

![0_image_0.png](0_image_0.png)

Intuitively, the interpretation means the following. If in T −
gso each observation is a
"system run", encoded by a stratified order of the event occurrences, which is observed by some implicit observer, then in T*pslcore* we explicitly describe this observer as an object. For our interpretation, we are particularly interested in objects that participate in a unique activity occurrence of each activity at a unique time point. In other words, observers are objects satisfying the following properties:

1. The time point in which an object participates with an activity occurrence of an activity is exactly the time when the object observes the activity.

2. The object observes every activity.

3. The object only observes each activity exactly once.
All of the other interpretations πearlier than, πnot later than and πnonsimultaneous can be easily determined from the observations that all observers observed.

Theorem 9. The interpretation π *defined in Definition 6 is correct.*
Proof. It is easy to check that under the interpretation π, every axioms of T −
gso is a theorem of T*pslcore*. Hence, π defined in Definition 6 is a correct interpretation. ⊓⊔

## 5 Conclusion

In this paper, we proposed in our knowledge the first version of a first-order theory for gso-structures in [2,5]. We avoid the difficulty of not being able to quantify over relations in first-order logic by introducing the relations observed before and observed simult which take an observation as one of their parameters.

Using model-theoretic ontological techniques introduced in [4], we classified all possible models of Tgso, where our key results are the satisfiability theorem and axiomatizability theorem for Tgso. In our opinion, the classification of models of T*spec*, which decomposes the earlier than, not later than and nonsimultaneous into smaller graphs, is especially insightful in understanding these three relations. Although the classification of observations using ranking structures is quite artificial, we could not figure out any simpler characterization.

We also give a very intuitive interpretation of the weaker theory T −
gso into T*pslcore*,
which shows that T*pslcore* is strong enough to prove most of the theorems in Tgso. The main philosophical difference between Tgso and T*pslcore* is that causality relations are treated as logical relations without mentioning the concept of time in Tgso while the