Dispositions are realizable entities, that is, properties that can be realized by certain types of processes, but that are present even when they are not realized [1,2]. The Basic Formal Ontology (BFO) defines a disposition as a realizable entity "that exists because of certain features of the physical makeup of the independent continuant that is its bearer" ( [2], p. 178). To be realized, dispositions need to be triggered by some process: for example, a fragile vase has a disposition to break (its realization) when undergoing a strong shock (its trigger). Dispositions are useful to represent causal relations in BFO, as they provide their bearers with certain causal powers. In particular, when a disposition is triggered by t and realized by r, this implies that t causes r.
Röhl and Jansen [3] propose a formalization of dispositions for single-track dispositions, that is, dispositions that have exactly one kind of realization given one specific bundle of triggering events and/or background conditions. Barton and Jansen [4] suggest first steps towards a formalization of multi-track dispositions, taking into account that there could be several kinds of triggers as well as several kinds of realizations. For this, they introduce a relation of parthood between dispositions. The present paper will extend those results by distinguishing several mereological relations that can be defined for dispositions.
In the following, terms for classes will be written in italic (e.g. "Breaking"), and terms for instances in bold (e.g. "vase 0 "). A statement of the form "disposition d of b to R when T" should be interpreted as expressing (in the vocabulary of [3]), when d is a single-track disposition, that (1) d inheres_in b, that (2) d realized_in only R, and that (3) d has_trigger only T. If a and b are particulars, a+b is defined as the mereological sum of a and b. Similarly, if A and B are classes or universals, A+B is defined as the class of mereological sums of one instance of A and one instance of B.
We begin in section 2 by introducing three case studies illustrating different kinds of disposition-parthood which we will axiomatize in section 3. Section 4 will demonstrate the expressive power of the combined use of these kinds of dispositionparthood. Section 5 discusses whether disposition-parthood relations are authentic mereological relations and conclude.
The dispositional properties of an object and the way they are realized are often quite complex. In order to distinguish different varieties of such complexity, we introduce three case studies: the fragility of a vase, the solubility of a tablet, and the vulnerability of a human to a certain kind of poison.
The case study (Vase) serves as an illustration for the kind of disposition-parthood that has been discussed in [4] under the more general label "d-part_of". We will call it "mod-part_of" here, in order to distinguish it from other varieties of disposition-parthood. This case study involves the fragile vase vase 0 , whose fragility can be formalized as a disposition. Three particular dispositions disp_break 0 , disp_crack 0 and fragility 0 inhere in vase 0 , defined as follows:
• disp_break 0 is the disposition of vase 0 to Breaking when Strong_shock, • disp_crack 0 is the disposition of vase 0 to Cracking when Light_shock, • fragility 0 is the disposition of vase 0 to [Breaking when Strong_shock] and [Cracking when Light_shock].
We can introduce a relation of disposition-parthood mod-part_of such that:
• disp_break 0 mod-part_of fragility 0 • disp_crack 0 mod-part_of fragility 0
We use the modifier "mod", derived from "mode", because fragility 0 has two possible pathways, or modes, of being realized: via disp_break 0 or via disp_crack 0 . We will now introduce two alternative kinds of disposition-parthood relations.
The case study (Tablet) involves a medication tablet tablet 0 , composed of two halves half-tablet 1 and half-tablet 2 -that are both part_of tablet 0 . Let's introduce the following dispositions:
• solubility 0 is the disposition of tablet 0 to Dissolving when Object_in_solvent,
• solubility 1 is the disposition of half-tablet 1 to Dissolving when Object_in_solvent, • solubility 2 is the disposition of half-tablet 2 to Dissolving when Object_in_solvent.
We can introduce another relation of disposition-parthoood add-part_of such that:
• solubility 1 add-part_of solubility 0
• solubility 2 add-part_of solubility 0
The modifier "add" is derived from "additive": the dissolving processes of both halftablet 1 and half-tablet 2 compose the realization of solubility 0 , and the triggers of the former two compose the trigger of the latter.
The We introduce another relation of disposition-parthoood chain-part_of such that:
• vulnerability_heart 0 chain-part_of vulnerability_heart_&_life 0
• heart_need 0 chain-part_of vulnerability_heart_&_life 0
We use the modifier "chain" because the realization of vulnerability_heart 0 triggers heart_need 0 , thus creating a causal chain.
As will become clear later, the relations add-part_of and chain-part_of share several commonalities. Thus, we introduce a relation aggreg-part_of, of which both are subrelations. We also introduce a general relation of dispositional-parthood noted disp-part_of, of which mod-part_of and aggreg-part_of are subrelations. All these relations are subrelations of the general part_of relation, which is axiomatized in the Relation Ontology [5] (cf. Table 1).
The respective relations of proper parthood will be denoted by adding "proper_" in the name -e.g., proper_mod-part_of. We define a disp-complex (resp. mod-complex, aggreg-complex, etc.) as a disposition that has some proper disp-part (resp. mod-part, aggreg-part, etc.). I.e., for "[x]"="disp", "mod", "aggreg", "add" or "chain":
The bearer of a disp-part is a (proper or improper) part of the bearer of the disp-complex:
(DISP-BEARER)
In (Tablet), the bearer of solubility 1 (resp. solubility 2 ) is half-tablet 1 (resp. half-tablet 2 ), which is a part of tablet 0 . In (Vase), the bearers of fragility 0 , disp_crack 0 and disp_break 0 are the same, namely vase 0 . Similarly, in (Poison), the bearers of vulnerability_heart 0 , heart_need 0 , and vulnerability_heart_&_life 0 are the same, namely Jones. However, we will see later in section 4.2 with the domino case study that the bearer of a mod-part (resp. chain-part) is not always identical to the bearer of a modcomplex (resp. chain-complex).
We will now turn to the axioms concerning the realization of a disp-complex. In the general case, if a disp-complex is realized in a process, then at least one of its proper disp-parts is realized in a part of this process:
As we will now see, both mod-part_of and aggreg-part_of satisfy more specific axioms.
A mod-complex is realized in a process if and only if at least one of its mod-parts is realized in this very same process:
In (Vase), fragility 0 is realized in a process (instance of Cracking or Breaking) if and only if either disp_crack 0 or disp_break 0 is realized in this process. Things are different for aggreg-parthood. If an aggreg-complex is realized in a process, then all its aggregparts are realized in a part of this process:
In (Tablet), if solubility 0 is realized in dissolving 0 , its aggreg-parts solubility 1 and solubility 2 are both realized, respectively in dissolving 1 and dissolving 2 (that are both parts of dissolving 0 ). In (Poison), if vulnerability_heart_life 0 is realized in cardiac_arrest 0 +death 0 , its aggreg-parts vulnerability_heart 0 and heart_need 0 are both realized, respectively in cardiac_arrest 0 and death 0 .
Note however that if an aggreg-part is realized, this does not imply anything for the realization of the aggreg-complex. For example, if half-tablet 1 dissolves, this does not imply the dissolving of tablet 0 -as half-tablet 2 may not dissolve (in case it is not put into a solvent). Similarly, in (Poison), Jones might undergo some cardiac arrest which is not due to the ingestion of poison, in which case heart_need 0 will be realized without vulnerability_heart_&_life 0 being realized (because it was not triggered by any ingestion of poison).
We will now turn to the axioms concerning the triggers of a disp-complex. In general, if a disp-complex is triggered, then at least one of its proper disp-parts is triggered.
Again, both mod-part_of and aggreg-part_of satisfy more specific axioms. A modcomplex is triggered by a process if and only if at least one of its mod-parts is triggered by this very process:
In (Vase), fragility 0 is triggered by a process (an instance of Light shock or Strong shock) if and only if either disp_crack 0 or disp_break 0 is triggered by this process. However, if an aggreg-complex is triggered, then all its aggreg-parts are triggered:
In (Poison), if vulnerability_heart_&_life 0 is triggered by ingestion_poison 0 , its aggreg-parts vulnerability_heart 0 and heart_need 0 are both triggered, respectively by ingestion_poison 0 and the cardiac_arrest 0 that followed. More specifically, if an addcomplex is triggered by a process, then any of its add-parts is triggered by a part of this process:
In (Tablet), if solubility 0 is triggered by tablet 0 _in_solvent, its add-parts solubility 1 and solubility 2 are both triggered, respectively by the processes half-tablet 1 _in_solvent and half-tablet 2 _in_solvent (that are both parts of the process tablet 0 _in_solvent).
Note that replacing add-part_of by chain-part_of in the axiom (ADD-TRIGGER) does not lead to a true statement: as a matter of fact, when vulnerability_heart_&_life 0 is triggered by poison_ingestion 0 , the disposition heart_need 0 is triggered by cardiac_arrest 0 , which is not a part of poison_ingestion 0 .
However, on top of (AGGREG-TRIGGER), we can state that the trigger of a chaincomplex is the trigger of one of its chain-parts:
In (Poison), the trigger of vulnerability_heart_&_life 0 is poison_ingestion 0 , which is a trigger of vulnerability_heart 0 .
Note, however, that if an aggreg-part is triggered, this does not imply anything for the triggering of the aggreg-complex. For example, in (Tablet), if solubility 1 is triggered by half-tablet 1 _in_solvent, this does not imply the triggering of solubility 0 -as the solubility 2 of half-tablet 2 may not be triggered (in case half-tablet 2 is not put into a solvent). Similarly, in (Poison), Jones might undergo some cardiac arrest which is not due to the ingestion of poison, in which case heart_need 0 is triggered without vulnerability_heart_&_life 0 being triggered.
We will now show how we can use disposition-parthood relations to combine basic dispositions into mereological complexes, by developing case studies involving dominos.
Let's consider five dominos domino i (for any integer iÎ[0,4]) vertically placed on their short edge, such that they will fall if pushed. Each domino i is thus the bearer of a disposition unstability ¬i that is realized by domino i falling to the left (a process that is an instance of the defined class Fall ¬i ) and triggered when domino i is pushed to the left (Push ¬i ); and the disposition unstability ®i realized by domino i falling to the right (Fall ®i ) when domino i is pushed to the right (Push ®i ). It also has a disposition unstability ®i /¬i to fall to either side (Fall i equivalentTo (Fall ¬i or Fall ®i )) when pushed (Push i equivalentTo (Push ¬i or Push ®i )). We can characterize this by stating:
• unstability ®i /¬i realized_in only Fall i • unstability ®i /¬i has_trigger only Push i However, those two relations do not express the information that a Push ¬i always leads to a Fall ¬i , and that a Push ®i always leads to a Fall ®i . Therefore, it is more informative to say that unstability ®i /¬i has the two mod-parts unstability ®i and unstability ¬i and to ascribe the respective realizations and triggers to the mod-parts:
• unstability ®i mod-part_of unstability ®i /¬i • unstability ¬i mod-part_of unstability ®i /¬i
The five dominos 0 to 4 are arranged as illustrated on figure 1a below (dominos 5 to 8 from figure 1b will be considered later):
Figure 1b domino 2 is placed to the right (when seen from the domino's side) of domino 1 , such that when domino 1 falls to the right, it pushes domino 2 to the right; therefore, in this configuration, the endphase of an instance of Fall ®1 is also an instance of Push ®2 . Let's define unstability ®1,®2 as the disposition of domino 1 and domino 2 to fall to the right (a process named Fall ®1Ù®2 = Fall ®1 + Fall ®2 ) when domino 1 is pushed to the right (Push ®1 ). We have:
• unstability ®1 ,®2 realized_in only Fall ®1Ù®2 • unstability ®1 ,®2 has_trigger only Push ®1 However, those two relations do not express the information that there is a causal chain. Therefore, it is more informative to write that unstability ®1,®2 is a chain-complex composed by two chain-parts unstability ®1 and unstability ®2 :
Suppose that domino 3 is placed next to domino 1 , such that when one of these falls, it does not push the other. Then, the system domino 1+3 = domino 1 + domino 3 has a disposition unstability ®1:®3 of both dominos falling to the right (Fall ®1Ù®3 = Fall ®1 + Fall ®3 ) when both dominos are pushed to the right (Push ®1Ù®3 = Push ®1 + Push ®3 ):
• unstability ®1:®3 realized_in only Fall ®1Ù®3 • unstability ®1:®3 has_trigger only Push ®1Ù®3 However, those two relations do not express the information that when an instance push ®1 + push ®3 causes an instance of fall ®1 + fall ®3 , it is push ®1 that causes fall ®1 , and push ®3 that causes fall ®3 . Therefore, it is more informative to write that unstability ®1:®3 is an add-complex with two proper add-parts unstability ®1 and unstability ®3 :
• unstability ®1 add-part_of unstability ®1:®3
• unstability ®3 add-part_of unstability ®1:®3 domino 4 is placed to the right of domino 3 (like domino 2 was placed to the right of domino 1 ), such that when domino 3 falls to the right, it pushes domino 4 to the right. The disposition to Fall ®3Ù®4 when Push ®3 is noted unstability ®3,®4 .
Finally, domino 0 is placed to the left of domino 1 , such that when domino 1 falls to the left, it pushes domino 0 to the left. The chain-complex disposition to Fall ¬1Ù¬0 when Push ¬1 is noted unstability ¬1,¬0 .
We can now combine these dispositions into more complex dispositions. domino 1,2 = domino 1 + domino 2 falls to the right (Fall ®1Ù®2 ) when domino 1 is pushed to the right (Push ®1 ), and domino 1,0 falls to the left (Fall ¬1Ù¬0 ) when domino 1 is pushed to the left (Push ¬1 ). We can introduce the corresponding disposition unstability (®1,®2) / (¬1,¬0) , which is the disposition of [Fall ®1Ù®2 when Push ®1 ] and of [Fall ¬1Ù¬0 when Push ¬1 ] (see figure 2). That is, unstability (®1,®2) / (¬1,¬0) is a mod-complex composed by the mod-parts unstability ®1,®2 and unstability ¬1,¬0 :
• unstability ®1,®2 mod-part_of unstability (®1,®2) / (¬1,¬0)
• unstability ¬1,¬0 mod-part_of unstability (®1,®2) / (¬1,¬0) Moreover, by transitivity of the relation disp-part_of (of which mod-part_of and chain-part_of are subrelations), we can state that unstability (®1,®2) / (¬1,¬0) is a disp-complex composed by the disp-parts unstability ®1 , unstability ®2 , unstability ¬1 and unstability ¬0 . We have pointed out earlier that the bearer of a mod-part (resp. chain-part) is a part of the bearer of a mod-complex (resp. chain-complex), but not necessarily the same bearer; as a matter of fact:
• The bearer of the chain-complex unstability ¬1,¬0 is domino 1,0 ; whereas the bearer of its chain-parts unstability ¬1 and unstability ¬0 are domino 1 and domino 0 , respectively. • The bearer of the mod-complex unstability (¬1,¬0) / (®1,®2) is domino 0,1,2 ;
whereas the bearers of its mod-parts unstability (¬1,¬0) and unstability (®1,®2) are domino 1,0 and domino 1,2 , respectively.
We could also express a large variety of dispositions thanks to the various parthood relations introduced here. For example, domino 1,3 has a disposition to [Fall ®1Ù®3 when Push ®1Ù®3 ] and to [Fall ¬1Ù¬3 when Push ¬1Ù¬3 ], which is the mod-complex unstability (®1:®3) / (¬1:¬3) , composed by the add-complexes unstability ®1:®3 and unstability ¬1:¬3 .
To illustrate other configurations, consider the domino arrangement on Figure 1b. domino 5 is placed on the left of domino 6 and domino 7 , in-between, such that if domino 5 falls to the right, then both domino 6 and domino 7 also fall to the right. There is an underlying disposition to Fall ®5Ù®6Ù®7 when Push ®5 , which is the add-complex unstability (®5,®6):(®5,®7) composed by the chain-complexes unstability ®5,®6 and unstability ®5,®7 . Now, domino 8 is placed on the right of domino 6 and domino 7 , such that if either domino 6 or domino 7 falls to the right, then domino 8 also falls to the right. The underlying disposition to [Fall ®6Ù®8 when Push ®6 ] and to [Fall ®7Ù®8 when Push ®7 ] is the mod-complex unstability (®6,®8) / (®7,®8) composed by the chain-complexes unstability ®6,®8 and unstability ®7 ,®8 .
Suppose now that domino 8 is more massive, such that it needs both domino 6 and domino 7 to fall to the right for domino 8 to also fall to the right. The underlying disposition to Fall ®6Ù®7Ù®8 when Push ®6Ù®7 is the chain-complex unstability (®6:®7) , ®8 composed by the add-complex unstability ®6:®7 as well as by unstability ®8 .
Finally, we could compose dispositions in more intricate ways, mixing mod-part_of, add-part_of and chain-part_of -consider for example the disposition unstability [(®6:®7) , ®8] / [(¬6:¬7) , ¬5] to [Fall ®6Ù®7Ù®8 when Push ®6Ù®7 ] and to [Fall ¬6Ù¬7Ù¬5 when Push ¬6Ù¬7 ].
This paper has presented different ways to compose dispositions together: mod-parthood and aggreg-parthood, and the subrelations of the latter add-parthood and chain-parthood. All of them are subsumed by disp-parthood, which satisfy some axioms (DISP-BEARER), (DISP-TRIGGER) and (DISP-REALIZATION); and each of those subrelations also satisfy more specific axioms.
All these relations of disposition-parthood are partial orders: they are reflexive, transitive and anti-symmetric. To be authentic mereological relations, however, they also need to satisfy axioms of composition and decomposition (see [4] for a few preliminary remarks concerning mod-part_of). It can easily be seen that composition axioms can be introduced at least for some disposition-parthood relations. For any two dispositions d 1 and d 2 (with bearers b 1 and b 2 ) there exists a mod-complex noted (d 1 /d 2 ) and an add-complex noted (d 1 :d 2 ), which inheres in b 1 +b 2 . On the other hand, there exist a chain complex (d 1 ,d 2 ) (also inhering in b 1 +b 2 ) only if the maximally specified realization class of d 1 has as part the class of triggers of d 2 . By introducing such axioms of composition, we commit ourselves to a lot of strange disposition complexes, such as complexes of dispositions of different bearers, that might even exist at different times. This, however, is no special feature of disp-parthood: the same phenomenon occurs in standard mereology. Moreover, the formal commitment to the existence of these mereological sums does not imply any commitment that all of them have causal or explanatory relevance.
Consider now axioms of decomposition. Although this would require further investigations, the following axiom of complementation [6] seems to be satisfied for P being any of the five relation of disposition-parthood mentioned above, and O the associated relation of overlapping: ¬Pyx → ∃z ∀w (Pwz ↔ (Pwy Ù ¬Owx)) Note that our taxonomy of relations of disposition-parthood is not meant to be exhaustive. For example, one could consider the disposition realized by (only) domino 2 falling to the right when domino 1 is pushed to the right -not to be confused with the above disposition unstability ®1,®2 realized by both domino 1 and domino 2 falling to the right when domino 1 is pushed to the right.
By way of conclusion we point out that we expect disposition-parthood to be a powerful tool to explore relations between medical entities that have been formalized as dispositions, such as diseases [7,8] or medical risks [9].
AB acknowledges financial support by the "Bourse de fellowship du département de médecine de l'université de Sherbrooke" and the CIHR-funded Quebec SPOR Support Unit.