>>20276272Flows of Time
Before we can start a discussion of various logics of time, it helps to look at
some standard mathematical models of time. When asked to think of time
in an abstract way, many people will form a picture of a line — only the sim
plest of the many spatial metaphors that people use for temporal concepts!
The mathematics of this picture is given by a set of time points, together
with an ordering relation and perhaps a metric measuring the distance be
tween two points. Later on, we will discuss some objections and alternatives
to this pointbased paradigm. For now, let us formally represent time as a
frame
; that is, a structure
T
= (
T,<
) such that
<
is a binary relation on
T
, called the
precedence relation
. Elements of
T
are called
time points
; if
a pair (
s,t
) belongs to
<
we say that
s
is
earlier than
t
. In the remainder
of this section we will discuss a number of more or less intuitive conditions
that have been imposed on such structures in order to make them useful
as models of time. (We will frequently use first and second order logic for
describing these properties; the
first order frame language
we use will have
only one dyadic predicate symbol, which is denoted by
R
and interpreted as
<
.)
Obviously, many frames will not qualify as intuitively acceptable repre
sentations of time. At a minimum one should require that
<
be irreflexive
and transitive. A frame satisfying these conditions will be called a
flow of
time
. Flows of time are known from mathematics as strict partial orders,
and in accordance with this we will use familiar notation like
s > t
for ‘
s
is later than
t
’ or
s
≤
t
for ‘either
s
=
t
or
s < t
’. For a point
t
, the set
{
s
∈
T

t < s
}
will be called the
future of
t
; the
past of
t
is defined likewise.
(In the sequel, we will omit definitions pertaining to the past if they mirror
an obvious counterpart for the future.)