This is one illustration that using reflexivity in the ontology relieves us from the burden of tailoring definitions to capture formal degenerate cases.

The next example exhibits the general case β1 6= β2 **in the pattern given**
above.

Example. Let O = {Heard bell →IS−A Heard **noise**} and

$$C=\left\{\begin{array}{c}{{\begin{array}{c}{{O n\_a l a r m\ c a u s e\ H e a r d\_b e l l}}\\ {{H e a r d\_n o i s e\ c a u s e\ D i s t u r b a n c e}}\end{array}}\end{array}}\right.$$
$d\_b e ll$  $\underset{\downarrow}{\operatorname{tr}}\underset{\downarrow}{\operatorname{b}}\underset{\downarrow}{\operatorname{a}}\underset{\downarrow}{\operatorname{tr}}\underset{\downarrow}{\operatorname{b}}\underset{\downarrow}{\operatorname{a}}\underset{\downarrow}{\operatorname{tr}}\underset{\downarrow}{\operatorname{b}}\underset{\downarrow}{\operatorname{a}}\underset{\downarrow}{\operatorname{tr}}\underset{\downarrow}{\operatorname{b}}\underset{\downarrow}{\operatorname{a}}\underset{\downarrow}{\operatorname{tr}}\underset{\downarrow}{\operatorname{b}}\underset{\downarrow}{\operatorname{a}}$

$$V\;\;s t a t e s\;\;t h e$$

W states the facts induced by C and O*, that is:*

$\mathbf{a}\cdot\mathbf{a}=\mathbf{a}$. 
$$W=\left\{\begin{array}{c}{{O n\_a l a r m\to H e a r d\_b e l l}}\\ {{H e a r d\_n o i s e\to D i s t u r b a n c e l}}\\ {{H e a r d\_b e l l\to H e a r d\_n o i s e}}\end{array}\right.$$
$C\:\:and\:\:Q\:\:\:that\:\:i$. 
So, for the inference relation ⊢C , C ∪ O ∪ W *infers the atom:*
On **alarm** *explains* **Disturbance** bec *poss* {On **alarm**}.

In the second form of transitivity, β1 is inherited from β2 **by going downward**
in the IS-A **hierarchy.**

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

If α causes β1, β2 **causes γ,**
β2 →IS−A β1, and W 6|= ¬(α ∧ β2),

![0_image_1.png](0_image_1.png)

then α explains γ bec *poss*{**α, β**2}.

Example. O = {Heard loud bell →IS−A Heard **bell**}
C =
On alarm causes Heard **bell**
Heard loud **bell causes Deafening** 

W =

![0_image_3.png](0_image_3.png)

Heard loud bell → Heard **Deafening**

$\sqsubseteq$ . 

![0_image_2.png](0_image_2.png)

On **alarm** *does not* cause Heard loud **bell** *(neither does it* cause **Deafening**), but it is an **explanation** for Heard loud **bell** by virtue of the downward scheme.

Due to the base case, Heard loud **bell** *is in turn an explanation for* **Deafening**.

In fact, On **alarm** *is an explanation for* **Deafening** *by virtue of transitivity.*
Considering a causal operator which is transitive would give the same **explanations but is obviously more restrictive as we may not want to endorse an**
account of causality which is transitive. Moreover, transitivity for **explanations** not only seems right in itself but it also means that our model of explanations can be plugged with any causal system whether transitive or not.

The preceding examples are here to introduce the general pattern for transitivity of explanations which is as follows:

If α explains β bec *poss* Φ,

β explains γ bec *poss* Ψ,

and W 6|= ¬V(Φ ∪ Ψ),

then α explains γ bec *poss* Φ ∪ Ψ.
