Issue 98028 Editor: Erik Sandewall 14.3.1998

# 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.

# Debates

## Pat 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  <tt with the point  t , since such an interval makes no 'space' when interposed between two others, ie if  <ab meets  <bb meets  <bc, then  <ab also meets  <bc; but it is also consistent, if one wishes, to distinguish  t  from  <tt, or even to forbid instantaneous intervals completely. It is also quite consistent to have arbitrary amounts of density, discreteness, etc.; for example, one can say that time is continuous except in a certain class of 'momentary' intervals whose ends are distinct but have no interior points. (Vladimir might find these more congenial that points as the intervals which things like flashes of lightning must occupy.)

(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.ps
One 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  <ab but false during  <bc, even if  b  is considered to be 'in' both the intervals.

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  <st and the switch is hit at  t . If the light stays on, we have two meeting intervals. If the light flashes and immediately burns out (put a 120 V bulb in a 230 V socket), one could say that there is a flash at  t , surrounded on both sides by extended intervals of darkness. Both stories are perfectly consistent. It follows, for example, that a random timepoint during a period of extended illumination is not a flash, in spite of its being a timepoint at which the light is on.

Pat Hayes