States of affairs and their relation to time

Let us consider a number of states of affairs:

1. $t$'s being green
2. Mary's walking uphill
3. Mary's climbing the hill
4. Mary's reaching the top of the mountain
5. Robert's knowing Kay

Are those states of affairs different and in what way? How do they behave in time?

The first expresses the property of an object, $t$. This is a fact at a point in time or during some time interval. This state of affairs may persist over several time-points, or exist at only one point in time. Or it may be viewed as existing through some time-interval, and cease to exist in the following.

The second expresses a property of an object, too. It states that Mary is walking uphill. A detailed analysis, using the ontology of GOL as a reference, reveales the following: There is an individual process $p$, the movement of an object in upward direction (towards the top of the hill). Mary is participating on this process $p$ through her act of walking, $z$. Now this participation-relation is a universal in GOL, say $R$. The state of affairs is formed by a relator $r$ with $r::R$, the process $p$ and the activity that relates Mary to $p$, $z$. So the state of affairs would be $\langle\langle r::R,p,Mary,z\rangle\rangle$. This state of affairs may exist in a time-interval, but not at a point in time At sub-intervals of this time, Mary is still walking uphill, so the state of affairs exists at any given sub-interval. Also, there is no natural end-point of this activity, there is no culmination.

The third state of affairs is different from the second. There may be sub-intervals, where Mary is not climbing the hill. Also, there is an end-point, a culmination. But again, this state of affairs exists at a time-interval.

The fourth state of affairs is instantaneous, it can only exist at a point in time. It is inherently culminating, because the underlying process is.

The fifth state of affairs expresses a state. It exists during some time interval, but it makes no sense to ask, how long it lasted or whether it culminated.

There may be even more distinctions between states of affairs. We wanted to show, that states of affairs may be related in a variety of different ways to time. Also, they are neither only occurrents nor only endurants, they may be both. We may classify states of affairs by this distinction.

Some philosophers would disagree with us and deny, that states of affairs have a temporal extension. They would rather call those entities events, while states of affairs are the combination of objects at a point in time. Some philosophers like Ingarden would prefer to call some of the above relations between objects processes in a general sense, and only states of affairs of the fourth type ``events''. Others, like lombard or kim1 refer to events as special states of affairs. All discussion about events is mainly done under the question of causality. Events may be causes or effects of others. But then, the problem arises: ``Sometimes an event is described as the cause of some quality in an object [...], or as the cause of a state of affairs [...]. Sometimes, again, an object or state of affairs is described as the cause of a quality in an object, or as the cause of a state of affairs, or as the cause of some event; and there are perhaps other sorts of causally related pairs.''harris Some of these problems are avoided when events are regarded as states of affairs, or at least constituted by states of affairs. As we do not wish to discuss causality further, please be referred to riker for a good account of causality, that is close to our approach.

We subsume ``events'' (and ``processes'', in another terminology) under states of affairs. ``Complex events'' involving multiple states of affairs like ``the Second World War'' or a ``marathon'' do not exist in states of affairs, but rather in the domain of situoids and situations. ``Primitive events'', like the movement of an object $a$ from the spatial point $p_1$ to $p_2$ (or, to use the proper GOL-terminology, the participation of the object $a$ in the process consisting in the movement of an object from the points $p_1$ to $p_2$) or the states of affairs in the beginning of this section exist in the domain of states of affairs events.

This leaves us with the problem of distinguishing different kinds of states of affairs; there are instantaneous and enduring states of affairs, some, that exist at a single point in time, some that exist during some time interval. States of affairs have a temporal extension. Their temporal extension is the set of time points (or time boundaries) or time intervals at which they exist.

Definition 5.6 (Temporal Extension of States of Affairs)   The temporal extension of a state of affairs $s$ is the set $S$ of time boundaries and time intervals, at which $s$ exists. The temporal extension of the state of affairs $s$ is denoted by $TExt(s)$.

Now we can say, what instantaneous and enduring states of affairs are.

Definition 5.7 (Instantaneous, enduring and ambiguous states of affairs)   A state of affairs $s$ is called instantaneous, $InstSOA(s)$, if every element of the temporal extension of $s$ is a time boundary: $\forall x \left( x \in TExt(s)\rightarrow tb(x)\right)$. $s$ is called enduring, $EndSOA(s)$, if every element of the temporal extension of $s$ is a chronoid (time interval): $\forall x \left( x \in TExt(s) \rightarrow chr(x)\right)$. $s$ is called ambiguous, $AmbSOA(s)$, if it is neither instantaneous nor enduring.

There may be two kinds of ambiguous states of affairs, without a temporal extension and with time boundaries as well as time intervals as a temporal extension. States of affairs without a temporal extension may be viewed as abstract entities, or somewhere existing outside of time, like, maybe, $\langle\langle <,1,2 \rangle\rangle$. We believe that a better interpretation is that this state of affairs exists throughout all time. States of affairs, like $\langle\langle Green, t\rangle\rangle$ may be viewed as existing at time points and time intervals. We rather believe, that there are two different states of affairs with different relations, one relating some property to an object at a point in time, another relating this property over some time interval. We take on the point of view that there are no ambiguous states of affairs in reality.

Axiom 5.8   All states of affairs are either instantaneous or enduring:

$$\forall x ( SOA(x) \rightarrow (InstSOA(x) \lor EndSOA(x)) \land \neg AmbSOA(x))$$

Another distinction may be between static and dynamic states of affairs; consider the states of affairs $\langle\langle Rest, Kay \rangle\rangle$ and $\langle\langle Walk, Kay \rangle\rangle$, Kay's resting and Kay's walking. The first involves a state, and no property is changed by it, while the second inheres a change. Changes are ordered pairs of instantaneous states of affairs. For a detailed discussion of processes and how they can be used to classify infons, please be refered to herre4.

We assume that pictural states of affairs are always wholly present at a point in time, at a time boundary. Therefore, infons are also always wholly present at a point in time.

Infons, taking pictural states of affairs as constituting parts, may have another argument, specifying a time boundary or a time interval. This infon describes the information about presence or absence of a state of affairs at some time entity (boundary or interval).

Definition 5.8 (Timed infon)   A timed infon consists of a pictural state of affairs $\phi$, a function $f:picSOA \mapsto \{0,1\}$ and a time entity $t$. Let $\phi=\langle\langle R,a \rangle\rangle$ be a pictural state of affairs. Than we note an appropriate timed infon $\langle\langle R,a;p;t \rangle\rangle$, where $p \in \{0,1\}$ and $tb(t)\lor chr(t)$.

Now we can extend the correspondence relation between infons and states of affairs to a timed correspondence.

Definition 5.9 (Timed Correspondence of infon and state of affairs)   Let $s$ be a state of affairs and $i=\langle\langle P,a;p;t\rangle\rangle$ be a timed infon. Then $s$ and $i$ stand in the relation of timed correspondence, $tcorr(i,s)$, if for the infon $i'=\langle\langle P,a;p \rangle\rangle$ $corr(i',s)$ obtains and $t \in TExt(s)$.

Now we have a rich enough formalism to start our discussion of situoids and situations. But before we start this, we will have to make some remarks about features we omitted here, because they can only be fully understood in the domain of situations and situoids.