Adrien Barton
email: adrien.barton@irit.fr.

Fumiaki Toyoshima

Jean-François Ethier
email: ethierj@gmail.com.

Clinical Documents and Their Parts

Keywords: Mereology, Document structure, Informational slot, Information content entity, Drug prescription, Granularity

The medical world is replete with documents that ontologies can help to analyze and disambiguate. This paper provides a mereological analysis of such documents as structured by informational slots and fillers, completing recent work by our teams inspired by Bennett. It introduces the notion of adequate filling of an informational slot, and the representation of the layered structure of a document. An application is provided on the Prescription of drugs ontology (PDRO).

Introduction

The medical world is replete with documents that ontologies can help to analyze and disambiguate, enabling better data sharing: diagnosis sheets [START_REF] Schulz | An Ontological Analysis of Reference in Health Record Statements[END_REF], drug prescriptions [START_REF] Ethier | An ontological analysis of drug prescriptions[END_REF], drug dispensing reports [START_REF] Barton | A classification of instructions in drug prescriptions and pharmacist documents[END_REF], laboratory test prescription and laboratory test reporting documents [START_REF] Barton | LABO: An ontology for laboratory test prescription and reporting[END_REF], consent forms [START_REF] Lin | Development of a BFO-based informed consent ontology (ICO)[END_REF], questionnaires and surveys [START_REF] Fabry | QUESTO -An ontology for questionnaire[END_REF], etc. Important information is typically extracted from such clinical documents, such as the drugs that a patient is likely to take, or the medical conditions he is likely to have. Nevertheless, the semantics of the extracted information is often derived from implicit knowledge about the structure of the document itself. This can lead to potentially serious errors of interpretation. Therefore, an ontological analysis of clinical documents can produce fruitful, practical and significant results in terms of quality of care. For example, it is important to be able to distinguish the information 'diabetes' when it plays the role of a therapeutic indication in a drug prescription from the information 'diabetes' when it describes a past medical condition of a patient: intuitively, both have something in common (the content 'diabetes'), and something different (two different clinical roles played by this content). Additionally, it would be desirable to be able to represent unfilled templates of documents, or partially filled document.

Ontological foundations for documents have been proposed [START_REF] Brochhausen | Towards a formal representation of document acts and resulting legal entities[END_REF][8] based on the Basic Formal Ontology (BFO) [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF] and the Information Artifact Ontology (IAO) [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF]. Recent work has axiomatized the mereological structure of informational entities [START_REF] Barton | The mereological structure of informational entities[END_REF], reconsidering classical mereological systems that are traditionally used in formal ontology. This paper will build upon this work by introducing additional important notions, and showing how those can be used to analyze the mereological structure of clinical documents. Those new notions include the distinction between the adequate and inadequate filling of documents or parts or documents; and the stratification of document parts in hierarchized sublevels, that enables to express some axioms on how specific kinds of clinical documents should be filled and understood. Those notions will be first introduced on a toy example of a past medical history document, and later applied to the ontology of drug prescriptions PDRO.

The Mereological Structure of Informational Entities

Former Work

The axiomatic system will be expressed in FOL, and translated in OWL 2 when possible, using the OWL Manchester Syntax. Classes from the Open Biomedical Ontologies (OBO) Foundry [START_REF] Smith | The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration[END_REF] will be maximally re-used, such as IAO:Information content entity (the prefix indicates the ontology from which the term is extracted).

Classical mereology systems rest on the idea that an entity can only have a part once [START_REF] Effingham | A mereological challenge to endurantism[END_REF]. However, some entities can have a part twice (or more) over, such as universals (e.g. the universal of methane having the universal of hydrogen as part four times [14] [15]). Informational entity particulars share this characteristic with universals: consider the chain of characters 'aa' that has the same letter 'a' twice over, or the sentence 'A cat is a cat.' that has the same word 'cat' twice over. To account for this, we proposed [START_REF] Barton | The mereological structure of informational entities[END_REF] an axiomatization of the mereological structure of documents adapted from Bennett's work [START_REF] Bennett | Having a part twice over[END_REF].

This system introduces the unary predicate IS = "is a slot" and IF = "is a filler", that will be transposed here in OWL as the classes Information slot and Information filler. In a way that is compliant with IAO [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF], fillers and slots are both information content entities, which are defined and brought into existence by some cognitive acts (usually coordinated inside a group whose members share a semiotic system [START_REF] Barton | The mereological structure of informational entities[END_REF] [START_REF] Bateman | Ontology, Language, Meaning: Semiotic Steps Beyond the Information Artifact[END_REF]; see also section 6.2).

It also introduces the binary predicate F, S and P, where Sty means that t is a slot of y (where both fillers and slots can have slots), Fxt means that x fills t, and Pxy means that x is a part of y. Those will be written in OWL respectively as the relations fills (inverse: filled_by), slot_of (inverse: has_slot) and part_of (inverse: has_part).

As explained in [START_REF] Barton | The mereological structure of informational entities[END_REF], this axiomatic system is built on a domain of fillers (with the associate predicate IFx:=def ∃t Fxt) and slots (with the associate predicate IS), which are disjoint: IFx → ¬ISx. Only slots are slots of something (Stx → ISt), but some slots may not be a slot of anything (on the other hand, by definition, every filler fills a slot). Therefore, only slots are filled, and only fillers can fill: Fxt → (ISt & IFx) (but not all slots need to be filled: there can be "empty" slots, as when a slot structure has already been decided for a document that is not filled yet). Proper parthood is defined as filling a slot of a filler (PPxy:=def IFy & ∃s (Sty & Fxt)) and parthood is defined the classical way on this basis (Pxy:=def IFx & [PPxy ∨ (x=y)]). Axioms are introduced to ensure that no entity fills any of its slots (¬(Stx & Fxt)), that there is at most one filler for a given slot (Fyt & Fzt → y=z), and that S is a strict order relation.

The following axiom (named "AX8" in [START_REF] Barton | The mereological structure of informational entities[END_REF]) was accepted: if a filler x fills a slot t, any slot of x is a slot of t, and vice versa: Fxt → (Sux ↔ Sut). From this, a theorem of slot inheritance could be derived, that states that slots of a part of an entity are slots of that entity too: (Stx & Pxy) → Sty. On this basis, P was proven to be a partial order relation, in line with the classical view of parthood [START_REF] Simons | Parts: A study in ontology[END_REF] (and P is also taken to satisfy a specific axiom akin to weak supplementation, but involving S).

Example

Suppose that at Princeton-Plainsboro Teaching Hospital (abbreviated PPTH), all past medical history documents have the same structure as pmhd1='patient1[] condition1[]', where 'patient1[]' is a particular slot supposed to be filled with a patient name, and 'condition1[]' is a particular slot supposed to be filled with the name of a medical condition (for example a disease, a disorder or a pathological process, following the ontology OGMS [START_REF] Scheuermann | Toward an ontological treatment of disease and diagnosis[END_REF]) that the patient got in the past (for the sake of simplicity, we omit slots that would realistically need to be present, such as a slot to be filled by the date at which the condition was supposed to hold, or by the name of the doctor who wrote the document; we also suppose that each patient referred in the system has only one past condition).

Consider now that pmhd1 is filled such that it reads 'John Doe / flu'. Both its structure and content could be described with the following notation: The apparatus of slots and fillers enables an informational entity to have a part twice over. Suppose for example that past medical history documents at PPTH would have an additional field to be filled with the name of the doctor writing the prescription, such that a particular document would read 'patient1['John Doe'] condition1['flu'] doctor1['Gregory House']'. Suppose now that Dr. House diagnoses that he has himself a curmudgeon personality, and fills accordingly pmhd2 = 'patient2['Gregory House'] condition2['curmudgeon personality'] doctor2['Gregory House']' (although such selfdiagnosis would not be possible in most real-world medical jurisdictions). Then the same information filler 'Gregory House' fills two slots, namely 'patient2[]' and 'doctor2[]'.

Adequate and Generalized Filling

A Taxonomy of Relevant Entities

We will introduce the classes as represented on Figure 1, that will be explained below (see [START_REF] Barton | The mereological structure of informational entities[END_REF] and Section 6.2 below for a discussion of instances of Information slot carrying aboutness, and thus this class being a subclass of Information content entity). Note that an IAO:Document is considered as an Information filler, and thus fills a slot. Indeed, this class is defined as "A collection of information content entities intended to be understood together as a whole", and the slot it fills is precisely what makes that this collection of information content entities is to be understood as a whole (see [START_REF] Barton | The mereological structure of informational entities[END_REF] for more discussion on this point).

Given what we said earlier, we can already constrain the slot structure of a PPTH past medical history document as follows:

PPTH past medical history document SubClassOf Document and has_slot exactly 1 Patient slot and has_slot exactly 1 Condition slot

We will now move to characterize slots depending on whether they are adequately filled or not.

IAO:Information content entity Information filler (IF) IAO:Document PPTH past medical history document Human name PPTH patient name Clinical condition name Information slot (IS) Adequately filled slot Patient slot Adequately filled patient slot Condition slot

Adequately filled condition slot

Adequate and Generalized Filling

The notion of filling needs to be refined. As a matter of fact, there are three different ways in which a document can be inadequately filled. Suppose that the past medical history documents at PPTH hospital have the structure presented on Figure 2 (namely the structure described earlier, with the additional constraint that the slot for the patient name has two slots, one for the first name and one for the last name). This is what we will call "structural inadequacy": a slot of the document is not filled. Second, suppose that Dr. House fills condition1[] with a filler that is not of the expected kind, such as 'Gregory House' (see section 6.2 for a more general discussion of what we mean here by "expected kind"). Here, 'Gregory House' is a name of a human person, whereas we would expect condition1[] to be filled with a name of medical condition. This is what we call "semantic inadequacy". Third, Dr. House might fill the document as pictured on Figure 1, but still commit a mistake, if, in fact, John Doe never had the flu in the past. This is what we call "descriptive inadequacy".

The notion F we introduced earlier should be understood as a notion of "adequate filling" in the structural and semantic senses mentioned above (for a short discussion whether it should include descriptively adequate filling, see section 6.2). However, it can be completed by a relation F G of generalized filling, used when a filler fills a slot, whether adequately or not (such that in particular, Fxt → F G xt). On this basis, we can introduce the notion of generalized information filler (IF G ) as something that generally fills a slot (rather than adequately fills it): IF We can adapt the axioms previously exposed for F to F G and state that no entity generally fills any of its slots (¬(Stx & F G xt)) and that there is at most one general filler for a given slot ((F G yt & F G zt) → y=z). However, we do not transpose the axiom AX8 for general fillers: a general filler and the slot it fills do not necessarily have identical slot structures (for example, 'Gregory House' and 'condition1[]' do not have the same slot structure when the former structurally inadequately fills the later; Section 5.2 will explain why this would be problematic). We accept axioms stating that P G is antisymmetric and transitive (and it is trivially reflexive), and thus is a partial order (we will not discuss here whether P G should satisfy supplementation axioms similar to those satisfied by P).

Our ontology admits empty slots: if the community at PPTH has decided that all past medical history documents should have one patient slot and one condition slot, then empty past medical history documents already have ipso facto this structure, even before they are filled [START_REF] Barton | The mereological structure of informational entities[END_REF]. Therefore, not all slots need to be generally filled (and obviously, also not adequately filled).

Axiomatically, we cannot state that a patient slot can only be generally filled by a human name, since as we saw above, an agent might by mistake fill a patient slot with, say, a condition name. However, we can state that a patient slot can only be adequately filled by a human name. More generally, we could introduce axioms constraining what could be adequate fillers for each kind of slot:

Patient slot SubClassOf filled_by only PPTH patient name Condition slot SubClassOf filled_by only Clinical condition name We can then introduce the class of Adequately filled slot:

Adequately filled slot EquivalentTo Information slot and (filled_by some Information filler) Note that an adequately filled slot is not the "composition" (in whatever sense of this term) of a slot with its adequate filling; it is a slot that is adequately filled. We have:

Adequately filled patient slot EquivalentTo Patient slot and (filled_by some PPTH patient name) Adequately filled condition slot EquivalentTo Condition slot and (filled_by some Clinical condition name) A reasoner would deduce from those two axioms that Adequately filled patient slot and Adequately filled condition slot are both subclasses of Adequately filled slot.

Direct Slots and Direct Parts

Definitions

To clarify the slot structure of a document, we can define a hierarchy of levels among slots. Consider a past medical history document as represented on Let's write the relation "direct slot of" S D in FOL and direct_slot_of in OWL. Then we can say that a direct slot of x (where x can be either a slot or a filler) is a slot of x that is not a slot of a slot of x:

(DEF1) Direct slot of S D tx:=def Stx & ¬[∃u (Stu & Sux)]
We will call the inverse relation "has direct slot" and write it "has_direct_slot" in OWL.

We can then define the relation "direct proper part of" P D ("direct_proper_part_of" in OWL) as adequately 3 However, a filled PPTH past medical history document can have other slots, as long as they are not direct slots -such as the slot 'last_name JD[]' inherited from the filler 'John Doe' that fills its patient slot.

Axiomatization of Direct Slot and Direct Parthood

Let us now propose an axiomatization of the new relations we introduced, namely S D (direct_slot_of) and PP D (direct_proper_part_of). Note that many of those axioms are not (fully) representable in OWL (see [START_REF] Keet | Orchestrating a Network of Mereo (topo) logical Theories[END_REF] for a discussion), so we will write them in FOL.

Axiomatization of Direct Slot

Since S is irreflexive and asymmetric (and S D implies S), S D also trivially is: (THE1) Direct-slot irreflexivity ¬S D tt (THE2) Direct-slot asymmetry S D ut → ¬S D tu However, although S is transitive, S D is anti-transitive: From AX8 that states that the slots of an adequate filler are identical to the slots of the filled slot, we can prove that the direct slots of a filler are identical to the direct slots of the adequately filled slots (that is, a slot and its adequate filler have the same direct slot structure): We have shown in [START_REF] Barton | The mereological structure of informational entities[END_REF] that a slot of a part of an entity is a slot of this entity. However, this slot cannot be a direct slot of this entity: (THE5)

Axiomatization of Direct Parthood

Note first that since S D is a subrelation of S, direct parthood trivially implies proper parthood: (THE6) Direct proper parthood implies proper parthood PP D xy → PPxy As we have seen, PP is irreflexive, asymmetric and transitive. Therefore, from THE6, we deduce that PP D (direct_proper_part_of) is also irreflexive and asymmetric: In addition to its usefulness for writing universal restrictions, such a non-transitive relation might be useful to mitigate undecidability issues when cardinality restrictions are used [START_REF] Baader | Expressive number restrictions in description logics[END_REF].

Application to Drug Prescriptions

Let us now illustrate how those notions can be useful on an actual ontology, the Prescriptions of Drug Ontology (PDRO) [START_REF] Ethier | An ontological analysis of drug prescriptions[END_REF]. PDRO is a candidate ontology to the OBO Foundry [START_REF] Smith | The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration[END_REF], built in compliance with the realist methodology. It specifies generally the structure of drug prescriptions through some relevant classes constrained by some axioms. We will describe here how PDRO could be enriched by using the notions of slots, adequate filling and direct slots and parts.

Using Slots to Structure Drug Prescriptions

Consider an example of prescription written by Dr. House to John Doe, similar to the one presented in [START_REF] Ethier | An ontological analysis of drug prescriptions[END_REF], but structured using slots as in Figure 3: 'Amoxicilin 500 mg PO q8h x 7 days'.

Using Universal Quantifier with Direct Slot and Direct Part

In practice, drug prescriptions are further constrained according to the areas (e.g. countries) in which they are written. In particular, in order to direct the implementation of e-prescribing in Quebec using an application ontology, we need to constrain what kind of parts a drug prescription can have.

We want to formalize that a drug product specification in Quebec (that we will call here a Q-Drug product specification) can be composed only with instances of Drug active ingredient specification, Drug excipient specification, Drug dose form specification, Drug product brand name, or Drug Identification Number. However, we cannot use the relation BFO:has_part to express this universal restriction; just as a Drug product specification like 'Apo-Metoprolol 50mg tablet' has as part a Drug product brand name like 'Apo-Metoprolol', it has as part e.g. 'mg'. Thus, it is difficult, if not impossible, to make a list of all the possible parts of a Q-Drug product specification.

To address this representational issue, we can introduce relevant slots, and use the relation This explains why we refused to adapt the axiom AX8 for generalized filling; otherwise, an instance of Q-Drug product specification slot that would be inadequately filled by, say, 'John Doe', would have the same slots as its filler, namely one for the first name and one for the last name, and therefore axioms such as AX_QS would not hold.

Combining this with the notion of adequate filling we saw earlier, we can then list all the (adequate) direct parts an instance of Q-Drug product specification can only have: Those can be combined with axioms using existential quantification, for example stating that an adequately filled drug product specification in Quebec has as part an active ingredient specification or a drug product brand name, in order to specify the drug:

Q-drug
Adequately filled Q-drug product specification SubClassOf has_direct_proper_part some (Drug active ingredient specification or Drug product brand name)

Discussion

A number of issues pertaining to the mereological structure of informational entities have been discussed elsewhere [START_REF] Barton | The mereological structure of informational entities[END_REF] (including the possible relaxation of the axiom AX8, the diachronic identity of informational entities, and the introduction of mereological sums).

We will add here a couple of points.

Direct Parthood and Granularity

We introduced above the has_direct_slot and has_direct_proper_part relations to stratify the relation of parthood (between informational fillers) into several levels. This is a way to address the vexed problem of how to represent consistently granular levels of reality, in the particular case of informational entities (see e.g. Vogt's [START_REF] Vogt | Levels and building blocks-toward a domain granularity framework for the life sciences[END_REF] BFO-inspired domain granularity framework for the life sciences). Note however that levels between informational entities can be determined by a community of users of a semiotic system [START_REF] Bateman | Ontology, Language, Meaning: Semiotic Steps Beyond the Information Artifact[END_REF], since such levels are defined by cognitive acts that can be coordinated in a community, as explained earlier; whereas those between material objects (e.g. between collections of cells and organisms) presumably are not defined by such cognitive acts.

Aboutness, Semantic Adequacy and Descriptive Adequacy

We can now suggest a few considerations on aboutness for slots and fillers, although this question deserves more attention in future works. Note that cognitive acts are considered in IAO as providing intentionality to information content entities [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF]. Since cognitive acts also bring information slots into existence, as mentioned earlier, it is not surprising that they can also provide intentionality to those, the same way they do for information fillers. In IAO [START_REF] Smith | Aboutness: Towards foundations for the information artifact ontology[END_REF], ICEs can be about individuals or classes: 'John Doe', for example, is about the human person John Doe; and 'Flu' would be about the Flu class. Similarly, information slots can be about a class. 'patient0[]', for example, would be about the class of patients from the PPTH hospital. This explains why we classified Information slot as a subclass of ICE.

This theory of aboutness need to be developed more systematically to express formally the normative constraints on adequate filling that were mentioned above. In particular, the semantic adequacy requirement for a filler was expressed as being about an entity of the "expected kind". We can clarify what we mean by this with two paradigmatic cases. A first case is that 'John Doe' is a semantically adequate filler of patient0[] because 'John Doe' is about a particular (the human person John Doe) who is an instance of the class referred to (we take "refer to" and "being about" as synonymous here) by patient0[], namely the class of patients from the PPTH hospital. A second case is that 'Flu' is a semantically adequate filler of condition0[] because it is about a class (the class of flu diseases) that has a non-empty intersection with the class referred to by condition0[] (namely the class of human medical conditions; note that the former is not a subclass of the latter: the class of flu diseases encompasses also non-human medical condition, as some non-human animals can get the flu). A more systematic theory of semantic adequacy will depend on the details of the endorsed theory of aboutness.

A theory of aboutness could also clarify the link between descriptive adequacy and semantic adequacy. A filler is said to be descriptively adequate if it describes a state of affairs that obtained in the past; for example, 'John Doe / Flu' is descriptively adequate if it refers to a state of affair of John Doe having got the flu in the past. But descriptive adequacy might be seen as a form of semantic adequacy. For example, the slot past_history0[] might be about the class of clinical state of affairs that occurred in the past; and 'John Doe / Flu' is a descriptively adequate filler if it refers to a state of affair (John Doe having had the flu) that is an instance of the class of clinical state of affairs that obtained in the past (assuming that we would accept an ontology of state of affairs, which is not the case currently in BFO). Thus, this would match with our definition of semantic adequacy. Here again, interpreting descriptive adequacy as a kind of semantic adequacy depends on the details of the theory of aboutness accepted. Moreover, even if descriptive adequacy can be interpreted as a kind of semantic adequacy, it is epistemically much more demanding to evaluate some descriptive adequacy than to evaluate, say, the semantic adequacy of a filler of condition0[]. Indeed, it is relatively easy to check whether a filler refers to a human condition (by having a dictionary or terminology of human conditions); on the other hand, it is much more difficult (or even impossible, if one is a strict Bayesian) to be sure that John Doe indeed got the flu in the past.

Conclusion

We have here shown how the mereology of informational entities proposed in [START_REF] Barton | The mereological structure of informational entities[END_REF] could be extended with a normative notion of adequate filling, and a hierarchical notion of direct slots and parts. Both are useful tools to represent clinical documents and their parts, and could also be used to represent documents in non-clinical domains. Systematic theories of aboutness of information fillers and information slots need to be developed in the future to represent axiomatically the notion of adequate filling. The deontic import of some slot (e.g., a signature slot) should also be investigated in the future. The articulation of this ontological system with relational databases could be investigated. As a matter of fact, a mapping might be made between an attribute in a relational database and a class of slots (e.g., an attribute describing therapeutic indications might correspond to a class of slots that each refer to the class of therapeutic indications); whereas identical values in a relational database's fields might correspond to the same filler (e.g. all values of 'diabetes' could be retrieved, independently of whether they describe a therapeutic indication or a past medical condition). Finally, similarities with the type theory in computer science might provide some insight for this articulation with computer systems.

  pmhd1='patient1['John Doe'] condition1['flu']', equivalent to the following facts: fills('John Doe', 'patient1[]') ; fills('flu', 'condition1[]') slot_of('patient1[]', pmhd1) ; slot_of('condition1[]', pmhd1) One can then deduce the following facts from the definition of part_of: part_of('John Doe', pmhd1) ; part_of('flu', pmhd1) To illustrate the above-mentioned axiom AX8 with this example, if 'John Doe' fills the slot 'patient1[]', and 'last nameJD[]' is a slot of 'John Doe' (filled by 'Doe'), then 'last nameJD[]' is also a slot of 'patient1[]'.

Figure 1 .

 1 Figure 1. A taxonomy of relevant classes

Figure 2 .

 2 Figure 2. A past medical history document at PPTH hospital First, suppose that Dr. House leaves the slot last_name1[] unfilled by simply filling medical_history1[] with 'John / Flu'.This is what we will call "structural inadequacy": a slot of the document is not filled. Second, suppose that Dr. House fills condition1[] with a filler that is not of the expected kind, such as 'Gregory House' (see section 6.2 for a more general discussion of what we mean here by "expected kind"). Here, 'Gregory House' is a name of a human person, whereas we would expect condition1[] to be filled with a name of medical condition. This is what we call "semantic inadequacy". Third, Dr. House might fill the document as pictured on Figure1, but still commit a mistake, if, in fact, John Doe never had the flu in the past. This is what we call "descriptive inadequacy".

  G x:=def ∃t F G xt. Trivially, an adequate filler is a generalized filler: IFx → IF G x. We can then define a new notion of generalized proper parthood (PP G ) as generally filling a slot (PP G xy:=def IF G y & ∃t (Sty & F G xt))), and define generalized parthood (P G ) between generalized fillers as being a generalized proper part or identical (P G xy:=def IF G x & [PP G xy ∨ (x=y)]).

Figure 1 .

 1 Note that by slot-transitivity, 'first_name1[]' and 'last_name1[]' are slots of 'medical_history1[]'. But we might want to state that the slots 'patient1[]' and 'condition1[]' are direct slots of 'medical_history1[]', whereas the slots 'first_name1[]' and 'last_name1[]' are not (they are direct slots of 'patient1[]', however).

(

  THE3) Direct-slot anti-transitivity S D xy & S D yz → ¬S D xz Proof: Suppose that S D xy & S D yz. Then, we do have Sxy & Syz. If S D xz, then for every u, ¬(Sxu & Suz): contradiction by taking u=y. Therefore, we have ¬S D xz.

( THE4 )

 THE4 Direct slots of an adequate filler are identical to direct slots of the filled slot Fxt → (S D ux ↔ S D ut) Proof: Suppose that Fxt. S D ux iff Sux & ¬[∃v, (Suv & Svx)] by definition of S D ux. iff Sut & ¬[∃v, (Suv & Svt)] by AX8. iff S D ut by definition of S D ut.

( THE7 )

 THE7 Direct proper parthood irreflexivity ¬PP D xx (THE8) Direct proper parthood asymmetry PP D xy → ¬PP D yx PP D is not transitive, as a direct part of a direct part of z may not be a direct part of z: consider a=t1[x] t2[u1[y] u2[z]] (where x, y and z are atomic), where z is not a direct part of a, but is a direct part of u1[y] u2[z], which is a direct part of a. However, PP D is also not anti-transitive, as a direct part of a direct part of b might be a direct part b: consider b=t1[x] t2[u1[x] u2[y]], where x is both a direct part of b and a direct part of a direct part of b -namely, u1[x] u2[y].

Figure 3 .

 3 Figure 3. The layered filled slot structure of a drug prescription.

  has_direct_slot to state: (AX_QS) Q-Drug product specification slot SubClassOf has_direct_slot only (Drug active ingredient specification slot or Drug dose form specification slot or Drug product brand name slot or Drug excipient specification slot or Drug Identification Number slot)

  product specification SubClassOf has_direct_proper_part only (Drug active ingredient specification or Drug dose form specification or Drug product brand name or Drug excipient specification or Drug Identification Number)

Note that we might have additionally defined generalized direct parthood as generally filling a direct slot, but such a notion will be less useful for our applied purposes that will be exposed in section 5.2.

Acknowledgements

We thank Laure Vieu, Ryeyan Taseen, Paul Fabry and Cédric Tarbouriech for useful discussions and insights on related topics, as well as audience at a seminar in Université de Sherbrooke in July 2020. We also thank two reviewers for their useful feedback. FT acknowledges financial support by the SPOR Canadian Data Platform (CIHR).