
## 2.2.5 Explanation Provisos And Their Simplifications

Explanation atoms are written α explains β because *possible* **Φ as the**
definition is intended to make the atom true just in case it is successfully checked that the proviso is possible: An explanation atom is not to be interpreted as a kind of conditional statement. Indeed, we do not write "if *possible"***. The** argument in "because *possible"* **gathers those conditions that must be possible**
together if α is to explain β (there can be others: α **can also be an explanation** of β **with respect to another set of arguments in** "because *possible"*).

Notice that the set of conditions in an explanation atom can often be simplified. Using V**Φ to denote the conjunction of the formulas in the set Φ, the**
following general scheme amounts to simplifying the proviso attached to an explanation atom.

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

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

As a simple motivating example for this general scheme, let us consider the case where we have α causes β and β causes γ.

Then we get α explains β bec *poss*{α} and β explains γ bec *poss*{β} by
§ **2.2.1, thus** α explains γ bec *poss*{α, β} **by the general pattern for transitivity**
§ 2.2.4. Now, we get α → β from α causes β, thus, W |= α **is equivalent to** W |= α∧β: the two sets of condition {α} and {α, β} **are equivalent here, which justifies** to simplify the explanation atom into α explains γ because *possible* {α}**. This**
provides a justification for the general pattern of simplification of **the set of**
condition where n = 1, Φ = {α} **and Φ**1 = {β}.

More complex examples may involve disjunctions, such as in the small following example:
Let us suppose that we have derived the following two explanation atoms:
α explains γ bec *poss*{α, β1} and α explains γ bec *poss*{α, β2}**. Let us** suppose that W contains α → (β1 ∨ β2). Then, as soon as α**, together with**
either β1 or β2, is possible, we get that α explains γ. Now, [W 6|= ¬(α ∧ β1**) or** W 6|= ¬(α∧β2)] is equivalent to [W 6|= ¬(α∧(β1 ∨β2**))] and, from** α → (β1∨β2),
this is equivalent to [W 6|= ¬α].

Thus, it is natural to get the explanation atom α explains γ bec *poss*{α}.

This is the general pattern for simplification of the conditions of explanation where n = 2, Φ = {α}, Φ1 = {β1} and Φ2 = {β2}**. Generalizing this example**
produces naturally the general pattern for simplification of the conditions.

## 2.3 A Formal System For Inferring Explanations

The above ideas are embedded in a short proof system extending classical logic: