Example. **Getting cold usually causes Mary to become active. I see Mary**
jogging. So, Mary getting cold might be taken as an explanation for her jogging.

C = {Getting cold(mary) causes Moving up(**mary**) }
O = {Jogging →IS−A **Moving** up}.

For now, W **does not contain any special information (only the consequences**
of the preceding causal and ontological atoms).

We get Jogging(**M ary**) →IS−A(aug.) Moving up(M ary**) from (2**ext
′).

Thus W = { Getting cold(mary) → Moving up(**mary**),
Jogging(mary) → Moving up(**mary**).}
"Mary getting cold" can be inferred as an explanation for "Mary is jogging".

Indeed, the causal theory entails Getting cold(**mary**)*explains* Jogging(**mary**)
bec *poss*{Getting cold(mary), Jogging(mary)}. (**EXP L**)
If now we add the fact that if the weather is not cold, then Mary cannot get cold, this explanation is no longer possible in warm weather.

Adding the following formulas to W **takes the new information into account:** W arm W eather, ¬(W arm W eather ∧ Cold **W eather**),
¬Cold W eather → ¬Getting cold(**mary**). Then, the causal theory fails to entail the explanation atom (**EXP L**).

## 4 About A Few Features Of The Formalism

The explanation inference follows the patterns presented before. The inference pattern (3a) in § 2.3 (or pattern (3a) in § **3.3) is important.**

1. In this inference pattern, the direction of the →IS−A links a →IS−A b and a →IS−A c **is important. Unexpected conclusions would ensue if other** directions, e.g., b →IS−A a and c →IS−A a**, were allowed.**
2. Also, there are good reasons for excluding conclusions not endorsed by this pattern. One reason is that it is better to limit the explanations to a minimum, otherwise an overwhelming set of "explanations" could result. As an example of conclusions not endorsed, notice that no explanation atom can be derived if it does not start with a ground atom which occurs somewhere on the left side of a causal atom.
As a short justification of these two points, let us introduce an example which comes from a real-world application [SMOC+**98, BC99], but has been drastically**
simplified and reduced. The causal model describes a physical system in which a sliding of the flywheel **(to be abbreviated as** SOF**) causes a** *step* **in the vibration** measurement signal. It is also known that *step* and *slow increase* **are two kinds** of evolution of the vibration measurement signal and that a *sharp step* **is itself**
a kind of step.

This is to be formalized as follows (the example has been simplified for the sake of conciseness and clarity, thus the propositional version of § **2 suffices).**