GFO: The General Formal Ontology

The category Time comprises (a.o.) instances called chronoids, which may be understood as closely similar to real-valued intervals with endpoints. The category Space, called phenomenal space, exhibits space regions as instances of primary importance. While every space region has an extension, there is an idealization of a greatest space region, containing any space region as part. Phenomenal space as such does not carry any metric. If we equip that idealized greatest space region with metrics, we can get various metric spaces, including Euclidian Space. We defend the position that the categories of phenomenal space and time should not be conflated into a homogeneous four-dimensional space-time continuum. The phenomena exhibited by space and time are different and there arises the problem of how these categories can be integrated into a uniform system. We believe that the integration law of GFO, see axiom A18 in Section 3.2, yields also a step forward to the solution of this integration problem.

To cover all relevant aspects of space-time that are fundamental in physics, a further space-time theory will be needed, as a four-dimensional space-time. The notion of space and time as occurring in physics differs from the phenomenal space and time in GFO. Yet there is a transformation from the theory GFO-Space (Baumann et al., 2016) into the mathematical space of physics, by extending GFO-Space to a pseudo-metric space from which, in turn, a metric space can be constructed. GFO 2.0 (the next version of GFO) will contain a further space-time module that is related to physics along these lines.4

Concrete individuals can be distinguished with respect to the type of relation that they have to time and space. According to the type of this relation, spatiotemporal individuals are classified into continuants (also called endurants), presentials and processes. Continuants persist through time and are wholly present at every time point of their lifetime.5

There is a complementary duality between processes and presentials: processes can never be wholly present at time points, whereas presentials possess this property. GFO provides a basic classification of processes that is related to the changes that occur in the process. A process – as a whole – cannot change, though it can possess changes, classifications of which have been examined earlier (Herre et al., 2007, Section 8.3; Herre, 2010, Section 14.4.4). Processes form a subclass of processual complexes, which are the most general kind of concrete individuals that have a temporal extension. The temporal extension of a processual complex is a mereological sum of a non-empty set of chronoids. Cohesive processes form an important subclass of all processes: a process is cohesive if any two process boundaries are causally connected, if their temporal boundaries coincide (Michalek, 2009, esp. Section 5.2). Some examples of processes include: the locomotion of a material object; sitting in front of a computer, viewed as a state extended in time; the development of a cancer; a hundred-meter-run.

There are many concrete entities that depend on processes in various ways.7

Material situations are parts of the material world (at the macroscopic level) that can be consistently comprehended as a whole. We distinguish three kinds of situations: object situations, presentic situations, and situoids. Object situations consist of material objects and relations between them. A presentic situation is a snapshot of an object situation, and a situoid (or processual situation) is a part of the spatiotemporal world that arises from an object situation if all objects are replaced by the corresponding processes. An example of a situoid is a football match, happening in time, and including all necessary participating entities, among them the players, the football, the goals and other entities, but also the localization and the corresponding environment. Finally, the notion of comprehended as a whole is used here in an informal manner. We consider this notion as primitive, which is not defined by other notions, but needs to be described in terms of axioms.

Complex individuals are composed of elementary ones or components. There are various relations of composition. One type of composition uses dependency relations that glue entities together, another uses the part-of relation and the construction of mereological sums.

OPEN IN VIEWER

Fig. 2.

Taxonomy of attributives (Herre, 2019, layout adapted), which serves as the basis for GFO-Data (Herre, 2016), an ontology for the foundation of an integrated data semantics based on GFO.

Elementary individuals that are related to other individuals by some kind of dependency relation are called attributives. They include, among others, qualities, relators, roles, functions, dispositions, and structural features. Material objects are wholes consisting of bundles of attributives;8

Categories, the instances of which are attributives, are called properties. According to the different types of attributives (relational roles, qualities, structural features, individual functions, dispositions, etc.) we distinguish corresponding types of properties, such as quality properties, role properties, etc. More recently, an additional perspective on attributives – illustrated in Fig. 2 – has been developed as a foundation for an integrated data semantics (Herre, 2016), since attributives (and their corresponding properties) present the semantic side of data.9

With respect to their bearers we distinguish between object attributives and processual attributives. Regarding object attributives, first we consider here only genuine/intrinsic attributives, i.e., attributives without further direct connections to other objects than their bearer (in contrast to extrinsic attributives). Moreover, among object attributives we differentiate presentic and non-presentic ones. A presentic object attributive is determined fully and as a whole by the object it inheres in and a time during which it exists. For example, an individual color of red inhering in some object can be wholly accessed at time points. In accordance with GFO’s integration law (see Section 3.2), a presentic object attributive thus gives rise to a continuant, on the one hand, which itself exhibits, at each time point of its lifetime, a wholly present attributive at that time (a presential). Non-presentic object attributives inhere likewise in objects, but they require something in addition to just their bearer and a time point. For example, the age of an object at a certain point in time is not only determined by that object and that very time point, but it requires something else; say, the initial point in time of the object’s existence. Besides, note that the composition of an object with some of its qualities yields more complex entities, called object facts.

Processual attributives have processes and process boundaries as bearers and they are classified into presentic and global attributives. Presentic processual attributives are associated with process boundaries; they must be wholly accessible at time points. The isolated presentic data of process boundaries do not need any reference to a process; they can be completely reduced to object qualities. These are typical qualities of objects that participate in the process. An example of a non-isolated presentic attributive of a process is the velocity of a moving body at a time point (even if it is zero, e.g. as an initial velocity). The velocity cannot be determined and specified without a temporally extended process.

The global attributives of processes present the richest class of attributives of processes. A systematic classification of these attributives is in its initial stage. Their main feature is that it is not meaningful to specify or refer to them at a process boundary. One type of global attributives is abstracted from time series in the form of diagrams (such as curve diagrams). Examples are electro-cardiograms and long-term blood pressure measurements. Additionally, we emphasize that intrinsic global attributives of processes are themselves processes, called attributive processes. A clear separation between an attributive process and a corresponding bearer-process needs further investigation and clarification. A water wave, for example, can be considered as an attributive process that occurs on the basic process of local movements of water particles. This basic process is the bearer of the wave. Finally for processual attributives, many other global attributives of a process are not derived from time series. Examples are the duration of a process and its occupied space. Physics provides many examples of this kind, for example the average velocity of a moving body. Further investigation of the global attributives of processes should take the distinction between intrinsic and extrinsic attributives into account.

There are further kinds of attributives, including relators, roles, functions, and dispositions, which are more abstract than phenomenal attributives; they cannot be directly perceived or measured. GFO adopts the general understanding of a role presented by Frank Loebe (2007), where a role individual is defined as a relational entity that links any entity, called the role player (or filler), with some context, in which the entity plays that role. The notion of role is essential for a broad spectrum of modeling areas, one of which supported by GFO is functional modeling. Its objective is to depict a domain in teleological/functional terms, as contrasted with other aspects, such as structural or behavioral ones. The notion of function is built upon that of role: we introduce a function as a category that captures a role played by some entity in the context of a goal achievement (GA), where the GA provides a teleological specification of transitioning to something that is intended to be achieved.

Concluding this part, let us emphasize again that the ontology of attributives provides a foundation for the semantics of data (Herre, 2019,2016). Moreover, we mention that interfacing between attributives and knowledge occurs through the transformation from facts to propositions,10

3.3.1. Dependence and attributives

There are various relations of dependence between entities, among which Section 2.3.3 briefly addresses existential dependence in a general sense as well as inherence. For example, an attributive cannot exist independently of a bearer and we may say that the bearer has higher degree of existential independence than the attributive. The notion of existential dependence is used throughout this paper in an informal manner. The basic schema of complexity is summarized as follows, cf. also the lower part of Fig. 1 (Section 2.3) and the categories highlighted in green: attributives/properties are constituents of material objects and material processes, resp., while those latter are included in material situations and situoids/situational processes.

We introduce some axioms w.r.t. the dependence relations depends ( x , y ) : = ‘the entity x depends on the entity y’ and on inherence, denoted by inh ( x , y ) : = ‘x inheres in y’.

Properties ( Prop ( x ) ) are categories of attributives ( Attr ( x ) ), the latter being understood as individuals. Attributives are divided into unary attributives, which need only one bearer, and multiply dependent attributives, which depend on more than one bearer. Relators, for example, depend on more than one player via their roles. Functions, dispositions and, most importantly in the sequel, qualities ( Qual ( x ) ) are unary attributives.

Besides the mere inherence of a quality, GFO relies on values to characterize a quality bearer (with some analogy to DOLCE (Masolo et al., 2003) in that respect). Insofar, the phrase “an individual color of red inheres in this apple” in Section 2.3.3 can be more precisely extended to “an individual color of the value red inheres in this apple”. In order to understand value attributions, we need to consider both the individual and the categorial level. The individual quality of the apple instantiates the property color, which is associated with a property value ‘red’. The latter is not only a category of quality values (themselves individuals), but typically also part of a measurement system or, more technically, a value structure. As categories, properties and their values can be considered independently of individuals. Properties correspond to determinables in the philosophical literature, e.g., described by Ingvar Johansson (1989), whereas property values equate determinates. Note that both notions need not be taken to be absolute, but both can depend on the assumed granularity. Finally, we use another dependence relation, namely value ( x , y ) : = ‘quality x has value y’, for linking qualities and their individual values.

On this basis, we selectively illustrate some connections in axiomatic form.

Attr ( x ) → ∃ y depends ( x , y )   Attributives depend on an entity.

3.3.2. Stuff

The concept of stuff (denoted by Stuff ( x ) ) has instances that are amounts of stuff. Accordingly,“Gold is a stuff”, for example, is read as “Every amount of gold is an amount of stuff”, Gold ( x ) → Stuff ( x ) . We assume that an amount of stuff has the character of a continuant, i.e., it persists through time. Furthermore, an amount of stuff is a dependent entity that is a part of a material aggregate.

Few examples of axioms follow, focusing on the dependence relation consistsOf .

Every material object consists of an amount of stuff.

3.3.3. Material objects, material boundaries, material aggregates and material object-processes

Solid material objects have a natural boundary. This does not apply to all natural objects, for instance, what are the natural material boundaries of a rock or of a mountain? The earth is a material object, while certain parts of it do not satisfy all conditions for material objects. For example, a mountain lacks the existence of a natural material boundary. Another example of a natural boundary of a material object is the separation of an airplane from the surrounding air. The latter has a boundary and the airplane as a material object has its material boundary; both boundaries belong to distinct material objects, can be clearly distinguished by different properties, and they touch.

Further notions that play a role in the analyses of the subsequent Section 4 are material aggregates, object-processes and the notion of solidity. A material aggregate is a collective of material objects; hence, it is a continuant. A material object-process satisfies the condition that every process boundary is an aggregate of material structures. This kind of processes is the counterpart of material aggregate. Solidity is a presentic quality of a material object, i.e., it derives from the presential with that quality and applies at each time point (even though the test for solidity requires a process).

MatAggr ( x ) → Cont ( x )   Material aggregates are continuants.

3.3.4. Material facts, situations, situoids and collectives

Considering relational facts, material objects can be connected by relators. Material facts, in turn, are composed of a relator and material objects playing the corresponding roles. Material facts are continuants. Material situations are composed of material facts, such that this system of material facts can be comprehended as a whole. Material situations are continuants, with a relation to collectives.15

by the material objects included in it,

4.1.1. Basic assumptions and relations to GFO

Every material object is composed of stuff; formally, consistsOf ( x , y ) expresses that ‘the material object x consists of the amount of stuff y’, cf. Section 3.3.2. We consider stuff as a macroscopic, spatiotemporal entity, which possesses phenomenal attributives. Macroscopic stuff has properties such as a specific weight, color, aggregate state (solid, fluid, gaseous), and other properties that can be perceived by the senses or can be measured by instruments. GFO postulates that any amount of stuff is a material continuant that does not carry unity. An amount of stuff existentially depends on an aggregate of material objects and is subject to the part-of relation.

4.1.2. Semi-formal description and domain specific signature

OPEN IN VIEWER

Fig. 3.

A timeline that surveys and relates temporally the material objects, aggregates and processes involved in the composition/constitution example.

Figure 3 indicates the temporal extensions of the individuals named in the case description (and a few more) and interrelates them w.r.t. several temporal phases (chronoids c i with s i and t i as starting and ending time points, resp., for i ∈ { 1 , … , 5 } ) that can be derived from analyzing the case. In particular, there is the table – denoted by t b and made of wood – that experiences differing stadiums. The first one is t b 1 , based on c 1 ,16

In order to express interrelations in connection with the replacement process p r , we introduce the domain-specific relation replaces ( p , x , y , z ) . It states that p is a replacement process in the course of which the part y of x is replaced by z. Similarly, for demolition, the predicate demolishes ( p , x , y ) says that in the process p the object x is demolished, yielding the material aggregate y. Observe that w.r.t. both predicates, all arguments are assumed to refer to continuants.

In addition, Table ( x ) , Leg ( x ) , and Wood ( x ) are employed as three unary predicates with the usual intended reading of their atomic formulas P ( x ) as ‘x is an instance of P’.

We gather compactly a first group of axioms on the basis of the previous sub section. Roughly following its structure, the case-specific signature as well as the constants (see Fig. 3) introduced there are used in combination with symbols of GFO (remember also the Appendix).

In particular, to capture the relation between the table t b and its sub continuants t b i in E2 we apply timerestr ( p , c , p ′ ) , which states that the process p ′ is the restriction of the process p to the chronoid c, with recourse to the continuant processes for t b and the t b i according to the integration law (A18). In contrast, in E8 we state the part-of relation between the table legs leg i by considering the presentials of t b and leg i , resp.17

Formulas E1–E10 represent the modeling case rather “literally” so far. Primarily E10 starts to reveal additional considerations, namely on how to understand “the wood is still there after the demolition” from the case description. Indeed, the formalization should be extended by further axioms in order to capture more of the intended semantics of the symbols introduced. We constrain ourselves to a number of sample formulas that come into question in this respect, starting with more general axioms that relate the domain level to the foundational level. Then a few proposals concerning the central predicates replaces and demolishes follow.

Many more potential axioms could be considered, while we conclude this overall case with two further points. First, regarding replacement there is the highly interesting question of whether and which requirements can be posed w.r.t. the relationship/similarity among the replaced part and its substitute. Would a similar shape/form suffice, perhaps extended with an appropriate material? (E.g., thinking of required stability in the case of a table leg.) Should one enforce that the substitute object can fulfill the same function? Those questions may easier be tackled in a more specific application context.

Secondly, this first modeling gets close to the puzzle of the ship of Theseus and its variants. Applied to the table case, how much can the table change to remain the same? Of what nature is the entity during the replacement? We see no absolute metaphysical solutions to these problems. There is no uniquely determined identity criterion, in our opinion. Instead, there are various equivalence criteria. What is important is the fact that a foundational ontology is expressive enough to distinguish these different forms of equivalence explicitly.

4.2.1. Basic assumptions and relations to GFO

The described situation clearly exhibits social aspects and we stipulate first that all mentioned terms (teacher, class, school and summer break) refer to, at least among others, kinds of entities in the social ontological region. The latter is one out of four major ontological regions distinguished in GFO and only briefly mentioned in Section 2.2. In parentheses, we list a few example categories for illustration here:

Considering entities w.r.t. these ontological regions is inspired by the theory of levels of reality (Poli, 2001). Poli uses the term ‘stratum’ instead of ‘region’, and strata are further divided into more fine-grained levels. For instance, the material region comprises levels such as the physical, chemical and the biological level. There are tight relations among the regions, e.g., where most entities at the mental-psychological region as well as at the social region rely on entities at the material region (more technically, overforming or building-above (Poli, 2001; Hartmann, 1964)). Put differently, one may say they have a material basis. Moreover, there are categories and individuals that involve aspects associated with multiple regions. A ‘human’, for instance, typically involves (at least) a ‘human body’ as well as a ‘human mind’. ‘Person’ may be understood to add social aspects on top of ‘human’, such as bearing a (socially agreed upon) ‘name’ and ‘human rights’.

A second line of work that is highly relevant for the situation described in the case is concerned with the notion of ‘role’. The approach to roles developed for GFO originates from an early analysis in 2003 (Loebe, 2003) and was covered in more detail later (Loebe, 2007). At the gist of it, there is a fairly general understanding of a role as a kind of entities that depend on (i) a context and (ii) a player. The context typically comprises of several roles (up to singular exceptions), which complement each other, such as teacher, student and subject. The player of a role is an entity that plays/fills/has/holds the role,18

The term ‘social role’ hints at another aspect of the GFO account of roles: contexts are utilized for distinguishing broad role types. Three types have been considered so far. We quote their informal short-hand characterizations as provided by Loebe (2007, p. 133):

Admittedly, especially the characterization of social roles simplifies matters significantly and the notion of ‘society’ may be rather vague as the context of social roles. ‘Social system’ would be another, perhaps slightly better choice. What remains hidden from that characterization is that social roles are, on the one hand, object-like entities that are equipped with their own properties and behavior. On the other hand, as roles they depend on their players, for which they are like (complex) properties. This dependence lets social roles appear also akin to attributives in GFO. The second dependence that social roles exhibit is that on a context. It will be more instructive, however, to view the concrete problem case at hand. Relational and processual roles are less relevant in that analysis and can be covered only briefly.

Let us start with the phrase “Mr. Potter is the teacher of class 2C at Shapism School […]”. We argue that ‘teacher’ is suggestive of a social role, but language use is not as simple as that. Mr. Potter should/can be viewed as an instance of the role category teacher. Instead, there is an individual social role TR (for Mr. Potter) that instantiates the role category of teacher. Along the lines of role theory as elaborated (mainly) in sociology, e.g. by Bruce J. Biddle (1979), the role category teacher can be expected to have an associated characteristic set of rights, duties, norms and expectations, including expected behavior. As stated above, TR depends on a player and a context. In our analysis, Mr. Potter is the player of the social role TR . Despite the complex nature of TR itself, the plays-relation is analogous to the inherence of a quality in an object insofar that the role player Mr. Potter actually embraces its role TR already, in the same sense that an object embraces all of its qualities. Notably, playing social roles cannot be reduced to assigning all properties of the social roles only directly to the player, because then the extra integration/bundling of those properties by the social roles is lost. Returning to the phrase “Mr. Potter is the teacher of class 2C at Shapism School […]”, we conclude first that ‘teacher’ is used here to refer to Mr. Potter as a role player (of a teacher role, namely TR ), but to Mr. Potter as a whole, which includes TR , but is not limited to it.

Besides depending on its player, TR hinges on a context. Thinking about the context of social roles is incomparably harder than in the cases of relational or processual roles. It is tempting to view simply the overall Shapism School as the context of TR – and, indeed, this simplification is made in the formalization provided below. This notwithstanding, we argue that a more detailed analysis yields a much more complex picture. The school is genuinely an institution (and thereby a social object) and it is more than merely a whole composed out of various role individuals. Rather, we think that actually a ‘school context’ SC emerges from the school, which involves (1) an abstract system of the school’s constituents and their interrelations, e.g., of functional nature and normative expectations, as well as (2) an individual, temporally extended situation (a situoid, in GFO terminology) that covers the actual development of the school and its constituents (including roles). In accordance with these views, there is Mr. Potter and a ‘teaches’ relator to the class 2C, which itself is a part of the school.

Regarding the replacements of Mr. Potter by Mrs. Bumblebee as well as of student Mary with student John (after a gap), we advocate the view that Mrs. Bumblebee plays her own social role TR ′ as a teacher at Shapism School, i.e., Mr. Potter and Mrs. Bumblebee do not play the very same role individual. However, at the level of the abstract system of the school context SC , TR ′ would likely exhibit pretty similar properties compared to TR . The teaches relator between Mr. Potter and 2C before the summer break disappears, while a new relator, between Mrs. Bumblebee and 2C is present after the summer break. Similarly, Mary’s membership relator towards 2C lapses at the beginning of the summer break, whereas another relator comes into being with John becoming a member of class 2C. The class itself, i.e., its identity, is here not considered to be affected by the exchange of a single student, albeit we see the need for further research on that matter. A difference between the teacher and the student cases could be accounted for if we assume that the school context requires a teacher (role) for each class. In that analysis, there is a vacant role after Mr. Potter has left and before Mrs. Bumblebee adopts the role. In contrast, there need not be a vacant student role after Mary has left.

4.2.3. Formalization

We can only partially formalize the given situation, where we start with two background axioms of GFO, followed by three exemplary specialization axioms, connecting domain with top-level categories.

Several predicates for domain-specific constellations should first be introduced explicitly as signature elements: class ( x , y ) : = ‘class x is a class at school y’, teacher ( x , y , z ) : = ‘x is a teacher at school y during chronoid z’, classTeacher ( x , y , z ) : = ‘teacher x is the class teacher of class y during chronoid z’, studentOf ( x , y , z ) : = ‘student x is a student in the class y during chronoid z’. Moreover, we use the constants 2 C for class 2C, P for Mr. Potter, B for Mrs. Bumblebee, M for Mary, J for John, and S T and S T ′ for the school term before and after the spring break, respectively. Based on this signature, major parts of the case description can already be stated in terms of (basically) logical literals.

Next the major part of the analysis in Section 4.2.2 must be established by further axioms. As the case description itself does not specify more explicitly, this can become a boundless enterprise. We illustrate matters by some specific axioms that relate to the analysis given, well aware that this is far from complete in any sense. The first two axioms formulate domain-specific conditions.

Another important kind of axiom links the “short-hand” predicates to a more detailed formal description. For instance, the ternary teacher predicate is intended to be bound to teacher as a social role, signified by TeacherRole . The latter, in turn, can be described in terms of properties and behavior (below with a very simplistic example).

Finally, being a class teacher ( classTeacher ) is analyzed as a temporalized relation with three roles, thus the class teacher role is seen as a relational role, in contrast to the social teacher role right above.

Having a class teacher at a school during some term implies that there is a corresponding relator with exactly three roles, played by class teacher, school and term.

4.3.1. A flower of changing color

In order to analyze the color statements in sub case (a), we need to consider both the individual and the categorial level. Colors of objects lead to qualities and thus to the realm of attributives (individuals), properties (categories of attributives), and property values in GFO, introduced in some detail in Section 2.3.3. We analyze color as a property and red as a property value, typically being part of a measurement system. When we turn to a particular flower, “the same” entities exist at the level of individuals. That means, the flower is an individual in which qualities can inhere (a.o., the color of that flower). A quality, in turn, has an individual value assigned, such as the particular red that is the value of the color of the flower and an instance of the property value red.

Accordingly, the change of the color of a flower is in fact a switch from one value of an individual quality to another quality value. In real life ontology engineering, however, the dynamics of modeled entities rarely involves a change of only a single property as in our case. Instead, usually the modeled entities undergo changes of multiple properties and the change can have diverse forms of temporal distribution – they can change simultaneously, with overlap, follow one other, etc. For example, a growing cell can both move and change its shape and size in parallel. The basic principles of modeling change explained just above can be applied to such diverse and complex scenarios, as well, but they can be beneficially augmented with modeling patterns that support a given scenario best. The performance of several patterns for modeling the change of qualities based on GFO is discussed by Burek et al. (2019a).

The next question in the case of the flower, call it F, is what sort of change it is. Clearly, it is not an instantaneous one, but a continuous change instead, during a part of the lifetime of F. If, in accordance with the integration law (A18, Section 3.2), we consider the flower F as a process CProc ( F ) , then the quality in question (color) is an isolated presentic attributive and the processual change can be considered as a continuous one.

Let us capture some aspects more formally, limiting ourselves to the fact that the color of F is red in summer and brown in autumn and only referring to F as continuant (which exhibits presentials). F is a material object with a certain lifetime. We consider a sub interval/chronoid y r of that lifetime, such that y r is exactly one year. Then there are two non-overlapping proper parts of y r such that the first corresponds to (the time of) summer and is before the second part, which corresponds to (the time of) autumn. Then we can express that F during any time point of that first part has the color red, while F during the second part has the color brown.

This can be formalized as follows, using F, autumn , summer , brown and red as constants and GFO signature (cf. the Appendix).

Importantly, we stipulate that changes are only possible with respect to the values of one property. For example, there is no property that subsumes the properties weight and color. Hence, these properties have distinct values and it is not plausible that a color value changes into one of the property color. One may still see a gap in this argumentation, however: Why is that plausibility not given? This gap can be closed by looking at the processes behind the material objects. First of all, according to the integration law (cf. Section 3.2), there is a process CProc ( F ) associated with the material object F. In addition, there are several attributive processual layers that are parts of CProc ( F ) , including a color process layer and a weight process layer of the process CProc ( F ) . We assume that all those considered processes are cohesive, which means that their process boundaries that are coincidence pairs are causally mediated (Michalek, 2009, esp. Section 5.2). We believe that a process boundary of the weight process layer cannot cause one that comprises a color quality with value red, and that the scrutinized implausibility can be supported by using further information and a deeper analysis.20

We claim that causality is a phenomenon defined for processes (that occurs in connection with processes), but not (directly) for continuants. A presentic attributive of a continuant can be considered without any process, and we reject the possibility that a causal relation between two coinciding presentials, exhibited by the continuant, can be established by the continuant alone. Causality is in the process, but not (in itself) in the set of the exhibited presentials of a continuant (Michalek, 2009).

The second sub-case of property change is (b) “A man is walking when suddenly he starts walking faster and then breaks into a run.” It will be necessary to consider the man M, categorized as a material object in GFO, in connection with the process CProc ( M ) behind the man, i.e., using the integration law of GFO (A18 in Section 3.2). For simplicity, note that we consider M as well as CProc ( M ) as temporally limited to the situation under consideration.

We see two options of interpreting the motions of “walking” and “running”. In GFO they can be seen (1) as global attributives of processes or (2) as processes themselves. Under both options the process CProc ( M ) is divided into three consecutive temporal parts, the first two being an instance of the processual concept of walking ( Walk ( x ) ), the third an instance of the processual concept of running ( Run ( x ) ).

The remaining problem for capturing the case is a specification of the concepts Walk ( x ) and Run ( x ) . A correct and complete axiomatic specification of Walk ( x ) implies that M is walking if and only if Walk ( CProc ( M ) ) , and analogously for Run ( x ) . We emphasize that the general problem of a fine-grained classification of the motions of humans is a difficult unsolved problem. One kind of process description is a declarative formalization, i.e., by a set of declarative sentences, capable of truth and falsity. Another form of specifiying processes is the representation by a sequence of activities. Then there occurs the problem how the declarative and the procedural specification are interrelated.

We note that these considerations refer mainly to the second option above, of viewing walking and running as processes. While we cannot elaborate an in-depth characterization, we explore this option a bit further, first considering those three temporal parts of CProc ( M ) on the physical level. During the first part p 1 the man M is moving with a constant velocity. During the second temporal part p 2 , M accelerates up to a maximal velocity at a certain point (where he starts running). One may further assume that this velocity remains constant during the third part p 3 of the whole process, the running. This can be formalized as follows, with c r 1 , c r 2 , c r 3 as constants for the time intervals during which p 1 , p 2 , p 3 happen. Moreover, the function velocity ( x , y ) yields a value (within a suitable measurement system that is left implicit) for the velocity of x at time point y21

However, the description according to physical parameters like velocity and acceleration is surely not enough to describe the whole process in detail. Such a description could likewise be attributed to, say, a driven car (constant velocity, acceleration to a maximum speed, etc.). The specific motions of walking and running can be further analyzed. A key difference between walking and running is that a walking man does not lose the contact to the ground. Running, in contrast, involves touching the ground and jumping in between, i.e., a runner is without ground contact for short time intervals. These intervals could be measured and then serve as a parameter for the description of the running. Of course, there is more to say about walking and running, for instance, there are certain coordinated movements of the man’s arms and legs. There is, for example, a parallel forward movement of the left leg and the right arm, or of the right leg and the left arm. There are consecutive sub-processes of the walking process that satisfy a well-defined regularity. For instance, the sub processes (left-arm, right-leg) and (right-arm, left-leg) occur alternately. Running allows for similar observations.

This sketches basic ideas and indicates a few directions towards a more detailed treatment – all limited to the material ontological region so far. The next case involves additional aspects.

4.4.1. Analysis focusing on the material region

First and from a physical point of view, one may distinguish various movements that occur. The first is a process, starting at a certain time point and being a movement towards the station. The station is not reached, though, instead there is another process that starts beforehand. That process can be divided into a processual segment of turning around, followed by a process of walking that is directed home. This is a description of process segments of the case at the physical level/the material ontological region.

Keeping the illustration of a corresponding, fairly straightforward formalization brief, we consider two processes that involve a material object – a man M. The first, say p 1 , is a movement of M to some location l and the other, p 2 , is a movement of M from l to the home location h. We can also represent the fact that one process follows the other immediately: their temporal extensions are two chronoids (time intervals) c 1 and c 2 that have coinciding boundaries, i.e., those chronoids meet.

However, the whole process cannot merely be described based on physical laws alone. In particular, there is no physical causality between the first and the second process segment. This means that the whole process is conducted by an actor that is more than a material object.

4.4.2. Analysis involving teleological aspects

For GFO, we introduced a layered architecture that permits to separate (1) non-functional descriptions specifying what is present in a domain, which structure it has and how it behaves from (2) a functional layer that represents which goals and rationale are behind (Burek, 2007; Burek et al., 2016). Section 4.4.1 elaborates a purely non-functional description, abstracting away from the teleological character of the above processes. In order to grasp their teleological character, we need to ask the question of why p 1 and p 2 occur. In other words, what are the goals or functions behind them.

In GFO the teleological character is depicted by a category of Goal Achievement (GA) understood as a purposeful transition, i.e., a transition to a situation that is (considered to be) a goal. That is founded on the notions of Goal, Situation and Transition. A goal is a teleological situation, i.e., a situation (considered to be) of utility.22

A situation is understood as a state of affairs that is composed of facts and can be comprehended as a whole, which can vary in its complexity depending, e.g., on the number of objects involved. In our case we speak about situations s 1 and s 2 involving the man M, station l s and home location l h , such that in s 1 M is at l s , whereas M is at l h in s 2 .23

The notion of utility is considered as primitive (and is denoted by Util C ). It is beyond the scope of this use case to model why and for whom a given situation is useful. But we mention that GFO distinguishes various utility functions and permits to model them adequately. For example, it could be useful for M to be at the station because that supports the next goal that M could have, say, taking a train.

We remark further that a goal as such is modeled first on the categorial level, before an individual situation can be identified as its instance. A goal can therefore also be captured adequately if it is not achieved. This applies in the use case: e.g., there is no situation (instance) in which M is at the station. We list five exemplary axioms formalizing transitions and GAs at the categorial and individual levels.

Overall, a goal achievement category accounts for a teleological specification of transitioning to what is intended to be achieved, hence, to a goal. It describes the context in which an item has (and possibly realizes) a function. Typically, GAs are realized by processes, and therefore certain process categories can be understood as goal achievement categories.

Returning to the use case, we have seen two levels of description – a non-teleological one of two processes p 1 and p 2 and a second, purely teleological specification of two goal achievements. The open question is how those two are interlinked. First, we observe that p 1 is not a goal achievement, as it is no causal transition that leads to the goal s 1 of M at l s . The second process p 2 , in contrast, is a goal achievement, since it causally transitions to a situation of M being present at home. This allows us to model the actual realizations of goal achievements and to answer which goals have been achieved.

What is still missing is an insight on the intentions behind, i.e., why those processes happened. This demands another layer of description and is represented in GFO by means of a family of has-function relations. These capture the intuitions that entities may have some functions for some agents due to certain reasons, e.g. because they have been designed, used, or desired to do something (Patryk Burek (2007, esp. ch. 6) developed a detailed classification). Notably, in GFO a function can be assigned not only to objects, but also to processes. That way we may say that the process p 1 of the movement of M to l s has an intended function of transporting M to the station and process p 2 has that of transporting M home. The main principle behind the GFO modeling of teleological entities is a separation of concerns. On the one hand, this allows for modeling non-functional entities such as objects and processes independent of goals and functions. On the other hand, it supports functional (and goal) modeling independent of realizations. Eventually, both worlds can be interlinked by realization and has-function relations.

As a final remark, functional/teleological modeling frequently requires a more advanced toolset for handling more detailed descriptions of goal achievements. These can involve (1) objects contributing to the goal achievement, which can be classified as doers, contributors, operands or instruments, depending on their role in the GA. Further aspects are (2) the mode of realization and (3) side effects as well as the modeling of interdependencies between goal achievements and between functions, such as functional decompositions (Burek et al., 2016,2009,2017).