# Formal theory of causality

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 Subject classification: this is an Philosophy resource.
 Educational level: this is a research resource.
Dominoes are a clear example of causality.
Billiard balls are also a great example of causality.

This is a first-order theory of causality, so some familiarity with first-order logic is assumed. The goal of this theory is not to prove anything useful or unexpected, but to describe the structure of causal systems and to develop an elegant terminology to talk about causality.

According to this theory, the structure of causal systems is probably that of a directed acyclic graph.[1]

This theory makes heavy use of the formal dictionary.

## Preliminaries

• The domain of discourse is the set of all events.
• The letters c, d and e (with or without subscripts) are used as variables for events, instead of the usual x, y and z. The letter e is meant to evoke the word "event" and sometimes the word "effect". The letter c is meant to evoke the word "cause", and the letter d is meant to evoke some intermediate event between c and e. So often, c will be the cause of d, and d the cause of e.
• Every definition, axiom and theorem is first stated it in natural language, then in formal language, and in the case of theorems, it's followed by a proof.
• For aesthetic reasons, external parenthesis and universal quantifiers are omitted.
• When defining a new term, the usual symbol is ":=", but here we use just ":" per being simpler, beautiful and consistent with dictionary practice.

## Primitives

### Big Bang

b is an individual constant with intended reading "Big Bang".

### Event

Event is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Event (philosophy) at Wikipedia.

Ex is a one-place predicate with intended reading "x is an event".

### Cause

Cause is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Cause at Wikipedia.

xCy is a two-place predicate with intended reading "x is a cause of y".

## Definitions

Visual representation of a causal split.
Visual representation of a causal tree.
Visual representation of a causal merge.

The definitions in this section are all taken from the Formal dictionary.

### Effect

Let c and e be events. Then e is an effect of c means: c is a cause of e.

${\displaystyle eEc:cCe}$

### Direct cause

Let c, d and e be events. Then c is a direct cause of e means: c is a cause of e and there is no d such that c is a cause of d and d is a cause of e.

${\displaystyle cDCe:cCe\land \lnot \exists d(cCd\land dCe)}$

### Indirect cause

Let c and e be events. Then c is an indirect cause of e means: c is a cause of e, but c is not a direct cause of e.

${\displaystyle cICe:cCe\land \lnot cDCe}$

### Causal independence

Let c and e be events. Then c is causally independent of e means: c is not a cause of e and e is not a cause of c.

${\displaystyle cCIe:\lnot cCe\land \lnot eCc}$

### First cause

Let e be an event. Then e is a first cause means: there is no event c that is a cause of e.

${\displaystyle FCe:\lnot cCe}$

### Full set of causes

Let e be an event and ε be a set of events. Then ε is a full set of causes of e means: for every event c that is a cause of e, c is an element of ε.

${\displaystyle \epsilon FSCe:cCe\to c\in \epsilon }$

### Causal chain

A sequence of events (e1, e2, e3 ..., en) is a causal chain means: e1 is a cause of e2, e2 is a cause of e3 and so on until en-1 is a cause of en

${\displaystyle CC(e_{1},e_{2},e_{3}...,e_{n}):e_{1}Ce_{2}\land e_{2}Ce_{3}...\land e_{n-1}Ce_{n}}$

## Axioms

### The Big Bang is an event

The Big Bang is an event.

${\displaystyle Eb}$

### The Big Bang is a first cause

The Big Bang is a first cause.

${\displaystyle FCb}$

### Events have effects

Every event has at least one effect.

${\displaystyle \exists e\ cCe}$

### Causality is asymmetric

If event c is a cause of event e, then e is not a cause of c.

${\displaystyle cCe\to \lnot eCc}$

### Causality is transitive

If event c is a cause of event d, and d is a cause of event e, then c is a cause of e.

${\displaystyle (cCd\land dCe)\to cCe}$

## Theorems

### Simple equivalences

Let c and e be events. Then:

• c is a cause of e iff e is an effect of c
• c is a direct cause of e iff e is a direct effect of c
• c is a direct cause of e iff c is not an indirect cause of e
• c is an indirect cause of e iff e is not a direct cause of c
• e is a direct effect of c iff c is not an indirect effect of e
• c is an indirect cause of e iff e is an indirect effect of c

### No event is a cause of itself

No event is a cause of itself (causality isn't reflexive).

${\displaystyle \lnot eCe}$

For a proof by contradiction, suppose some event e is a cause of itself. Then by the axiom of asymmetry, e is not a cause of itself. But this is a contradiction, so no event can be a cause of itself. QED

### Direct causes of the same effect are causally independent

If event c and event d are both direct causes of event e, then c and d are causally independent.

${\displaystyle (cDCe\land dDCe)\to cCId}$

For a proof by contradiction, suppose c and d are both direct causes of e, but are not causally independent. Then c must be a cause of d, or d must be a cause of c, or both. It cannot be both, because causality is asymmetric. So suppose c is a cause of d. Then c is an indirect cause of e, as well as a direct cause. But this is a contradiction. The same happens if we suppose d is a cause of c. Therefore, c and d must be causally independent. QED

### The Big Bang is in every full set of causes

The Big Bang is in every full set of causes.

${\displaystyle \epsilon FSCe\to b\in \epsilon }$

## Notes

1. Proof needed.