EVENTS AND TIME IN A FINITE AND CLOSED WORLD
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Nordic Journal of Philosophical Logic, Vol. 5, No. 1, pp. 3–24. © 2000 Taylor & Francis.
4
francis y. lin
Most formal theories of time, such as the ones presented in Lin (1991,
events given to us. The idea of a closed world is similar to that of nonmono-
tonic reasoning in artificial intelligence, which tells us that jumping to conclu-
impose restrictions on a formal theory of time (it is to be noted that the
two criteria are independent of each other). They require i) that the axiom-
figure 1 has the following axiom:
Ye(Ze∞(e∞<e))
then this theory will violate the assumption that the world in figure 1 is closed (because it assumes that, for instance, there is an event before e1 in
1994), presume an infinite number of events. But in this paper I concentrate
on the construction of time from a finite number of events. The motivation
for doing so is this. On the one hand, we often need to reason about a
atisation and the construction should not assume that there are an infinite
number of events, and ii) that they should not assume that there are events
outside the world in question. Thus, if a theory of time for the world of
sions when faced with incomplete information is an important part of our
common-sense reasoning (Gabbay et al. 1993).
A formal theory of time consists of (at least) two parts: an axiomatisation
1. Introduction
Various theories have been devised in the past to explain what time is. Broadly speaking, they fall into two categories: absolute theories and relational theories (see Lin 1991 for a detailed exposition). Absolute theories take time elements, such as instants or intervals, as primitive, and regard events (occurrences, or processes, etc.) as happening in time. These theories can be further divided into two types. One type treats instants of time as primitive, as exemplified in Newton (1934) and McDermott (1982). The other type takes intervals of time to be elementary, e.g. Newton-Smith 1980, Hamblin (1972), Dowty (1979), and Allen and Hayes (1985, 1989). Relational theories of time, on the other hand, hold that our concept of time is derived (abstracted) from our experiences of events and thus regard events as more fundamental than time. Exponents of such theories include Whitehead (1920), Russell (1926, 1956), Kamp (1979, 1980), Bach (1981), Turner (1984), and Thomason (1984, 1989). In this paper I take the relational view of time, not only because it is psychologically plausible (Zward 1976, O’Connor and Hermelin 1978, Piaget 1954, Michon 1989) but also because it is practically useful (Bach 1981, Davidson 1967, Kamp 1979, 1980, Parsons 1990).
finite number of events; on the other hand, our concept of time is derived
from the events we perceive, which are finite in number. Let us define a
world as consisting of a set of events and their temporal relationships. We
the universe) which are not included in this world. For example, there are
beli2eo.uBonuudttsitifdotehbetehweevopernrldetssiesbncetlfoowsreoedrl,ed1w., eeInvsheonotuthsldearfntweortotrehd2,isn,akwnadebeoshvueotnustulsdcbhoenetvwlyeenedtnes,aewl1 whaiincthhd
EVENTS AND TIME IN A FINITE AND CLOSED WORLD
FRANCIS Y. LIN
There are numerous occasions on which we need to reason about a finite number of events. And we often need to consider only those events which are given or which we perceive. These give rise to the Criteria of Finiteness and Closedness. Allen’s logic provides a way of reasoning about events. In this paper I examine Allen and Hayes’ axiomatisation of this logic, and develop two other axiomatisations based on the work by Russell and Thomason. I shall show that these three axiomatisations are weakly equivalent, and that only the last two meet the Criteria of Finiteness and Closedness (to different degrees). I shall then examine two ways of constructing instants of time in a finite and closed world, i.e. the Russell construction and the Thomason construction. I shall prove that these two constructions are equivalent under certain conditions.
In this the event
world, of my
there are only three events, having breakfast, the event
eo1 ,f
em2 yanddriev3i,ndgetsocrwiboinrkg,,
say, and
the event of my listening to the radio. But of course there are events (in
of the events and their temporal relationships in a world, and a method
of ructing instants of time out of events. The criterion that the world
should be finite and the criterion that the world should also be closed both
Figure 1. Events in a closed world.
events and time in a finite and closed world
5
that world). Furthermore, if time in this world is not circular, then the above axiom will imply that there are an infinite number of events. And this will violate the assumption that the world in figure 1 is finite.
In this paper I construct a theory of time for a finite and closed world, that is, a theory of time that satisfies the criteria of Finiteness and Closedness. The task naturally divides into two parts. First, I must axiomatise the events of a finite and closed world and the temporal relationships among the events. I examine the axiomatisation by Allen and Hayes (1985, 1989), and develop two others based on the work of Russell (1926, 1956) and Thomason (1984, 1989). As we shall see, only the latter two meet the criteria of Finiteness and Closedness (to different degrees). The second part of our task is to construct instants of time on the basis of the Russell and Thomason axiomatisations. In undertaking this task I reveal some logical relationships among the three axiomatisations, and between Russell’s and Thomason’s constructions of instants out of events. All theorems contained in this paper are original.
are thus interested in events and time in a finite world here. Apart from
finiteness, we also want the world to be closed. To explain the concept of
a closed world, let us consider the world depicted in figure 1.
4
francis y. lin
Most formal theories of time, such as the ones presented in Lin (1991,
events given to us. The idea of a closed world is similar to that of nonmono-
tonic reasoning in artificial intelligence, which tells us that jumping to conclu-
impose restrictions on a formal theory of time (it is to be noted that the
two criteria are independent of each other). They require i) that the axiom-
figure 1 has the following axiom:
Ye(Ze∞(e∞<e))
then this theory will violate the assumption that the world in figure 1 is closed (because it assumes that, for instance, there is an event before e1 in
1994), presume an infinite number of events. But in this paper I concentrate
on the construction of time from a finite number of events. The motivation
for doing so is this. On the one hand, we often need to reason about a
atisation and the construction should not assume that there are an infinite
number of events, and ii) that they should not assume that there are events
outside the world in question. Thus, if a theory of time for the world of
sions when faced with incomplete information is an important part of our
common-sense reasoning (Gabbay et al. 1993).
A formal theory of time consists of (at least) two parts: an axiomatisation
1. Introduction
Various theories have been devised in the past to explain what time is. Broadly speaking, they fall into two categories: absolute theories and relational theories (see Lin 1991 for a detailed exposition). Absolute theories take time elements, such as instants or intervals, as primitive, and regard events (occurrences, or processes, etc.) as happening in time. These theories can be further divided into two types. One type treats instants of time as primitive, as exemplified in Newton (1934) and McDermott (1982). The other type takes intervals of time to be elementary, e.g. Newton-Smith 1980, Hamblin (1972), Dowty (1979), and Allen and Hayes (1985, 1989). Relational theories of time, on the other hand, hold that our concept of time is derived (abstracted) from our experiences of events and thus regard events as more fundamental than time. Exponents of such theories include Whitehead (1920), Russell (1926, 1956), Kamp (1979, 1980), Bach (1981), Turner (1984), and Thomason (1984, 1989). In this paper I take the relational view of time, not only because it is psychologically plausible (Zward 1976, O’Connor and Hermelin 1978, Piaget 1954, Michon 1989) but also because it is practically useful (Bach 1981, Davidson 1967, Kamp 1979, 1980, Parsons 1990).
finite number of events; on the other hand, our concept of time is derived
from the events we perceive, which are finite in number. Let us define a
world as consisting of a set of events and their temporal relationships. We
the universe) which are not included in this world. For example, there are
beli2eo.uBonuudttsitifdotehbetehweevopernrldetssiesbncetlfoowsreoedrl,ed1w., eeInvsheonotuthsldearfntweortotrehd2,isn,akwnadebeoshvueotnustulsdcbhoenetvwlyeenedtnes,aewl1 whaiincthhd
EVENTS AND TIME IN A FINITE AND CLOSED WORLD
FRANCIS Y. LIN
There are numerous occasions on which we need to reason about a finite number of events. And we often need to consider only those events which are given or which we perceive. These give rise to the Criteria of Finiteness and Closedness. Allen’s logic provides a way of reasoning about events. In this paper I examine Allen and Hayes’ axiomatisation of this logic, and develop two other axiomatisations based on the work by Russell and Thomason. I shall show that these three axiomatisations are weakly equivalent, and that only the last two meet the Criteria of Finiteness and Closedness (to different degrees). I shall then examine two ways of constructing instants of time in a finite and closed world, i.e. the Russell construction and the Thomason construction. I shall prove that these two constructions are equivalent under certain conditions.
In this the event
world, of my
there are only three events, having breakfast, the event
eo1 ,f
em2 yanddriev3i,ndgetsocrwiboinrkg,,
say, and
the event of my listening to the radio. But of course there are events (in
of the events and their temporal relationships in a world, and a method
of ructing instants of time out of events. The criterion that the world
should be finite and the criterion that the world should also be closed both
Figure 1. Events in a closed world.
events and time in a finite and closed world
5
that world). Furthermore, if time in this world is not circular, then the above axiom will imply that there are an infinite number of events. And this will violate the assumption that the world in figure 1 is finite.
In this paper I construct a theory of time for a finite and closed world, that is, a theory of time that satisfies the criteria of Finiteness and Closedness. The task naturally divides into two parts. First, I must axiomatise the events of a finite and closed world and the temporal relationships among the events. I examine the axiomatisation by Allen and Hayes (1985, 1989), and develop two others based on the work of Russell (1926, 1956) and Thomason (1984, 1989). As we shall see, only the latter two meet the criteria of Finiteness and Closedness (to different degrees). The second part of our task is to construct instants of time on the basis of the Russell and Thomason axiomatisations. In undertaking this task I reveal some logical relationships among the three axiomatisations, and between Russell’s and Thomason’s constructions of instants out of events. All theorems contained in this paper are original.
are thus interested in events and time in a finite world here. Apart from
finiteness, we also want the world to be closed. To explain the concept of
a closed world, let us consider the world depicted in figure 1.