General Formal Ontology (GFO)
The GFO approach of time is inspired by Brentano's ideas
(11) on continuum, space and time.
Following this approach, chronoids are not defined as sets of points, but as
entities sui generis.11Every chronoid has exactly two extremal and
infinitely many inner time boundaries which are
equivalently called time-points.
Time boundaries depend on chronoids (i.e., they have no independent
existence) and can coincide.
Starting with chronoids, we introduce the notion of
time region as the mereological sum of chronoids,
i.e., time regions consist of non-connected intervals of time.
Time entities, i.e., time-regions and time-points, share certain
formal relations, in particular the
part-of relation between chronoids and time regions, the relation of being
an extremal time-boundary of
a chronoid, and the relation of coincidence between two time-boundaries.
Dealing with the coincidence of time boundaries is especially useful if two
to be modeled as ``meeting'' (in the sense of Allen's relation
``meets''). In our opinion, there are at least three conditions that a
correct model must fulfill:
- there are two processes following one another immediately,
i.e., without any gaps,
- there is a point in time where the first process ends, and
- there is a point in time where the second process begins.
If, as is common practice, intervals of real numbers are used for modeling
time intervals (with real numbers as time points), there are four
possibilities for modeling the meeting-point:
In contrast, the approach via the glass continuum allows for two
chronoids to follow immediately, one after another, and to have proper
starting- and ending-``points'' by allowing their boundaries to coincide.
The coincidence relation entails that there is no time difference
between the coinciding time boundaries, while maintaining their status as
two different entities. This way, conditions (a), (b) and (c) are
fulfilled. Let us consider additional examples below.
- The first interval is right-closed and the second is
left-closed. This allows for two options with regard to the overlap of
- The intervals do not overlap. This conflicts with
conditon (a), because a new interval can be placed between the
final point of the first and the starting-point of the second
- The intervals overlap at the meeting-point. This raises,
however, the possibility of contradictions between properties of the first,
and properties of the second process (cf. the examples
- The first interval is right-open and the second one is
left-closed. However, this conflicts with condition (b).
- The first interval is right-closed and the second one
left-open. This conflicts with condition (c).
- The first interval is right-open and the second left-open. This
variant fails on both conditions (b) and (c).
``She drew a line with her fountain pen until there was
no more ink left.'' What do the conditions (a) - (c) mean in this
- There is no gap where there is no ink in the pen or ink
in the pen.
- There is a final point where the pen is not empty.
- There is an initial point where the ink pen is empty.
``Student changed his course of study from
physics to computer sciences by filling out the appropriate form.''
What do (a) - (c) mean in this example?
- There is no gap where studies nothing.
- There is a final point where is a student
- There is a first point where is a student
of computer sciences.