In the following, α, β, . . . denote the propositional atoms and Φ, Ψ**, . . .**
denote sets thereof.

## Atoms

1. *Propositional atoms*: **α, β, . . .**. 2. *Causal atoms*: **α causes β**.

3. *Ontological atoms*: α →IS−A β.

4. Explanation atoms: α explains β because *possible* Φ.

An ontological atom reads: α **is a** β. An explanation atom reads: α is an explanation for β because Φ is possible.

Notation: **In order to help reading long formulas, explanation atoms are**
sometimes abbreviated as α expl β bec *poss* Φ.

## Formulas

1. *Propositional formulas***: Boolean combinations of propositional atoms.** 2. *Causal formulas***: Boolean combinations of causal or propositional atoms.**
The premises of the inference ⊢C, namely C ∪ O ∪ W**, consist of propositional and causal formulas, and ontological** *atoms* **(no ontological formulas).**
Notice that explanation atoms cannot occur in the premises.

The properties of causal and ontological formulas 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.