq there exists some e P Ω such that C I “ teu. If I meets
ι
the definite descriptions presupposition of φ, then we also say I is suitable for φ.
ι
A free semantics for ALCO should respect the basic law regulating definite descrip-
tions, i.e. Lambert’s Law discussed above. This leads to the following twofold extension
of the various semantics for free ALCO:
• VI˚ pφq is only defined if I is suitable for φ;
• p CqI “ e iff C I “ teu, if I is suitable for φ
ι
g
Because of the suitability requirements, the valuation functions VI` , VI´ , VI on ALCO
ι
are partial functions. Further, because of the uniqueness of C, tC Ď K, Cp Cqu is un- ι ι
satisfiable and Cp Cq, Cpaq |ù a “ C. Since definite descriptions are singular terms the
ι ι
` ´ ´
differences between ALCO f , ALCO f and ALCO f with respect to existential presup-
positions apply to them (see lines 2 and 3 of Table 2). Jp Cq |ù Cp Cq is a logical truthι ι
`
in all systems and Kp Cq |ù Cp Cq in ALCO f .
ι ι
Artale et al. extend ALCO with definite descriptions in a similar way, but use the
approach of partial interpretations on names [18]. The system is called ALCO ι . In this
scenario, a nominal ttu is stipulated to denote the empty concept ∅ whenever the term
t does not denote (is not defined), both if t is a name and a definite description. This
follows the negative free logic paradigm and has the advantage to avoid truth-value gaps.
However, the approach has the consequence that ttu Ď C and ttu Ď C are true, for any
C and any empty term t. ALCO ι is reduced to ALCO U , i.e. standard ALCO over total
interpretations with the universal modality, and therefore has an EXPTIME-complete
satisfiability problem [19].
5. Discussion and Outlook
We could here only sketch the basic semantical distinctions that arise in the context of ap-
plying free logic paradigms and definite descriptions to description logics. An intriguing
landscape emerges when analysing the interactions between domain of quantification,
negation, and definite descriptions and nominals.
Of course, standard free-logical semantics could be simply applied to the standard
translation of DL into FOL, obtaining some free-logical DL. However, the more interest-
ing free-logical phenomena emerge when taking the status of DL as a concept language
seriously. This becomes perhaps most obvious when noting the non-equivalence in neg-
ative free logic of sentence negation vs. concept membership negation. Similarly, DLs
require a new approach to definite descriptions since Russell’s approach to unfold them
into more complex FOL expressions is simply not available.
One hidden use case of free logic is the validation of ontologies. Many ontologies
do not contain explicit existential commitments. E.g., they may contain a hierarchy of
classes of animals, but nothing in the ontology requires the existence of these animals.
Thus, an ontology may be consistent in spite of the fact that in all interpretations that
satisfy the ontology only a small minority of concept names is non-empty. However, the
interpretations that are intended by ontology developers satisfy the vast majority (or even
all) concept names of an ontology. In the consistency proof for Dolce we captured this
intention by distinguishing between those concept names that may be empty and those
that have to be non-empty. To use this information we represented Dolce in a multi-sorted
logic. Essentially we introduced a free and inclusive semantics for the ‘possibly empty’
categories whilst applying classical semantics to the remaining ones [20].
Studying the expressive power and reducibility between systems, as also begun by
[18], is important future work. In addition to the sentence and predicate negation as
sketched above, another important language extension concerns the distinction between
general concept inclusions that quantify on the meta-level only over the existing entities,
as presented in this paper, and those that quantify over the entire domain of existing and
non-existing objects. The latter allows inclusions that are valid universally, both for real
and fictional/non-existing entities, preventing at the same time fictional counterexamples
(for instance, all triangles will have three edges both in the inner and in the outer do-
main). In contrast, the former subsumption applies only to the inner domain, allowing
for fictional counterexamples, e.g. Santa Claus is a man, all men are mortal, but Santa
Claus is not mortal.
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