The premises of the inference ⊢C, namely C ∪ O ∪ W**, consist of propositional**
and causal formulas, and ontological *atoms* **(no ontological formula). Notice**
that explanation atoms cannot occur in the premises.

The properties of causal and ontological formulas we consider are **as follows.**

## 1. **Properties Of The Causal Operator**

(a) *Entailing [standard] implication*: If α causes β**, then** α → β.

## 2. **Properties Of The Ontological Operator**

(a) *Entailing implication*: If α →IS−A β**, then** α → β.

(b) *Transitivity*: If a →IS−A b and b →IS−A c**, then** a →IS−A c.

(c) *Reflexivity*: c →IS−A c.

Reflexivity is an unconventional property for an IS-A **hierarchy. It is included here because it helps keeping the number of inference schemes low (see**
later).

W is supposed to include (whether explicitly or via inference) all the implications induced by the ontological atoms. For example, if Heard loud **bell** →IS−A
Heard bell is in O then Heard loud bell → Heard bell is in W**. Similarly,** W **is supposed to include all conditionals induced by the causal statements**
in C. For example, if Flu causes F ever T emperature is in C, then Flu → F ever T emperature **is in** W.

## 2.2 Patterns For Inferring Explanations

A set of patterns, introduced in [BCM07], is proposed to infer explanations from premises C ∪ O ∪ W. Before providing the rules (see § **2.3 below), let us**
motivate these rules by listing the main patterns that we consider as **desirable.**
The base case explains β from α whenever α causes β (§ **2.2.1). More elaborate** cases explain β from α whenever α causes **some** β
′ **ontologically related with**
β (§ 2.2.2, 2.2.3). Finally, explanations should be transitive (almost) (§ **2.2.4).**

## 2.2.1 The Base Case

A basic idea is that what causes an effect can always be suggested as **an explanation when the effect happens to be the case:**

|  If     | α causes β           | then   | α explains β   | because possible {α}.   |
|-----|----------|--------|----------------|-------------------------|
| and | W 6|= ¬α |        |                |                         |

Example. *Consider a causal model such that* W 6⊢ ¬Flu and O *is empty* whereas C = {Flu causes F ever **T emperature**}.

Then, the atom Flu *explains* F ever **T emperature** because *possible* {Flu}
is inferred. That is, Flu *is an explanation for* F ever **T emperature**.