A good number of writers have constructed algebraic systems for dealing with intervals in connection with the logic of time. If we have two time intervals a, b, and 'a' ends before b begins, then this is obviously sufficient for 'a' to be earlier than b. I have not seen any writer who does not make this a necessary condition as well. 
If this is done, however, an awkwardness arises in practice. Two successive days are always considered to share an instant of time at their common midnight. This will now be taken into account in defining 'earlier than' for two intervals, provided neither reduces to a point. The result is that any set of intervals ordered in this way form a partially ordered set (poset). In fact these posets are also 'interval orders'.
The application I have in mind is to historical events and periods, which requires both point events and extended events. The idea was suggested to me, as far as I can remember, by the term 'topological dating' used by Charles Hockett in a book on general linguistics. This book is in the library of St Stephen's College, Delhi.
2. Definitions________(Contents Defns)
The underlying structure is the continuous time-line, linearly ordered by the relation 'earlier than' (<) on its instants (see also a technical note on the real time line). We then define an event 'a' as a closed interval:
DEFINITION 1. An event 'a' is a closed interval of instants t given by
B(a) <= t <= E(a) , with (1) B(a) <= E(a),
where B(a) is the beginning and E(a) the end of the event. If B(a)=E(a) we have a point event.
The relation 'earlier than' on a set of events H may now be defined.
DEFINITION 2. a<b iff E(a)<B(b) or B(a)<E(a)=B(b)<E(b).
The different ways in which a<b are therefore:
o o o o----- -----o o -----o o----- -----o----- a b a b a b a b a b
For some purposes it is enough to know that
(2) a<b ==> E(a)<=B(b).
Another result we shall use is
(3) a<b ==> E(a)<E(b).
Proof. If a<b then either
E(a)<B(b) by definition <=E(b), by (1) or E(a)=B(b)<E(b). by definition
The result follows in either case.
A set of events H ordered by 'earlier than', is written (H,<).
Lemma 1. (H,<) is a poset.
Proof: We have to prove that the relation < given by Def.1 is (i) irreflexive and (ii) transitive.
(i): suppose a<a for some event a. By Def.1 this requires either E(a)<B(a) or E(a)=B(a)<E(a), both of which are impossible. (ii): if a<b, b<c, then by (3), (2) we have E(a)<E(b)<=B(c). Hence E(a)<B(c), which gives b<c.
An interval order is defined as follows .
If a, b, x, y are any elements of an interval order, then
a<x, b<y ==> a<y or b<x.
Lemma 2. Any set of events ordered by < is an interval
Proof. Since the time-line underlies the system of events, we have either (i)E(a)<B(y), or (ii)B(y)<E(a), or (iii) E(a)=B(y). In case (i), it is immediate that a<y. In case (ii) we have
E(b) <= B(y) by b<:y < E(a) case (ii) <= B(x), by a<x
so b<x. In case (iii), as before,
E(b)<= B(y) by b<y = E(a) case (iii) <= B(x). by a<x
Hence we again have b<x, unless there is equality all through. In that case, however, Def.1 entails that a, b, x, y are all extended events. Then a<y, b<x and the events are arranged like this:
a x ------------|---------- -----------|----------- b y
It frequently happens in historical research that the data is incomplete. Let K be the relation 'known to be earlier than', on events. Obviously,
(4) aKb ==> a<b.
Lemma 3. (H,K) is a poset.
Proof. The negative of (4) with b=a gives: not a<a ==> not aKa. If aKb & bKc, then we deduce a<c using (4). The deduction itself tells us that aKc.
The result (4) means that (H,K) is contained in (H,<). This shows formally that we may set up (H,K) and develop it as information becomes available, all the time getting closer to (H,<). The final result need not be linearly ordered, because overlapping events are relatively unordered by <.
1. A. BOCHMAN, 'Concerted Instant-Interval Temporal Semantics I: Temporal Ontologies', Notre Dame Journal of Formal Logic 31, 403-414 (1990);
J. P. BURGESS, 'Axioms for Tense Logic II. Time Periods', Notre Dame Journal of Formal Logic 23, 375-383 (1982);
A. GALTON, 'Time and Change for AI', in D. M. GABBAY, C. J. HOGGER, and J. A. ROBINSON (eds), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol.4, Epistemic and Temporal Reasoning, Clarendon Press, Oxford, pp. 175-240 (1995);
C. L HAMBLIN, 'Instants and Intervals', Studium Generale, 24, 127-134 (1971);
I. L. HUMBERSTONE, 'Interval Semantics for Tense Logic: Some Remarks', Journal of Philosophical Logic 8, 171-196 (1979);
M. JIXIN and B. KNIGHT, 'A General Temporal Theory', The Computer Journal, 37, 114-123 (1994);
P. RÖPER, 'Intervals and Tenses', Journal of Philosophical Logic, 9, 451-469 (1980);
M. J. WHITE, 'An 'Almost Classical' Period-Based Tense Logic', Notre Dame Journal of Formal Logic, 29, 438-453 (1988).
Norbert WEINER, Proceedings of the Cambridge Philosophical Society 17, 441-449 (1914), was perhaps the first to give the definition of 'earlier than' for intervals, as 'completely earlier than'. His paper makes extensive use of Russell's notation for symbolic logic. (back to interval order)
2. P. C. FISHBURN, Interval Orders and Interval Graphs.
Wiley, 1985. (back)
Last updated: 8 March 2008