p([n], part, universal(X), universal(X)).

For example, in BFO the universal *object* is defined as a "maximal causally unified material entity". In the trope ontology, if my watch is asserted as an object, we would use the following predicates to represent this:

<sup>7</sup> BFO version 2 would define such a property more precisely as continuant\_part\_of, but our point remains the same.

<sup>8</sup> We have to take care not to fall afoul of self-referential regress, but will not address that detail here.

<sup>9</sup> Actually, it is not entirely clear whether BFO counts "part of". The specification says such relations are not first-class citizens (i.e., entities), but does not say what that means in terms of existence.

p([1], part, 'my watch', objects).

p([2], part, universal(object), universal(object)).

The reflexive declaration of the universal *object* simply establishes the existence of the relation under which the individuals (like my watch) are part of the mereological sum of objects. As such, there is an unstated, but implied and axiomatic inference from the universal (noted as a singular term, *object*) to the corresponding mereological sum (noted as a plural term, like *objects*). The rule can be schematically stated as follows.

$$p(\\_\text{part}, \text{universal}(X'), \text{universal}(X')) \longrightarrow p(\\_\text{part}, X, \\_\text{) or } p(\\_\text{part}, \\_\text{X}). \tag{3}$$

where *X'* is the singular form of *X*10.

In other words, the trope ontology keeps track of the extension of the universal through the corresponding mereological sum. Thus, the transitive property of the subtype relation of the universal is inferable through the transitivity of parthood relations. For instance, a further assertion that "An object is a material entity", would add the following relations to the ontology:

p([3], part, objects, material-entities).

p([4], part, universal(material-entity), universal(material-entity)).

It can be readily seen that:


Therefore, it also the case that, by transitivity of the subtype relation, my watch belongs to the universal *material-entity*.

These assertions of the existence of special relations enables us to consistently match BFO's universals with mereological sums in the trope ontology. Note that BFO, at least in its OWL 2 implementation, uses *classes* in a similar way to keep track of the transitive properties of universals [22].

The account so far enables us to match (mereological) tropes to corresponding general relations and universals. Next, we need to look at the specific universals that BFO commits to.

## **4. Accounting for Continuants and Occurrents**

BFO carves entities into *continuants*, which retain their identity over time, and *occurrents*, which have temporal parts. For example, John (a particular human) remains himself over time (i.e., John is a continuant), but John's life (an occurrent) has di fferent parts from time to time. That does not mean that John exists all the time. John can *exist at* a particular time, but whenever John exists, then he exists entirely as John. On the other hand, John's life has di fferent parts in each time period: he is born, he eats a meal, he has a birthday party, etc. Each of these experiences is an *occurrent part of* his life. To complete this picture, it may also be the case that John (the continuant entity) has di fferent parts at di fferent times. For example, he may have a beard today, but be clean shaven tomorrow—so, something may be *part of* a continuant *at a time*. Occurrents like processes or events may also have parts, but we do not need to specify "at some time", because occurrents come with time built-in, so to speak. In other words, a process stretches over time and any part of that process occupies a fragment of that time. Lastly, we make the connection between continuants and occurrents by allowing an occurrent to *have a participant*. So, John participates in John's life.

<sup>10</sup> Note that the distinction between plural and singular is not formally necessary, but a naming convention maintained for consistency with BFO's naming convention of universals as singular.

By contrast, the trope ontology asserts a primitive causal relation that may exist between entities, or more precisely between relations of entities11. So, John's life comprises the sum of causally linked stages of John, where those stages are defined by individual relations. Note that in the trope ontology, a continuant can be the mereological sum of all its parts, without the causal separation of those parts. For example, there may be a relation of "John having a natural right hand" and another of "John having an artificial right hand", with a causal link between them (where one relation replaces the other). In this case, if we *ignore* the causal link, then the mereological sum of John (i.e., the continuant) has both a natural right hand, as well as an artificial right hand—but that seemingly strange sum in the trope ontology only exists when one "sums up" without regard for the causal relations. This overly broad summing up would be tantamount to saying, "what are all the right hands that John has, regardless of time".

The existence of an occurrent without regard of time cannot match to BFO's view of an occurrent existing in its entirety through time. The reason is that parthoods that occur at particular times in BFO are always specified with the temporal designator (e.g., "John's hand is part of John at time t"). As such, one can only ge<sup>t</sup> a sum of *John at a certain time*, ensuring the impossibility of a sum of John with *and* without an artificial hand in BFO. Given this disparity in views of what constitutes the sum of John, the matching from the trope ontology to BFO should either exclude parts that are impermanent, or always attach a temporal property to parts.

To illustrate the preceding treatment of events as named causal sequences, consider the following sentences (to avoid noisy punctuation, we are using lowercase names for entities):

john is part of melbourne, because john was part of sydney;

and john intended that john will be part of melbourne.

The process from john was part of sydney upto john is part of melbourne is the move of john.

These sentences translate into predicate form in the ontology text as follows (where terms like "the move of john" are transformed into functions like *move*(*john*) and we assume the existence of certain universals):

**p**([5], part, john, sydney).

**p**([1], part, john, melbourne).

**p**([30:32], part, john, melbourne).

**p**([31], part, intending(30), john).

**p**([33], cause, [5], [1]).

**p**([34], cause, [31], []).1

**p**([35], part, 'cause\*'[[5]],[[1]]), move(john)).

**p**([37], part, move(john), moves).

**p**([39], part, universal(move), universal(move)).

Predicate [35] represents the notion that the causal chain is part of an occurrence called the "move of john" (i.e., *move*(*john*) in functional form). We use a special predicate called 'cause\*' to indicate that the first argumen<sup>t</sup> refers to a chain of relations starting with predicate [5] and ending with predicate [1]. Once the chain of events is a "named" entity, that entity can then be treated like any other entity. For example, the last two predicates reflect that the "move of john" is part of the mereological

<sup>11</sup> Keeping in mind that tropes are an integral complex of a relation and the entities it relates.

sum of *moves*. In other words, John's move is a *move*, where *move* is a universal that ultimately would be an *occurrent* in BFO's terms.

Note that the trope ontology does not use a time argumen<sup>t</sup> in its occurrences. Rather, causal chains are demarcated by the events they span. One can calculate a time metric based on the lengths of chains, but we can speak about occurrences entirely without needing explicit time parameters. This means that matching from the trope ontology causal chains to BFO occurrence membership is straightforward, but not so much for matching to participation or parthood relations that have time parameters. At the minimum, we would need to pick a time reference point, but we might also need an explicit causal relation in BFO (if that could be countenanced). Moreover, causal chains allow different time metrics along different chains (think of the mind experiment of the astronaut on a fast extra-terrestrial trip who experiences a different length of time than people on earth). In other words, while we can match to occurrence universals in a straightforward way, the matching of corresponding relations depends on the specific scope of the ontology.

On the other hand, occurrents as causal chains does provide for a straightforward way to implement process boundaries and regions. That is, the relations that demarcate the causal chain can also be used to demarcate process boundaries. The further matching of regions would mirror the matching of immaterial entities that is discussed in the next section.
