Issue 98028 | Editor: Erik Sandewall | [postscript] | ||
14.3.1998 |
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Today |
Today, Pat Hayes answers Jixin Ma and argues in favor of an ontology for time where intervals are the only elementary concept and timepoints play a secondary role.
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Debates |
Ontologies for timePat Hayes:Im largely in agreement with Jixin about points and intervals, although I dont think its got anything to do with knowledge, and I think there's a simpler way to say it all.
First, just forget about whether intervals are open or closed.
This issue arises only if we insist (as the standard mathematical
account of the continuum does) that an interval is a set of points.
But if we take points and intervals as basic, there is no need to do
this. Points, as
Allen suggested long ago, can be thought of as places where intervals
meet each other, not as the substance out of which intervals are
constructed. It took me a long time to see how powerful this idea is.
The question of which interval 'contains' the meeting point is
meaningless. This gives a very simple, elegant formulation in which
points are totally ordered, intervals are uniquely defined by their
endpoints (which are also the points they fit between) and two
intervals meet just when the endpoint of the first is the startpoint of
the second. That's all the structure one needs. Truths hold during
intervals. One can allow instantaneous intervals, whose endpoints are
identical and which have no duration. One can, if one wishes, identify
the interval (Instantaneous intervals have the odd property of meeting themselves, by the way; in fact this is a way to characterise them without mentioning points explicitly. It is also perfectly consistent to have 'backward' intervals whose end is earlier than their beginning, and which have negative durations. Axiomatic details can be found in a rather long document available as two postscript files http://www.coginst.uwf.edu/~phayes/TimeCatalog1.ps http://www.coginst.uwf.edu/~phayes/TimeCatalog2.psOne can cast the whole theory in terms of a single three-place relation MEETS-AT between two intervals and a point, much as Allen's original theory can be cast in terms of MEETS.)
In this theory, to talk of the set of points 'in' an interval requires
one to specify what it means for a point to be 'in' an interval. If a
point is later than the beginning and earlier than the end, its clearly
in the interval, but we have some freedom with the endpoints. One could
insist that interval endpoints are 'in' the interval. But this is now
OK, since truths hold not at points but during intervals, so the
apparent contradiction of the light being both on and off at the
splitting point simply doesnt arise. The light isn't either on or off
at a single point: if you want to know whether the light was on or not,
you have to say which interval you are talking about. P may be true
during
Jixin says that "one cannot talk about anything about the switching
point P, which is intuitively there anyway." Well, the point is
certainly there, and we can talk about it (for example, its relation to
other points and intervals) but the question is whether it makes sense
to say that something is true at it. Some truths may be instantaneous,
ie true only at points; others make sense only when asserted to hold
during noninstantaneous intervals. Lights being on or off, for example,
might be enduring, while changes in illumination, or isolated flashes,
can be instantaneous. So for example suppose it is dark during
interval Pat Hayes
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