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 interpreted 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.
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),
realizable 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
ontologies 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 [
However, the identity conditions of PPs have been little studied (but see [
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.
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 property particulars. In the above situation, on top of the object a, we have an additional individual, 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 inheres 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) ! 9a(INH(p; a))
In a framework containing PPs, the property P can be associated with a type of PPs represented by the predicate P¯. Therefore, ‘a is P’ is represented by 9p(INH(p; a) ^ P¯ (p)). For instance, the fact that the rose r is red can be represented by 9p(INH(p; r) ^ RE¯ D(p)), i.e., being red is read as ‘having an inherent RE¯ D-PP’. Given the fact that PPs can be interpreted in several ways, we introduce P¯ without making any further commitment (e.g., theories of tropes usually reduce P¯ to an equivalence class of similar PPs).
3To 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; : : :
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 applies 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 P¯ the set of FOL-predicates defined on PPs representing the properties in P , we assume one-to-one mappings between P , P, and P¯ . Given the property P 2 P , we note P 2 P the corresponding predicate defined on objects and P¯ 2 P¯ 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 RE¯ D in P¯ . As said, the redness of the rose r can be formally represented by RED(r) (“the rose r is of type red”) or by 9p(INH(p; r) ^ R E¯D(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 P¯ 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.
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 2 P specializes the property Q 2 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 2 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 S¯ ). The specializations relations in S can be represented by introducing the axiom (cspec), stating that if A specializes B, i.e., (A; B) 2 S , then an object that is A is also B.
(cspec) V(A;B)2S 8a(A(a) ! B(a))
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 4Note 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 A¯ inhering in an object a, then there is a PP q (which might be different from p) of type B¯ inhering in a. (spec) V(A¯;B¯)2S¯ 8pa( A¯(p) ^ INH(p; a) ! 9q( B¯(q) ^ INH(q; a))) (spec) is very permissive on the existence of PPs. Suppose from now on that ( P¯; Q¯) 2 S¯ . (spec) is compatible with the facts that either (i) P¯-PPs are included in Q¯-PPs or (ii) P¯ -PPs are disjoint from Q¯-PPs (it is also compatible with a proper subset of P¯-PPs being included in Q¯-PPs, but we will not consider this option here). (spec) can then be refined 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 S¯ is the set of proper specializations, i.e., S¯ = S¯ n f( A¯; A¯) j A¯ 2 P¯ g. (incl spec) V(A¯;B¯)2S¯ 8p( A¯(p) ! B¯(p)) (disj spec) (spec) ^ V(A¯;B¯)2S¯ :9p(A¯ (p) ^ B¯ (p))
According to (incl spec), if a PP is associated with a property, then it is also associated with all the properties it specializes, e.g., a P¯ -PP is also a Q¯-PP. This amounts to require the identity between p and q in (spec). In the rose example, all RE¯ D-PPs are also CO¯ L, but it is possible to have CO¯ L-PPs that are not RE¯ D.
According to (disj spec), PPs are specific to a given property: if P (properly) specializes Q, a P¯ -PP is not a Q¯ -PP. In the rose example, it is not possible to have PPs that are instances of both RE¯ D and CO¯ L. However, (disj spec) (as well as (incl spec)) allows for the existence of several RE¯ D-PPs and of several CO¯ L-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.
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 2 f1; : : : ; ng) 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 C¯ are the corresponding sets of triples of properties in, respectively, P and P¯ .
Classically, one could represent the being covered of Q by P1 and P2 by considering the predicates Q; P1; P2 2 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 introduce V(A¯;B¯)2S¯ 8pa(A¯(p) ^ INH(p; a) ! 9q(q 6= p ^ SD(q; p) ^ B¯(q) ^ INH(q; a))). This would allow to specify the links between the different PPs.
(ccov) V(A;B1;B2)2C 8a(A(a) ! (B1(a) _ B2(a)))
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 introducing 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: (cov) V(A;B1;B2)2C¯ 8pa(A¯ (p) ^ INH(p; a) ! 9q(( B¯1(q) _ B¯2(q)) ^ INH(q; a))) 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) V(A;B1;B2)2C¯ 8p( A¯(p) ! (B¯ 1(p) _ B¯ 2(p))) (disj cov) (cov) ^ V(A;B1;B2)2C¯ 8p(:9p( A¯(p) ^ (B¯ 1(p) _ B¯2(p))))
– according to (incl cov), when we have C O¯L(pr) we also have R E¯D(pr) _ YE¯ L(pr), i.e. pr represents also either the being red of r or its being yellow. – (disj cov) allows for general PPs that satisfy CO¯ L but neither RE¯ D nor YE¯ L; 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.
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 8a(Q(a) $ (P1(a) _ P2(a))).
For the predicates defined on PPs we obtain two consistent options: – f(incl spec), (incl cov)g reduces to V(A;B1;B2)2C¯ 8p( A¯(p) $ (B¯ 1(p) _ B¯ 2(p))), – f(disj spec), (disj cov)g 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) ^ A¯ (p) ^ A¯ (q) ^ p 6= q.
Second, (incl spec) guarantees that the PPs that instantiate a given PP-predicate A¯ also instantiate all the PP-predicates specialized by A¯, i.e., all the predicates B¯ such that ( A¯; B¯ ) 2 S¯ . However, it allows for PPs that instantiate a given predicate without instantiating any more specific PP-predicate. The addition of (incl cov) rules out this last possibility 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 RE¯ D-PPs and only one CO¯ L-PP (inhering in the same object). 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; pro perty). The next section will therefore discuss which principle of restriction on the identity of PPs could be endorsed to limit this multiplicativism.
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 P¯ . However, intuitively, 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 intuition one could introduce the axiom (idpp).
(idpp) (INH( p; a) ^ INH(q; a) ^ WA¯2P¯ ( A¯( p) ^ A¯(q))) ! p = q
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 RE¯ D(pr), it follows that C O¯L(pr) and, by (idpp), pr is the unique CO¯ L-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 f(incl spec), (idpp)g there is only one PP pr that satisfies RE¯ D(pr) ^ CO¯ L(pr), (2) in f(disj spec), (idpp)g there are two PPs pr 6= qr such that RE¯ D(pr) ^ CO¯ L(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 MA¯SS
and that there exists a predicate M¯C such that MA¯SS and CO¯ L specialize M¯C, that is,
(MA¯SS; M¯C); (CO¯ L; M¯C) 2 S¯ . If CO¯ L and MA¯SS are disjoint (as they should intuitively be),
there could not be two PPs ca and ma inhering in the same object a such that CO¯ L(ca) and
6It is logically possible to have CO¯ L(pr) ^ :RE¯ D(pr), but this implies that the rose has no RE¯ D-PP, i.e.,
:9p(RE¯ D(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 CO¯ L(pr) ^ :RE¯ D(pr) would imply YE¯ L(pr).
MA¯SS(ma): otherwise, we would deduce by (incl spec) that M¯C(ca) and M¯C(ma), 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 [
An alternative view to (idpp) would be to allow the existence of multiple instances (inhering in the same object) of a given PP-predicate only when such predicate is specialized by other PP-predicates. In the case of the red rose, one would have a single RE¯ D-PP pr, that would also be a CO¯ L-PP; and a second, different CO¯ L-PP qr, that would not be an instance of any specialization of C O¯L.7 This kind of ontology would be compatible 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) ^ VA¯2P¯ ( A¯(p) $ A¯(q))) ! p = q
First note that (idall) is weaker than (idpp): (idall) (but not (idpp)) allows for two different PPs pr and qr such that CO¯ L(pr), R E¯D(pr), CO¯ L(qr), :R E¯D(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 VA¯2P¯ (:A¯ (p) ^ : A¯(q)). In this case (idpp) vacuously hold, but (idall) does not hold. However in the reasonable assumption that all PPs are classified under at least one predicate in P¯ , (idpp) implies (idall). Consider PPs p and q and object a such that INH(p; a), INH(q; a) and VA¯2P¯ ( A¯(p) $ A¯(q)). By our assumption, all PPs are classified under at least one predicate in P¯ . Suppose A¯(p) holds. From the equivalence just mentioned, we can deduce A¯ (q). And from A¯(p) and A¯ (q), we deduce by (idpp) that p = q. Thus, we have proved (idall).
Third, f(disj spec), (idall)g is equivalent to f(disj spec), (idpp)g when all the predicates in P¯ are disjoint and each PP is classified under at least one predicate.
To summarize, we can distinguish at least four theories: T1 = f(incl spec), (idpp)g, T2 = f(disj spec), (idpp)g, T3 = f(incl spec), (idall)g, and T4 = f(disj spec), (idall)g 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 epistemic vs. realist view, we will not consider T4, which is quite close from T2.
To further illustrate T1-T3, suppose that the rose r is not only red, but more precisely it is scarlet (SCA) and that (RE¯ D; C O¯L); (SC¯ A; RE¯ D) 2 S¯ . 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 1 some of those PPs are identical or not. According to T1, there can be only one PP sr 7A weaker version would be to accept a first CO¯L-PP qr that would be specifically existentially dependent on pr, in the sense discussed in footnote 5; and a second CO¯L-PP q0r that would not be specifically existentially dependent on any instance of any specialization of CO¯L. 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 characterizing 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
information, observations or cognitive categorizations; and a realist perspective, where
PPconstants are intended to refer to entities existing in the world (most of which are not
informational, although some of them may be so). We will not discuss in detail the
distinction between the two approaches (see, e.g., [
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 specificity. 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 R E¯D(r2r) as expressing that r2r is the most specific entity warranting 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 (CO¯ L; RE¯ D; YE¯ L); (RE¯ D; SC¯ A; CR¯ I) 2 C¯ (including the color crimson CRI). It is easy to see that, in the rose example, there is only one possible configuration of PPs, namely a single PP s1r characterizing at the same time the scarletness, the redness and the colored character of r, i.e., SC¯ A(s1r), R E¯D(s1r) and CO¯ L(s1r). 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.
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 P¯ . 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 2 PPT the PP-type corresponding to P 2 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 . (ptspec) V(a;b)2SPT a v b
Crucially, in this framework it is possible to make the difference between the abovementioned direct classification and the indirect classification (iCF) defined in (d1).
d1 iCF(p; t) , 9u(dCF(p; u) ^ u v t) The general case where ‘a is P’ is then modeled using indirect, rather that direct, classification, i.e., by 9q(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 existence 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 RE¯ D(pr) ^ C O¯L(pr) together with :R E¯D(qr) ^ CO¯ L(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 C O¯L while pr, but not qr, is also standardly classified under RE¯ D. Here all the atomic assertions have the same ‘status’ and one can establish the level 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 different kinds of classification of PPs (different classification modalities) by means of several relations defined between PPs and PT-instances. Furthermore, the fact that the specialization relation is also represented by a relation (v) leaves space for a more intensional
9Given the reflexivity of v, dCF(p;t) ! iCF(p;t), i.e., direct classification is a limit case of indirect classification. characterization of this relation. Future work should investigate the possible benefits of the reification strategy.
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 reification 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 theories about 1) what counts as a property 2) which of those properties are linked by a specialization 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 specific investigations about the identity of complex properties such as roles, functions and dispositions.