Formal theory of causality

Dominoes are a clear example of causality.

Billiard balls are also a great example of causality.

This is a first-order theory of Template:W, so some familiarity with Template:W 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 Template:W.

This theory makes heavy use of the formal dictionary.

Preliminaries

• The Template:W is the set of all events.
• The letters c, d and e (with or without subscripts) are used as Template:W 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 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 Template:W 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 Template:W with intended reading "Big Bang".

Event

Ex is a Template:W with intended reading "x is an event".

Cause

xCy is a Template:W with intended reading "x is a cause of y".

Definitions

Branching causal relation.

Lineal causal relation.

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.

Axioms

The Big Bang is an event

The Big Bang is an event.

$Eb$

The Big Bang is a first cause

The Big Bang is a first cause.

$FCb$

Events have effects

Every event has at least one effect.

$\mathrm{\exists }ecCe$

Causality is asymmetric

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

$cCe\to \mathrm{\neg }eCc$

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.

$(cCd\wedge dCe)\to cCe$

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).

$\mathrm{\neg }eCe$

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.

$(cDCe\wedge dDCe)\to cCId$

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.

$ϵFSCe\to b\in ϵ$