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 nonexisting, 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 dualdomain 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.
According to conceptualists, ontologies are about concepts, including possibly fictional concepts. Rather obviously, fictions often use names that fail to refer to anything existing: 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 become 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 exists (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 [
Fictions, scientific hypotheses, and idealisations are not the only origin of
nonreferring 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
realises 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. [
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
nonreferring 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
presupposition that singular terms denote something in the domain of quantification. Although
an important subject in its own right [
In particular, we discuss here three different philosophical intuitions about the
truthvalues 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
non2Specifically, [
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 Section 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.
We summarise some of the basic semantic distinctions that can be made. More detail can
be found in [
Classical Logic requires that each singular term (i.e., a constant or functional
expression) 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” [
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 [
3Sometimes ‘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 semantics 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 singular 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. Candidates are sentences about fictions like “Pegasus is a horse” and “Pegasus is the offspring of Poseidon” or about idealisations “The ideal gas is a gas”. In PFL these sentences may be true without requiring the existence of Pegasus or the ideal gas. The equation 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. Therefore, for any atomic sentence P npt1; : : : ; tnq and any of its arguments ti , the sentence P npt1; : : : ; tnq Ñ E!ti is a logical truth in NFL. Since the negation in NFL is classical, it follows that P npt1; : : : ; tnq 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 [
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 interpretation function is partial and, thus, the denotation of non-referring terms is undefined. For this reason, the interpretation function on its own does not determine the truthvalue 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 alternative 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
handle complex formulae and compositionality. For instance, given two formulas and ,
where is truth-valueless while is true, what should the truth value of _ be? In [
One of the main applications of Free Logic is the treatment of definite descriptions. Definite descriptions are expressions which consist of a combination of the definite article ‘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’.
According to Russell [
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 [
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
containing the free variable x, the singular term x represents the definite description
The x such that . E.g., our running example would be represented as Baldp x FKingpxqq.
Typically, free logics with definite descriptions obey Lambert’s law [
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.
In the following we will present free-logical variants of description logic, based semantically on the dual-domain approach.
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 [
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 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
Cpt1q
Cpt1q
t1Rt2
pt1Rt2q
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
different 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 [
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 4We will extend the grammar of ALCO by adding definite descriptions as singular terms below. 5Note 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 DR:tbu, which holds in a model iff paRbq, etc. ttuI ttI 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 pDR:CqI td P 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 T D) if CI DI for every model I of T . We write C T D when C T D and D T 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 CI is not empty. A knowledge base K entails an inclusion or an assertion , K |ù , if is satisfied in every model of K.
An advantage of the dual-domain semantics that we will introduce below is that it allows 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 neutral, 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 quantifiers 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 ALCOf. 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.
a triple I p I ; I ; I q, where I I is a (possibly empty) inner domain, I a nonempty outer domain, and I is a function mapping every individual name a to an element of I , each concept name to a subset of I , and each role name to a binary relation on
I I .
We clarify the terminology regarding terms:
vidual name. We call an element d P the denotation of t if I ptq d. If d P , we call the individual name t referring, and otherwise (if d P z ) 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.
the extension of complex concepts is defined inductively as follows: pKq
I p AqI pDR:CqI pJq
I pA [ BqI
I
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, VI ; VI ; VIg 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 semantics which however allows for truth-value gaps. Moreover, we write VI for any of VI ; VI ; VIg , in case a definition is uniform across all three semantics, as in the case of defining the subsumption.
VI pA Bq
I : if e P AI then e P BI ; otherwise VI pA Bq
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 assumption 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 ALCOf )
VI pCptqq VI pRpt1; t2qq
VI pt1 t2q
q VI p f
True iff t1I P CI True iff pt1I; t2Iq P RI True iff t1I
t2I True iff VI p q
otherwise VI pCptqq otherwise VI pRpt1; t2qq otherwise VI pt1
t2q
VI p q
False iff VI p q In ALCO the valuation function is therefore a total and bivalent function to tTrue; Falseu on the above ABox statements. The semantics for the negative assertions is classical and is the same for ALCOf , ALCOf , and ALCOfg . Next, in the negative semantics, atomic ABox assertions can only be true of existing things, and are false otherwise: Definition 6 (Negative Semantics of ALCOf )
VI pCptqq
True iff tI P CI and tI P VI pRpt1; t2qq
VI pt1 t2q
True iff pt1I; t2Iq P RI and t1I; t2I P True iff t1I t2I and t1I; t2I P
otherwise VI pCptqq otherwise VI pRpt1; t2qq otherwise VI pt1 t2q
In ALCOf , 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, i.e. VIg is partial. g Definition 7 (Gapped Semantics of ALCOf )
VIg pCptqq
VIg pCptqq VIg pRpt1; t2qq VIg pRpt1; t2qq
VInpt1
g VI pt1 t2q t2q
True iff tI P CI and tI P False iff tI R CI and tI P True iff pt1I; t2Iq P RI and t1I; t2I P False iff pt1I; t2Iq R RI and t1I; t2I P True iff t1I False iff t1I t2I and t1I; t2I P t2I and t1I; t2I P
otherwise VIg pCptqq is undefined otherwise VIg pRpt1; t2qq is undefined otherwise VIg pt1 t2q 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).
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 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 I (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 ALCOf 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 K is still logically true in ALCOf , because the semantics of 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 I , they behave classically (e.g., p CqI CI ; p pC [ DqqI p C \ DqI , pC \ CqI I ).
A characteristic feature of classical logic as well as the classical version of description logics such as ALCO is the duality and interdefinability of the existential and universal 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 Bu is satisfiable in ALCOf . Because subsumption is defined only over the elements of the inner domain, but it specifies nothing about the element of
I z I . Conversely, the negation is defined over the outer domain, and behaves classically. 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 I z I , A B could be true while A shares some elements with B in I z I . To better visualise the situations, see the Fig. 1.
Analogously, we have that Cpaq; C D * Dpaq in ALCOf . The subsumption requires that the element of I that are in pCqI must be elements of the set pDqI . But it could be the case that aI P I z I . In this case Dpaq would be true in ALCOf . On the other hand, if we assume the existence of a we have a different result: Cpaq; Jpaq; C D |ù Dpaq in ALCOf . 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.
In contrast to ALCOf , in ALCOf and ALCOfg the truth of Cpaq requires a to be in . Thus in these logics, Cpaq |ù Jpaq. Consequently, Cpaq; C D |ù Dpaq in ALCOf and ALCOfg .
In ALCOf the two negations differ in their behaviour. In Cpaq the complex concept C is asserted of the individual a. Hence, in ALCOf 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 ALCOf 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 ALCOf .
Since a may not exist, a a is not a logical truth in ALCOf and ALCOfg , unless Jpaq is assumed.
This section sketches some of the difficulties encountered when adding the idea of definite description to description logics. ALCO extends ALCO syntactically by adding the basic machinery for definite descriptions as follows. For any given (complex) concept C of ALCO, we introduce C as a ALCO definite description. Intuitively, we want C to pick out, in accordance with Lambert’s Law, the unique object that is a C, if it exists.
The set of ALCO singular terms is then the union of NI with the set of ALCO definite descriptions.
6A similar observation was made by Lambert in [
Artale et al. extend ALCO with definite descriptions in a similar way, but use the
approach of partial interpretations on names [
We could here only sketch the basic semantical distinctions that arise in the context of applying 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 interesting 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 negative 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 [
Studying the expressive power and reducibility between systems, as also begun by
[