=Paper=
{{Paper
|id=Vol-2518/paper-FOUST8
|storemode=property
|title=The Identity of Property Particulars
|pdfUrl=https://ceur-ws.org/Vol-2518/paper-FOUST8.pdf
|volume=Vol-2518
|authors=Claudio Masolo,Adrien Barton
|dblpUrl=https://dblp.org/rec/conf/jowo/MasoloB19
}}
==The Identity of Property Particulars==
https://ceur-ws.org/Vol-2518/paper-FOUST8.pdf
The Identity of Property Particulars
Claudio MASOLO a,1 and Adrien BARTON b,1
a Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy
b Institut de Recherche en Informatique de Toulouse, CNRS, France
Abstract. A property particular is a particular that characterizes the satisfaction of
a property by an object, for example the ‘redness’ of a specific rose. We consider
three theories of identity for property particulars, illustrating them on a common
example. We discuss their strengths and weaknesses, and whether they can be in-
terpreted in an epistemic or realist perspective. We introduce some first steps of a
theory of reification of property types and discuss the benefits that it could bring.
Keywords. Property, identity, realism, epistemology, existential dependence
1. Introduction
Properties are one of the most basic building block of ontologies. A variety of kinds of
properties has been proposed by ontologists: qualities (e.g. the redness of a rose), real-
izable entities like dispositions (e.g. the fragility of a glass), functions (e.g. the function
of a heart to pump blood), roles (e.g. a doctor role), etc. Several foundational ontolo-
gies introduce into the domain of quantification the individualization of properties, e.g.,
the redness of a particular rose or the weight of a given car, which are specific of (they
inhere in) the rose and the car, respectively. For instance, BFO [1] considers dependent
continuants, DOLCE [2] individual qualities, GFO [3] property individuals, and UFO [4]
moments. We focus on individualizations of properties, called here property particulars
(PPs), corresponding to the satisfaction of a property by an object. In a philosophical per-
spective, PPs are interpetrable in several ways, e.g., as tropes [5] or truth-makers [6], and
they are intertwined with more complex entities like states of affairs [7] or facts [8]. The
introduction of PPs into the domain of quantification brings a variety of representational
advantages, such as analyzing the relations between simple PPs and the complex ones
they compose (enabling, for instance, a finer analysis of causality as in [9]), or modeling
some meta-properties via properties of PPs.
However, the identity conditions of PPs have been little studied (but see [10] for an
investigation into the identity of dispositional properties). This question if deeply inter-
twined with the epistemic or realist position one takes to interpret an ontological frame-
work. Since PPs are intuitively interconnected with propositions, they might be epis-
temically interpreted as pieces of information about objects, observations or cognitive
1 Corresponding Authors: LOA, Via alla Cascata, 56, 38123 Trento TN, Italy. IRIT, Université
Toulouse III, 118 Route de Narbonne, 31062 Toulouse cedex 9, France; E-mail: masolo@loa.istc.cnr.it;
adrien.barton@irit.fr. Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
categorizations. A theory could then admit a PP corresponding to the red character of a
rose without introducing a PP corresponding to its specific shade of red, e.g., its specific
scarletness. This would represent the situation where the only information at hand con-
cerns the fact that the rose is red, and where no information is available about its specific
shade of red. On the other hand, a realist framework might impose that the redness of a
rose would be identical with (or at least would depend on) its scarletness. An important
theoretical work would therefore be to clarify the identity conditions for PPs and their
adequacy to capture a realist vs. epistemic stance.
The question of the identity of PPs is linked with two other questions that are still
matter of debate and that we will not discuss here: first, whether and how PPs can persist
through time; second, which kinds of properties are associated with PPs, and which kinds
are not. Instead, we will analyze identity criteria in a static scenario where a given set of
properties (and associated PP-types, see Sect. 2) are taxonomically organized.
2. Property Particulars
Suppose one wants to represent the situation where ‘the object a has the property P’,
shortly, ‘a is P’ or, as usually stated in the philosophical literature, ‘a instantiates P’.
In first order logic (FOL) one would typically represent the object a by the individual
constant a, the property P by the unary predicate P, and the instantiation of the property
P by the object a by the proposition P(a). For example, the property being red would be
represented by a predicate RED, and the red character of a rose r would be represented by
the proposition RED(r).2
An alternative way to represent (also in FOL) this kind of situations relies on prop-
erty particulars. In the above situation, on top of the object a, we have an additional indi-
vidual, a PP inhering in a that intuitively stands for ‘a’s being P’. The inherence relation
between PPs and objects, a kind of specific existential dependence, is represented by the
primitive INH that is minimally characterized by axioms (a1)-(a3), where PP(p) stands
for ‘p is a property particular’ and OB(a) for ‘a is an object’. Because a PP always in-
heres in a unique object (a2)—but an object may have several PPs inhering in it—we
note pa a PP inhering in a.3
a1 INH(p, a) → PP(p) ∧ OB(a)
a2 INH(p, a) ∧ INH(p, b) → a = b
a3 PP(p) → ∃a(INH(p, a))
In a framework containing PPs, the property P can be associated with a type of PPs
represented by the predicate P̄. Therefore, ‘a is P’ is represented by ∃p(INH(p, a) ∧ P̄(p)).
¯
For instance, the fact that the rose r is red can be represented by ∃p(INH(p, r) ∧ RED(p)),
¯
i.e., being red is read as ‘having an inherent RED-PP’. Given the fact that PPs can be in-
terpreted in several ways, we introduce P̄ without making any further commitment (e.g.,
theories of tropes usually reduce P̄ to an equivalence class of similar PPs).
2 We assume here that the rose r has a uniform color.
3 To improve the readability of formulas we assume the following conventions: (i) individual constants are
noted using the typewriter typestyle font; (ii) variables are noted in italic; (iii) variables/constants ranging
over objects are noted a, b, . . . a, b, . . .; (iv) variables/constants ranging over PPs are noted p, q, . . . , p, q, . . .
3. Specialization and Covering
We consider a scenario where the user (that could have a realist or epistemic stance)
wants to represent the taxonomical organization of a finite set P of properties of objects
using a FOL-framework. We assume that the user commits to the set of properties P and
to their taxonomical organization, and we want to analyze the adequacy of alternative
FOL-frameworks (especially the ones based on PPs) for representing this commitment.
This scenario may be extended to denumerable sets of properties and relations, but here
we stick to the finite case of properties.
Classically, each property in P would be represented by a unary predicate that ap-
plies to objects. We note P the set of FOL-predicates defined on objects representing the
properties in P. Alternatively, as seen in Sect. 2, properties can be represented by means
of unary predicates that applies to PPs. Calling P̄ the set of FOL-predicates defined on
PPs representing the properties in P, we assume one-to-one mappings between P, P,
and P̄. Given the property P ∈ P, we note P ∈ P the corresponding predicate defined on
objects and P̄ ∈ P̄ the one defined on PPs. For example, the property RED (being red)
in P would be represented by the predicate RED in P or by the predicate RED ¯ in P̄. As
said, the redness of the rose r can be formally represented by RED(r) (“the rose r is of
¯
type red”) or by ∃p(INH(p, r) ∧ RED(p)) (“the rose r has a property particular of type
redness inhering in it”). Notice that, in principle, some of the predicates in P and P̄ could
be FOL-definable in terms of other predicates. We do not consider this interesting aspect
that requires a deeper analysis of the links between the considered properties, here we
focus only on taxonomical relations between properties.
3.1. Specialization
Specialization, e.g., the property being red specializes the property being colored (COL),
is one of the main taxonomical relation. Suppose one wants to represent the fact that the
property P ∈ P specializes the property Q ∈ P, i.e., intuitively, the fact that an object
being P is also Q. We analyze below different options to model this fact and, in Sect. 4.2,
how these options are intertwined with the question of the identity of PPs.
In FOL specialization is usually represented by material implication, i.e., one would
consider the predicates P, Q ∈ P corresponding to the properties P and Q and introduce
an implication axiom from P to Q. Let S be the set of couples of properties in P that
the user assumes to be linked by a specialization relation4 and S the set of couples of
predicates in P corresponding to the couples in S (similarly for S̄). The specializations
relations in S can be represented by introducing the axiom (cspec), stating that if A
specializes B, i.e., (A, B) ∈ S , then an object that is A is also B.
(A,B)∈S ∀a(A(a) → B(a))
V
(cspec)
Having PPs in the domain of quantification opens new options. One can start from
a basic requirement where the specialization is seen as a generic existential dependence
4 Note that we will not address in this article the question of determining which properties should enter in
P, and in particular which predicates should be linked by a specialization relation (that is, which couple of
predicates should be included in S ); in particular, we will not address the question of determining whether we
can associate a predicate to each predicative linguistic expression, and whether a PP should be associated to
each predicate. In addition, we assume here that specialization is at least a partial order relation with no loops.
between PPs inhering in the same object, i.e., one can introduce the axiom (spec) (where
p and q range over PPs), stating that if there is a PP p of type Ā inhering in an object a,
then there is a PP q (which might be different from p) of type B̄ inhering in a.
(Ā,B̄)∈S̄ ∀pa(Ā(p) ∧ INH(p, a) → ∃q(B̄(q) ∧ INH(q, a)))
V
(spec)
(spec) is very permissive on the existence of PPs. Suppose from now on that
(P̄, Q̄) ∈ S̄. (spec) is compatible with the facts that either (i) P̄-PPs are included in Q̄-PPs
or (ii) P̄-PPs are disjoint from Q̄-PPs (it is also compatible with a proper subset of P̄-PPs
being included in Q̄-PPs, but we will not consider this option here). (spec) can then be re-
fined by adopting one of the two (incompatible) options, i.e., the specializations relations
in S can be represented by introducing the axiom (incl·spec) or, alternatively, the axiom
(disj·spec) where S̄∗ is the set of proper specializations, i.e., S̄∗ = S̄ \ {(Ā, Ā) | Ā ∈ P̄}.
(incl·spec) (Ā,B̄)∈S̄ ∀p(Ā(p) → B̄(p))
V
(disj·spec) (spec) ∧ (Ā,B̄)∈S̄∗ ¬∃p(Ā(p) ∧ B̄(p))
V
According to (incl·spec), if a PP is associated with a property, then it is also asso-
ciated with all the properties it specializes, e.g., a P̄-PP is also a Q̄-PP. This amounts to
¯
require the identity between p and q in (spec). In the rose example, all RED-PPs are also
¯ ¯
COL, but it is possible to have COL-PPs that are not RED.¯
According to (disj·spec), PPs are specific to a given property: if P (properly) spe-
cializes Q, a P̄-PP is not a Q̄-PP. In the rose example, it is not possible to have PPs that
are instances of both RED ¯ and COL.¯ However, (disj·spec) (as well as (incl·spec)) allows
¯ ¯
for the existence of several RED-PPs and of several COL-PPs inhering in the rose.5
An interesting extension of the proposed framework would be to assume that the
user distinguishes specialization from correlation. These two relations could then be
represented differently, for instance, one could consider (incl·spec) for specialization and
(disj·spec) for correlation.
3.2. Covering
A second important taxonomical relation is covering. While specialization concerns the
way more specific properties are related to more generic properties, covering is intended
to cope with the opposite mechanism. Intuitively, when a property Q is covered by n
properties Pi (with i ∈ {1, . . . , n}) then all the objects having the property Q also have (at
least) a Pi -property. For simplicity, consider the case with n = 2, i.e., Q is covered by
P1 and P2 both different from Q. For example, in the case of the rose, assume that being
colored is covered by being red and being yellow. Similarly to the case of specialization,
C represents the set of triples of properties linked by the covering relation, while C and
C̄ are the corresponding sets of triples of properties in, respectively, P and P̄.
Classically, one could represent the being covered of Q by P1 and P2 by considering
the predicates Q, P1 , P2 ∈ P and by adding an implication from Q to the disjunction of P1
and P2 , i.e., one would add the following axiom (ccov) (where a ranges over objects).
5 An additional interesting possibility is to have a specific existential dependence SD between PPs: SD(p, q)
stands for ‘the PP p specifically depends on the PP q’, where SD(p, q) ∧ SD(p, r) → q = r. One could then intro-
duce (Ā,B̄)∈S̄∗ ∀pa(Ā(p) ∧ INH(p, a) → ∃q(q 6= p ∧ SD(q, p) ∧ B̄(q) ∧ INH(q, a))). This would allow to specify
V
the links between the different PPs.
(A,B1 ,B2 )∈C ∀a(A(a) → (B1 (a) ∨ B2 (a)))
V
(ccov)
Note that this does not mean that all the material implications with form A(a) →
(B1 (a) ∨ B2 (a)) are cases of covering. As a matter of fact, covering has an intensional
aspect that is not captured by (ccov) (indeed, this is also true in the case of (cspec) for the
specialization relation). For instance, one could think that being colored is not covered
by being 1kg heavy and not being 1kg heavy.
Considering PPs, similarly to specialization, covering can be modeled by introduc-
ing a generic existential dependence, i.e., by introducing (cov) according to which the
existence in the object a of a PP associated with a given property implies the existence
of a PP inhering in a associated with one of its covering properties:
(A,B1 ,B2 )∈C̄ ∀pa(Ā(p) ∧ INH(p, a) → ∃q((B̄1 (q) ∨ B̄2 (q)) ∧ INH(q, a)))
V
(cov)
Similarly to (spec), (cov) can be refined in the two following ways: (i) (incl·cov)
guarantees that a PP associated with a property is also associated with one of its covering
properties; (ii) (disj·cov) assures that the existence of a PP associated with a property
implies the existence of a different PP associated with one of its covering properties:
(incl·cov) (A,B1 ,B2 )∈C̄ ∀p(Ā(p) → (B̄1 (p) ∨ B̄2 (p)))
V
(disj·cov) (cov) ∧ (A,B1 ,B2 )∈C̄ ∀p(¬∃p(Ā(p) ∧ (B̄1 (p) ∨ B̄2 (p))))
V
In the example of the rose:
– according to (incl·cov), when we have COL(p¯ ¯
r ) we also have RED(p ¯
r ) ∨ YEL(pr ),
i.e. pr represents also either the being red of r or its being yellow.
– (disj·cov) allows for general PPs that satisfy COL¯ but neither RED¯ nor YEL;
¯ that is,
PPs associated only with the general colored character of the rose. However, from
the existence of this general PP, we can deduce the existence of a different, more
specific PP associated with either the redness or the yellowness of the rose.
3.3. Combining Specialization and Covering
Suppose now to represent the situation where Q is covered by P1 and P2 and where P1
and P2 are both specializations of Q. For the predicates defined on objects, by combining
(cspec) with (ccov), we obtain ∀a(Q(a) ↔ (P1 (a) ∨ P2 (a))).
For the predicates defined on PPs we obtain two consistent options:
– {(incl·spec), (incl·cov)} reduces to (A,B1 ,B2 )∈C̄ ∀p(Ā(p) ↔ (B̄1 (p) ∨ B̄2 (p))),
V
– {(disj·spec), (disj·cov)} reduces to a mutual existential dependence between the
PPs that are instances of a property and the (different) PPs that are instances of
one of the covering properties.
These positions have several important features. First, in all the positions considered,
independently of the level of specificity of the PPs, it is possible to have different PPs
(inherent in the same object) instantiating the same PP-predicate, i.e., there could be p,
q, and a such that INH(p, a) ∧ INH(q, a) ∧ Ā(p) ∧ Ā(q) ∧ p 6= q.
Second, (incl·spec) guarantees that the PPs that instantiate a given PP-predicate Ā
also instantiate all the PP-predicates specialized by Ā, i.e., all the predicates B̄ such that
(Ā, B̄) ∈ S̄. However, it allows for PPs that instantiate a given predicate without instanti-
ating any more specific PP-predicate. The addition of (incl·cov) rules out this last possi-
bility for those PPs associated with a covered property.
Third, (disj·spec) does not allow a PP to be an instance of two different PP-predicates
one specializing the other. Actually, (disj·spec) does not even guarantee that the number
of the instances of a given PP-predicate is greater than the number of the instances of
its specializations: it is possible to have different specific PPs for only one generic PP,
¯
e.g., there could be different RED-PPs ¯
and only one COL-PP (inhering in the same ob-
ject). To avoid that, one could specialize (disj·spec) by introducing a specific existential
dependence as done in footnote 5. Similar considerations hold for (disj·cov).
All the options discussed are compatible with a quite strong multiplicativism of
PPs—that is, many PPs can be associated with the same pair (ob ject, property). The
next section will therefore discuss which principle of restriction on the identity of PPs
could be endorsed to limit this multiplicativism.
4. Restrictions on the Identity of Property Particulars
4.1. A Strong Principle of Identity
As observed at the end of Sect. 3, nothing prevents the possibility to have different PPs
(inherent in the same object) all instantiating the same predicate in P̄. However, intu-
itively, one could think that any object can satisfy any property in at most one way, i.e.,
there is a maximum of one PP for each couple (object, property). To formalize this intu-
ition one could introduce the axiom (idpp).
(idpp) (INH(p, a) ∧ INH(q, a) ∧ Ā∈P̄ (Ā(p) ∧ Ā(q))) → p = q
W
In general, (idpp) is consistent with all the options introduced in Sect. 3. When
combined with (incl·spec), it prevents the possibility to have several PPs inherent in the
¯
same object at different levels of specificity. In the rose example, if we suppose RED(p r ),
¯
it follows that COL(p ¯
r ) and, by (idpp), pr is the unique COL-PP inhering in r, i.e., in that
case it is impossible to have a PP associated with the colored character of r that would
not be associated with the redness of r.6
In the case of the red rose r we are then left with two options:
¯
(1) in {(incl·spec), (idpp)} there is only one PP pr that satisfies RED(p ¯
r ) ∧ COL(pr ),
¯ ¯
(2) in {(disj·spec), (idpp)} there are two PPs pr 6= qr such that RED(pr ) ∧ COL(qr ).
In (1) pr concerns both properties being colored and being red. Consequently,
(idpp) forces a one-to-one mapping between PPs and (object, property) couples only
in the presence of (disj·spec). Another constraint brought by (1) concerns the relation
of specialization. Assume, for example, that having a mass is represented by MASS ¯
and that there exists a predicate MC¯ such that MASS¯ and COL ¯ specialize MC,¯ that is,
¯ MC),
(MASS, ¯ (COL,¯ MC)
¯ ∈ S̄. If COL
¯ and MASS
¯ are disjoint (as they should intuitively be),
¯ a ) and
there could not be two PPs ca and ma inhering in the same object a such that COL(c
6 It is logically possible to have COL(p
¯ ¯ ¯
r ) ∧ ¬RED(p r ), but this implies that the rose has no RED-PP, i.e.,
¯
¬∃p(RED(p) ∧ INH(p, r)), and this ontology would not formalize adequately the redness of the rose. Adding
(incl·cov) would even worsen the situation because COL(p¯ ¯
r ) ∧ ¬RED(p ¯
r ) would imply YEL(p r ).
¯
MASS(m ¯ a ) and MC(m
a ): otherwise, we would deduce by (incl·spec) that MC(c ¯ a ), and by
(idpp) we would have ca = ma . A suggestion could be that the relation of specialization,
as well as the relation of covering, should be reserved for pairs of PP-types that are of
kind determinate/determinable [11], an interesting point left for future work.
4.2. A Leibnizian Principle of Identity
An alternative view to (idpp) would be to allow the existence of multiple instances (in-
hering in the same object) of a given PP-predicate only when such predicate is special-
¯
ized by other PP-predicates. In the case of the red rose, one would have a single RED-PP
¯
pr , that would also be a COL-PP; ¯
and a second, different COL-PP qr , that would not be
¯ 7 This kind of ontology would be compatible
an instance of any specialization of COL.
with the axiom (idall) below, a sort of restricted Leibniz principle that assures that two
PPs (inhering in the same object) are identical only when they are indistinguishable by
means of PP-predicates.
(idall) (INH(p, a) ∧ INH(q, a) ∧ Ā∈P̄ (Ā(p) ↔ Ā(q))) → p = q
V
First note that (idall) is weaker than (idpp): (idall) (but not (idpp)) allows for two
¯
different PPs pr and qr such that COL(p ¯
r ), RED(p ¯
r ), COL(q ¯
r ), ¬RED(qr ), and pr 6= qr .
Second, in general (idpp) does not imply (idall). Consider PPs p and q and object
a such that INH(p, a), INH(p, a), p 6= q, and Ā∈P̄ (¬Ā(p) ∧ ¬Ā(q)). In this case (idpp)
V
vacuously hold, but (idall) does not hold. However in the reasonable assumption that all
PPs are classified under at least one predicate in P̄, (idpp) implies (idall). Consider PPs
p and q and object a such that INH(p, a), INH(q, a) and Ā∈P̄ (Ā(p) ↔ Ā(q)). By our
V
assumption, all PPs are classified under at least one predicate in P̄. Suppose Ā(p) holds.
From the equivalence just mentioned, we can deduce Ā(q). And from Ā(p) and Ā(q), we
deduce by (idpp) that p = q. Thus, we have proved (idall).
Third, {(disj·spec), (idall)} is equivalent to {(disj·spec), (idpp)} when all the predi-
cates in P̄ are disjoint and each PP is classified under at least one predicate.
To summarize, we can distinguish at least four theories: T1 = {(incl·spec), (idpp)},
T2 = {(disj·spec), (idpp)}, T3 = {(incl·spec), (idall)}, and T4 = {(disj·spec), (idall)} to
each of which we can add an axiom to deal with the cases of covering, (incl·cov) or
(disj·cov). For our discussion about the adequacy of these theories for modeling the epis-
temic vs. realist view, we will not consider T4 , which is quite close from T2 .
5. Realist vs. Epistemic Interpretations of the Three Theories of Identity
To further illustrate T1 -T3 , suppose that the rose r is not only red, but more precisely it
is scarlet (SCA) and that (RED,¯ COL),
¯ (SCA,¯ RED)¯ ∈ S̄. To represent this situation, a theory
should accept at least one PP associated with its colored character, one PP associated
with its redness, and one PP associated with its scarletness. The next question is whether
some of those PPs are identical or not. According to T1 , there can be only one PP s1r
7 A weaker version would be to accept a first COL-PP
¯ qr that would be specifically existentially dependent
¯
on pr , in the sense discussed in footnote 5; and a second COL-PP q0r that would not be specifically existentially
dependent on any instance of any specialization of COL.¯
characterizing at the same time the scarletness, the redness and the colored character of
r. According to T2 , there must be three different PPs: s2r characterizing its scarletness
only (but neither its redness nor its colored character), r2r characterizing its redness only,
c2r characterizing its colored character only. Finally, T3 is compatible with the existence
of three PPs: s3r characterizing its scarletness, redness and colored character, c3r charac-
terizing its redness and colored character, r3r characterizing its colored character only (it
is also compatible with a more economical ontology including only s3r , but it would then
collapse to the same ontology as the one implied by T1 ).
There are at least two broad perspectives to interpret PP-constants: an informational
or epistemic perspective where PP-constants are intended to refer to pieces of infor-
mation, observations or cognitive categorizations; and a realist perspective, where PP-
constants are intended to refer to entities existing in the world (most of which are not in-
formational, although some of them may be so). We will not discuss in detail the distinc-
tion between the two approaches (see, e.g., [12,13] for a related debate), but instead com-
ment on which theory seems to fit with which perspective, acknowledging that definitive
conclusions require more extensive investigations.
T1 seems to make more sense when interpreted in a realist perspective than in an
epistemic perspective: s1r would represent the unique color of the rose, in its full speci-
ficity. T2 may fit more naturally an epistemic perspective: s2r , r2r and c2r would represent
pieces of information on the color of an object acquired by measurement instruments
with different resolutions. The inference from the existence of s2r to the existence of r2r
could amount to the deduction of the information that the rose if red from the information
that the rose is scarlet. It might also be possible to interpret T2 in a realist perspective, if
¯
one sees a statement like RED(r 2 2
r ) as expressing that rr is the most specific entity warrant-
ing the inference that the rose is red: in this ontology, the two other PPs s2r and c2r would
be respectively the most specific entity warranting the inference that the rose is scarlet,
and the most specific entity warranting the inference that the rose is colored. T3 seems to
be interpretable in both perspectives. In a realist perspective, c3r could be viewed as the
general structure responsible for the colored character of r, whereas r3r (resp. s3r ) could
be viewed as the more specific structure responsible for its red (resp. scarlet) character,
which is also responsible for its colored (resp. red and colored) character. In epistemic
terms, c3r could represent the information collected by a device that could only detect
the colored character of an object, r3r to the information collected by a device resolving
being red and categorizing the redness as a color, and s3r to the information collected by
a device that would resolve being scarlet, and categorize it as a redness (and as a color).
As we said, covering axioms can be added to those theories. Suppose that we add
(incl·cov) to T3 and that (COL,¯ RED,
¯ YEL),
¯ (RED,
¯ SCA,
¯ CRI) ¯ ∈ C̄ (including the color crim-
son CRI). It is easy to see that, in the rose example, there is only one possible configu-
ration of PPs, namely a single PP s1r characterizing at the same time the scarletness, the
redness and the colored character of r, i.e., SCA(s¯ 1 ¯ 1 ¯ 1
r ), RED(sr ) and COL(sr ). That is, in
presence of (incl·cov) and covering axioms between the relevant properties, the ontology
compatible with T3 will collapse to the ontology compatible with T1 .
6. The Reification of PP-types
We will now sketch an alternative representation method that reifies PP-predicates and
is able to represent direct vs. indirect classification under a property. In an epistemic
view, this method allows to clearly separate the information acquired from measurement
devices from the one deduced from such information by using some general knowledge.
Up to now, the PPs corresponding to a property P has been collected by means of
the PP-predicate P̄. We now reify PP-types into the domain of quantification. For this, we
introduce a new kind of entities—PT(t) stands for “t is a PP-type”—and we assume that
there is a one-to-one correspondence between the properties in P and the PT-constants
in the set PPT . We note p ∈ PPT the PP-type corresponding to P ∈ P.8
The general idea is that PPs are classified by PP-types. In particular we consider the
primitive relation of direct classification between PPs and PP-types: dCF(p,t) stands for
“the PP p is directly classified under the PP-type t” (a4). We assume that PPs may have
a unique direct PP-type, i.e., they can be directly classified under a single PP-type (a5).
a4 dCF(p,t) → PP(p) ∧ PT(t)
a5 dCF(p,t) ∧ dCF(p, u) → t = u
The specialization relation between properties is represented by the partial order
v defined between PP-types: t v u stands for “the PP-type t is a specialization of the
PP-type u”. The specialization relations in S are modeled by introducing (ptspec ) where
SPT is the set of couples of PT-constants corresponding to the couples in S .
(a,b)∈SPT a v b
V
(ptspec )
Crucially, in this framework it is possible to make the difference between the above-
mentioned direct classification and the indirect classification (iCF) defined in (d1).
d1 iCF(p,t) , ∃u(dCF(p, u) ∧ u v t)
The general case where ‘a is P’ is then modeled using indirect, rather that direct, clas-
sification, i.e., by ∃q(INH(q, a) ∧ iCF(q, p)), where p represents the property P.9 In the
example of the rose, dCF(pr , red) together with red v col imply iCF(pr , col). The
indirect classification iCF allows then to explicitly represent classifications grounded on
knowledge about the specialization relation. First note that iCF does not require the ex-
istence of any additional PP. Second, this framework is compatible with the existence
of PPs pr and qr (both inhering in the rose) such that iCF(pr , col) and dCF(qr , col),
i.e., the existence of a PP directly classified under a given PP-type does not exclude the
possibility to have a different PP directly classified under a more general PP-type.
This theory is similar to T3 where the previous situation may be represented by
¯
RED(p ¯
r ) ∧ COL(p ¯
r ) together with ¬RED(q ¯
r ) ∧ COL(qr ). Note however that in T3 we don’t
really have a notion of indirect classification, we just have the PPs pr and qr that are both
standardly classified under COL ¯ while pr , but not qr , is also standardly classified under
¯ Here all the atomic assertions have the same ‘status’ and one can establish the level
RED.
of resolution of a given assertion only by looking at the other assertions present in the
theory. On the other hand, by reifying PP-types, it is possible to direclty manage differ-
ent kinds of classification of PPs (different classification modalities) by means of several
relations defined between PPs and PT-instances. Furthermore, the fact that the special-
ization relation is also represented by a relation (v) leaves space for a more intensional
8 Variables/constants ranging over P-types are noted t, u, . . . , t, u, . . .
9 Given the reflexivity of v, dCF(p,t) → iCF(p,t), i.e., direct classification is a limit case of indirect classi-
fication.
characterization of this relation. Future work should investigate the possible benefits of
the reification strategy.
7. Conclusion
We have thus identified three theories of identity among PP, and characterized with which
perspective (epistemic or realist) they appear to be compatible. We have presented a reifi-
cation of property-types that enables to formalize how PPs can be directly or indirectly
classified under property-types. Future work should investigate more closely various the-
ories about 1) what counts as a property 2) which of those properties are linked by a spe-
cialization relation and 3) how the distinction determinate/determinable can shed light on
the application of our formalization to various theories of properties and specialization.
It should also investigate further the benefits that the reification of property types could
bring. The question of the diachronic identity of PPs will also need to be addressed:
PPs are generally considered as continuants, that persist in time by being fully present
at each instant; but what are the criteria for two PPs at different times to be the same
entity? This investigation on the identity of PPs should also be refined into more spe-
cific investigations about the identity of complex properties such as roles, functions and
dispositions.
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