=Paper=
{{Paper
|id=Vol-2708/foust5
|storemode=property
|title=Refining OntoClean. Identity Criteria and Grounding
|pdfUrl=https://ceur-ws.org/Vol-2708/foust5.pdf
|volume=Vol-2708
|authors=Massimiliano Carrara,Ciro De Florio
|dblpUrl=https://dblp.org/rec/conf/jowo/CarraraF20
}}
==Refining OntoClean. Identity Criteria and Grounding==
https://ceur-ws.org/Vol-2708/foust5.pdf
Refining OntoClean.
Identity Criteria and Grounding1
Massimiliano CARRARA a,2 and Ciro DE FLORIO b
a FISPPA Department, University of Padua, Padua Italy
b Faculty of Economics, Catholic University of Milan, Milan Italy
Abstract. In this paper we introduce some logical and philosophical refinements
to OntoClean, first by developing some formal constraints on identity criteria, sec-
ondly by specifying a kind of identity criteria, two level identity criteria, whose
role is to explain an identity among some entities referring to some other, more ba-
sic, entities. Using such refinement we add a formal constraint to the stock of On-
toClean meta-constraints (OC+). We, then, observe that two level identity criteria
have an intuitive reading in terms of dependence of a kind of entities on some other
entities, possibly specified in terms of a grounding relation. Are identity criteria
grounding principles? In the second part of the paper we discuss this option.
Keywords. OntoClean, identity criteria, constraints on logical adequacy of identity
criteria, two-level identity criteria, grounding
1. Introduction
OntoClean (see [9], [10] and [11]) is surely one of the main important contributions to
a formal foundation for ontologies in knowledge representation. It has been developed
and applied in a variety of papers and researches.3 OntoClean analyzes ontologies us-
ing some formal, domain-independent properties, i.e. metaproperties such as identity,
rigidity, unity and dependence.
Aim of this paper is to introduce some logical and philosophical refinements to
OntoClean, refinements we hope could be useful for new developments and research of
this important tool for ontologies. Specifically, we concentrate here on identity criteria
and on a related topic, grounding.
The paper is organized as follows. First (2) we briefly introduce identity criteria– an
OntoClean formal, domain-independent property of identity. Then (3), we describe some
formal constraints on identity criteria so that one can say what does it mean they are
logically adequate. They are specified on the basis of the logical form of identity criteria
and some properties induced by it. Moreover, we observe (4) that identity criteria have
to play an explanatory role for identity. To this purpose we introduce a specific kind of
identity criteria: two level identity criteria, whose role is just to explain an identity among
1 Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0).
2 Corresponding Author email address: massimiliano.carrara@unipd.it
3 Google scholar reports 2.340 items just searching the word "OntoClean"
some entities referring to some other, more basic, entities. We enrich OntoClean with
a principle for this kind of criteria, (OC+). A first reading of two level identity criteria
is in terms of a relation of dependence of a kind of entities on some other entities. Is it
possible to read such dependence relation in terms of a grounding principle? In (5) we
analyze this option. We give a largely negative answer to the above question.
2. Identity Criteria
The credit for introducing identity criteria is usually attributed to Frege. In his Founda-
tions of Arithmetic, he introduces this idea in a context where he wonders how we can
grasp or formulate the concept of number (see [8], sec. 62). This is the standard Fregean
quotation:
If we are to use the symbol a to signify an object, we must have a criterion for
deciding in all cases whether b is the same as a, even if it is not always in our power
to apply this criterion. (see [8], sec. 62)
Two famous examples of identity criteria provided by Frege are as follows:
• Directions: If a and b are lines, then the direction of line a is identical to the
direction of line b if and only if a is parallel to b.
• Hume’s principle: For any concepts F and G, the number of F-things is equal to
the number of G-things if and only a one-to-one correspondence exists between
F-things and G-things.
In ([9], [10] and [11]) Guarino and Welty suggest that an identity criterion, associ-
ated to a property K, has to answer the following question:
(OQ) If a and b are Ks, what are necessary and sufficient conditions for the object
a to be identical with b?
In Guarino and Welty’s opinion identity criteria are «the criteria we use to answer
questions like, “is that my dog?”» ([9]: 154). That means identity criteria should be
useful to justify identity judgements. A general, formal way to represent identity criteria
(hereafter: IC) is as follows:
∀x∀y((K(x) ∧ K(y)) → (x = y ↔ R(x, y))) (IC)
R represents the identity condition; R holds between a pair of Ks x and y, iff x and y
are identical. Given that x = y is an equivalence relation, the right side of the biconditional
R must be an equivalence relation. Following the Fregean example of directions (IC) is
also formulated in the following way (without a reference to K):
∀x∀y(x0 = y0 ↔ R(x, y)), (IC*)
where “x0 ” and “y0 ” are terms representing entities of the kind K suitably connected
with x and y. Frege’s criterion of identity for directions is an example of (IC*):
∀x∀y((o(x) = o(y) ↔ P(x, y))) (O)
where x and y range over lines, o stands for ‘the direction of’ and P means ‘is parallel
to’. As an example the direction of line a is identical to the direction of line b if and only if
a is parallel to b. In (O), the identity sign is flanked by terms constructed with a functional
letter, and the right-hand side of the biconditional introduces a relation among entities
(lines) different from the entities for which the criterion is formulated (directions).
As we will see in the following, the logical structure of IC raises some interesting
questions about their applicability in ontology. Specifically, we will take into exam the
so-called cases of two-level Identity Criteria like (O), that is, cases in which the identity
of certain entities is explained by the reference to a particular (equivalence) relation be-
tween other, more basic, entities. Before that, in what follows, we consider some formal
constraints on (IC).
3. On some formal constraints for identity criteria
An implicit assumption of OntoClean treatment of identity criteria (see, for example,
[9]) is that logical adequacy is a necessary constraint for them. What does it mean to say
that identity criteria are logically adequate?
First, observe that those formal constraints concern the logical adequacy of the re-
lation R in (IC), which is the identity condition of the criterion. In other words, given
an identity statement a = b, R is a relation that holds between a and b. R is other than
identity and analyzes what it is for the referents of a and b to be identical. How should it
be characterized so to have logically adequacy? In what follows we propose some logical
properties of R.
Non-vacuousness: the identity condition should not be vacuously satisfiable. Con-
sider the following example (see [12], 32-33): let PO be the set of physical objects, S
the set of relevant abstract objects, R(x,y) the identity condition for PO, and R’(x,y) the
identity condition for S. Then let (IC) be defined as:
∀x∀y(x ∈ PO ∨ x ∈ S) ∧ (y ∈ PO ∨ y ∈ S)) → (x = y ↔ (R(x, y) ∨ R0 (x, y)))) (1)
The condition given above for the identity of x and y is not associated with a kind
of entity in a metaphysically interesting sense, since the members of the alleged kind
physical objects or set do not share any essential property.
In a strong metaphysical realist perspective the identity condition R should identify
instances of the same kind of those objects that share all the essential properties associ-
ated with that kind. From such a perspective, the identity condition can be thought of as a
property of properties. Lombard calls such a property determinable since it determines a
class of properties, or determinates, having that property. An example of a determinable
is ‘being a spatio-temporal property’, which can be considered a good candidate for an
identity criterion for objects: if o and o’ are physical objects, then they are identical iff
they are alike with respect to all the properties that are spatio-temporal properties. A cri-
terion of identity for K-objects, to be acceptable, must provide a determinable such that
it makes non-vacuously sense to attribute determinates falling under the determinable to
each K-object.
Informativeness: R should contribute to specifying the nature of the kind K of objects
for which R acts as an identity condition. The identity condition does not completely
characterize the nature of instances of K: to decide identity questions concerning items of
K we need the concept of K, which is not provided by the identity criteria. Nevertheless,
an identity criterion should specify some non-trivial essential properties of objects of
kind K. This means that the form of the relation cannot be tautological, for instance, of
the form:
S(x, y) ∨ ¬S(x, y) (2)
where S(x,y) is an arbitrary binary predicate.
Partial exclusivity: an identity condition for a kind K of objects cannot be so general
that it can be applied to other kinds of objects. The example provided by Lombard is the
following:
If x and y are both non-physical objects, then x and y are identical iff they have the same
individual essence.
Now, properties falling under the wide property ‘having an individual essence’ do
not apply only to non-physical objects, and can be part of the identity conditions for many
kinds of objects. Living beings, for example, instantiate properties usually considered
individual essences, being an individual x generated by gametes y and z for example, but
they are not non-physical objects.
Minimality: the identity condition for K-objects is required to specify the smallest
number of determinables such that the determinates falling under them turn out to be nec-
essary and sufficient to ensure identity between two objects of kind K. The determinables
specified in the identity condition cannot be superfluous. Consider the following example
( [12], 38):
If x and y are both sets, then x and y are identical iff they have the same members and
are liked by the same people.
The above criterion suggests that it is part of the very idea of sets that they are liked
by people. But, clearly, it is not so. In order to rule out such cases the formal requirement
of minimality for identity criteria is introduced.
Non-circularity: the identity condition for K-objects cannot make use of the con-
cept of identity itself; otherwise it is circular. There has been a long debate about the
circularity of IC. Consider the criterion of identity for events proposed by Davidson [3]:
If x and y are events, then x is identical to y iff x and y have the same causes and effects.
One could argue that the above formulation is not formally circular, since the identity
predicate does not occur in the right part of the bi-conditional. However, it has been
observed that whether an event e has the same causes and the same effects as an event
e’ can depend on the solution to an identity question concerning entities of the same
kind. On the right side of the bi-conditional, causes and effects are mentioned; since
those are considered to be events, the identity criterion for events turns out to involve
identity between events. In fact, to determine whether two events are the same, one is
first required to determine the identity of the events taken as their causes or effects. One
can thus conclude that identity is already presupposed. This criticism corresponds to
the denial that it is possible to give an explicative criterion of identity for objects of a
certain kind, such as events. In fact, the formulation of such a criterion would involve
a quantification over all objects for which the criterion is specified, and quantification
presupposes the determinacy of the identity of the objects quantified (see on this [13]).
Non-totality: given at least two objects belonging to some kind K, R cannot be a
property that every two K-objects share. Formally:
(C1) R ⊂ K × K
(C1) says that the relation R is a proper subset of the set K × K: that is, there is some
pair of K-objects such that the objects of the pair are not in the extension of R.
K-Maximality: R must be maximal with respect to K. In other words, R is required
to be the widest dyadic property that makes an identity condition true.
A dyadic property G is wider than a property G’ iff for any x and y, if G’(x,y) then
G(x,y), but not vice versa.
In other words, the ordered pairs of G’ are a subset of the set of ordered pairs of G. In
such a way we always obtain a condition for an ultimate kind or ultimate sortal (concept)
K (here, for the sake of simplicity, we use the term ‘kind’ and ‘sortal’ as synonymous.
The reason for introducing the formal constraint of K-maximality is this. Consider
what Wiggins calls “a structure comprising only sortals” where sortals or kinds stand in
relation to one another and have common members. Take two sortals C(1) and C(2). We
have three cases.
Case 1. Neither C(1) nor C(2) is a restriction of any other sortal and each is an ulti-
mate sortal. If they have common members, then, “because they will cover iden-
tities relating to these common members [...] C(1) must be identical with C(2) or
extensionally equivalent to it” ([17], 67).
Case 2. Either C(1) or C(2) is an ultimate sortal and the other is not. In this case, if there
are common members the non-ultimate one gives a restriction to the other.
Case 3. C(1) and C(2) have common members but no concept subsumes the other. In this
case we have cross-classification, but some ultimate sortal will subsume both C(1)
and C(2). “This picture of things”, Wiggins argues, “founded in the nature of sor-
tal[s] . . . and the absoluteness of identity, concedes everything that deserves to be
conceded to the over-stringent demand that sortal[s] . . . should form a hierarchy”
([17], 67).
Uniqueness: R is unique with respect to K. This means that if there are relations
R1 , R2 , . . . Rn such that (i) each Ri satisfies IC and (ii) each Rk is independent of each R j
– that is, every Rk is neither narrower nor wider than each R j – then at most one among
the relations in R1 , R2 , . . . , Rn provides a correct identity criterion for K-objects.
Equivalence: R must be an equivalence relation. On the left side of the bi-conditional
in IC there is an identity relation that is an equivalence relation; consequently, the right
side of the conditional is supposed to present an equivalence relation as well. R must then
be reflexive, symmetric and transitive.
Congruence: a is the same K as b iff the way in which a is K-related to b via R is
sufficient for whatever is true of a to be true of b and for whatever is true of b to be true
of a.
4. Identity Criteria and Functional Entities
As we have seen before, the use of identity criteria in OntoClean entails a series of logical
requirements which must be satisfied.
It is important to notice, however, that since the introduction of IC an important
IC feature is the explanatory role which the identity condition plays. In other terms, IC
should provide an explanation of why items belonging to K are identical.
Notice that there are at least two general forms of IC. Frege’s preferred example is
the so called Identity Criterion for directions. Roughly, the idea is to give the necessary
and sufficient condition for two directions are the same:
∀x∀y((o(x) = o(y) ↔ P(x, y))) (O)
As it is easy to notice, (O) is an instance of the following general schema:
∀x∀y(x0 = y0 ↔ R(x, y)), (IC*)
where “x0 ” and “y0 ” are terms representing entities of the kind K suitably connected with
x and y.
(IC*) is the logical form of a two-level identity criterion (see [18], 145-146). The
crucial point is that in the case of two-level identity criteria the conditions of identity
concern objects which are not of the same kind of objects for which the identity criterion
is provided. On two-level identity criteria Williamson points out that: «The idea of a
two-level criterion of identity has an obvious advantage. No formula could be more basic
(in any relevant sense) than ‘x = y’, but some might be more basic than ‘ox = oy’, by
removing the symbol ‘o’ and inserting something more basic than it». ([18], 147)
If identity criteria play an explanatory role, Williamson’s remark seems to be plau-
sible: in order to explain an identity we have to refer to some other, more basic, level of
entities.
It is worth to notice that, in this case, (O) has not only a control-function on the
inflation of our ontology (according to Quine’s slogan: No Entity without Identity) but
it is to introduce new entities in the domain. So, two levels identity criteria seem to be
explanatory; however, this entails that entities on which identity is defined are presented
as functions. Let us ask a question similar to Guarino and Welty’s one:
Given that x and y are directions, is x the same direction of y?
The answer is that x and y are the same direction if and only if the lines of which
they are directions are parallel. This confirms Williamson’s remark: the explanation of
the identity is genuine since it exploits more basic entities, i.e., lines. In other terms, it is
because the lines are parallel that the direction of a is identical to the direction of b. But
this means that some kinds of entities are intrinsically functional: they are always items
of something. OntoClean could be, then, enriched by a principle as follows (OC+):
x ∈ K f ↔ ∃y(x = f (y)) (OC+)
that is, x is an entity which belong to a certain functional kind iff there exists an-
other entity of which x is the value of the function f ; x is an entity belonging to kind
directions iff there exists a line which x is the direction of.
This equivalence is an admissibility principle that can be added to OntoClean general
framework: a certain kind (for instance, directions) is admissible in dependence of the
existence of other entities. In order to underline the dependence relation surrounding
identity criteria, let us consider these two scenarios related to the well known identity
criterion for directions:
∀x∀y(L(x) ∧ L(y) → (d(x) = d(y) ↔ P(x, y))) (3)
Within S1 there is no line and therefore it is meaningless to refer to P(a, b), given
that there is no entity which satisfies the general property of being a line. So: no lines,
no directions. Within S2 , there are two items which satisfies the property of being a line
and moreover, it holds that ¬P(a, b). But then, through identity criteria for directions
we have two "new" entities, since d(a) 6= d(b). But how to specify and characterize the
intuitive idea of dependence introduced in our model of two-level identity criterion as
explanatory principles? In the next paragraph we propose a good candidate for this job:
the grounding relation.
5. Grounding and Identity Criteria
Grounding is, for many scholars in metaphysics, the new black. In the last two decades,
there has been a noteworthy amount of works dedicated to grounding (although this
concept is not entirely new, as often happens in philosophy. One can argue that grounding
originates within Aristotle’s notion of αιτια).
Grounding is one of the most discussed notions in contemporary philosophy.
Roughly, grounding is a type of non-causal, primitive relation or operation, such that the
grounded entities are somehow explained, determined or constituted by the grounding
entities.
The “grounding revolution” ([15]: 91) contributed to clarify concepts such as pri-
ority and fundamentality, usually conveyed by locutions as “in virtue of” and “because”
(see [6], [2], and [14] – at the moment the most complete overview of the logic of ground-
ing). As crucial as they are in the philosophical debate, these expressions were rarely
analyzed in a systematic way.
More recently, grounding has been applied, and is expected to be applicable, in a
vast range of philosophical disciplines, from metaphysics to philosophy of mind, of logic
and of language, and metaethics. One can draw several instances of grounding from these
disciplines: for example, the fact that a flower is scarlet is taken to ground the fact that it
is red; mental facts are taken to be grounded in neuro-physiological facts; and the truth
of “snow is white” would be grounded in the fact that snow is white, and not vice versa.
The closeness between the notion of grounding and that of dependence is straight-
forward and, therefore, it is natural, at least prima facie, to consider identity criteria as
grounding principles.4
In general, to say that at a certain time t, x and y are distinct particulars or the same
particular items seems to imply that there is something in virtue of which x and y are
distinct particulars/are the same particular, i.e. a fact that grounds the distinctness of the
two particulars at play.
In our case, the condition of identity explains, grounds, the identity at issue. By
using > to indicate grounding relation, we have that:
∀x∀y(L(x) ∧ L(y) → (P(x, y) > d(x) = d(y))) (4)
Now, we have all the ingredients for the refinement we propose to OntoClean; iden-
tity criteria, construed as grounding methodological principles, are able to account for
kinds the instances of which depend on other entities. For that reason, the identity of
these items is not, so to speak, primitive but it is explained by the suitable identity rela-
tions. Therefore, we can establish a form of admissibility criterion which concerns some
categories of entities. In other words, we set a criterion that, if satisfied, allows a cer-
tain kind of entities to be included in an ontology. From an intuitive point of view, the
4 On this point, literature is still thin; see, for instance, [7], [1], [4].
criterion is inspired by Quine’s slogan quoted at the beginning of this article: no entity
without entity.
This is just one half of the story since the grounding construal of identity criteria
does not simply allow the admission of entities (for instance, numbers or directions or
events or people) but it provides an illustration of the grounding relations between the
"new" entities and the fundamental ones in virtue of which the former can be introduced.
So, let us see how a provisional form of this criterion could be laid down for the well
known case of directions:
I(d, D) ↔ ∃o1 ∃o2 (R(o1 , o2 ) > d(o1 ) = d(o2 )) (5)
Here, I is a meta-predicate that expresses the relation of instantiation between an
individual x and a kind K. So, in our example, the entity d is an instance of kind D (i.e.
Directions) iff there exist at least two objects (o1 and o2 ) and their parallelism relation
explains (grounds) the fact that the direction of o1 is identical to the direction of o2 .
So far, so good. There are, of course, many questions on the table. In the following,
we will take into account just one problem concerning the inner logical structure of
grounding. The problem concerns the discrimination of specific facts connected by a
grounding relation. The majority of the accounts on grounding state that grounding is
an irriflexive relation: in other terms, if the fact A grounds the fact B, then A must be
distinct from B. Otherwise we would have a self-grounding fact. But then, we have a sort
of puzzle. Let us consider, with Rosen, a classical case of definition of square in terms of
an equilateral rectangle:
After all, if our definition of square is correct, then surely the fact that ABCD is a square and
the fact that ABCD is an equilateral rectangle are not different facts; they are one and the
same. But then the grounding-Reduction Link must be mistaken, since every instance of it
will amount to a violation of irreflexivity. ([16], 124)
But the same problem affects also the identity criteria, interpreted as grounding prin-
ciples. Consider the case of directions and let us reflect on what it means that it is a fact
that Par(a, b). Very likely, this is a (geometrical) scenario in which at least two items are
in a certain spatial relation. But in this specific scenario, the direction of a is the direction
of b; in other words, it seems that there is not a further grounded fact based in the identity
of the directions.
However, if the two alleged facts (about respectively, parallelism and directions)
are, actually, the same geometrical scenario, then they cannot be stuck together by a
grounding relation. There are some possible answers to this problem; one is sketched by
Rosen himself:
We can resist this [critique] by insisting that the operation of replacing a worldly item in a
fact with its real definition never yields the same fact again. It yields a new fact that ‘unpacks’
or ‘analyzes’ the original. ([16], 124)
Of course this train of thought should be accompanied by a more careful analysis
of the operation of "unpacking". Another path is proposed in [1] where an account of
conceptual grounding has been provided. In the following we want to sketch a view
similar to the latter.
OntoClean is a methodology for validating the ontological adequacy of taxonomic
relationships. It means that, within OntoClean various ways "to carve reality at its joint"
can be discussed and compared. The use of identity rriteria with a grounding interpreta-
tion is a path for showing the grounding relations between various modes of presentation
of bunch of entities.
The key concept is here that of mode. The same scenario can be intended and rep-
resented in many different ways; the extended example of Guarino and Welty ([9], 8ff)
clearly shows how to confront an intuitive and largely imprecise taxonomy with a far
more refined, cleaned taxonomy. Through identity criteria as grounding principles we are
able to fix the admissible ways in which one can represent a certain scenario: of course
we can say that, within S1 the lines a and b are parallel. But we can introduce another
kind of entities, that is, directions; in other words, the same scenario can be adequately
described in another mode: in a direction-mode in addition to a parallelism-mode.
Let us notice that, since the identity condition at issue is understood in terms of
grounding, the two modes (direction-mode and parallelism-mode) are not, so to speak,
on a par; on the contrary, the latter is more fundamental than the former. It explains, and
so it justifies also, why we can safely speak about directions.
6. Conclusions
In this paper we first have developed some formal constraints on identity criteria such
that, following OntoClean advice, one can say when they are logically adequate. More-
over, we have introduced a specific kind of identity criteria, two level identity criteria,
whose role is just to explain an identity among some entities referring to some other,
more basic, entities. Using them we have added a formal constraint to the stock of On-
toClean meta-constraints (OC+). Two level identity criteria have an intuitive reading in
terms of dependence of a kind of entities on some other entities, a reading possibly spec-
ified in terms of grounding. Are identity criteria grounding principles? In the second part
of the paper, we argued that there are problems with a grounding construal of identity
criteria; we sketched a possible way out, considering grounding in a more conceptual
fashion.
Some further developments of the paper regard (a) the connection between the
grounding relation and the dependency relation and (b) an analysis of the role of modes
in OntoClean, including a formal framework able to characterize dependence relations
between modes of presentation of the same scenario.
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