A first-order theory of causality

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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. My goal with this theory is not to prove anything useful or unexpected, but to describe the structure of causal systems and to introduce an elegant terminology to talk about causality.

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

This theory makes use of the formal glossary of philosophy.

Preliminaries[edit]

  • The domain of discourse will be the set of all events.
  • I will use the letters c, d and e, with or without subscripts, 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. All very neat.
  • When defining a new term, the usual symbol is ":=", but I prefer to use just ":" because it's simpler, beautiful, and consistent with dictionary practice.
  • For every definition, axiom and theorem, I will first state it in natural language, then in formal language, and in the case of theorems, I will then include a proof.
  • For aesthetic reasons, I will omit external universal quantifiers.

Primitives[edit]

Big Bang[edit]

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the meaning of the term on Wikipedia, here.

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

Event[edit]

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the meaning of the term on Wikipedia, here.

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

Cause[edit]

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the meaning of the term on Wikipedia, here.

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

Definitions[edit]

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

Effect[edit]

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

Direct cause[edit]

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.

Indirect cause[edit]

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.

Causal independence[edit]

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.

First cause[edit]

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

Full set of causes[edit]

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.

Causal chain[edit]

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

Axioms[edit]

The Big Bang is an event[edit]

The Big Bang is an event.

The Big Bang is a first cause[edit]

The Big Bang is a first cause.

Events have effects[edit]

Every event has at least one effect.

Causality is asymmetric[edit]

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

Causality is transitive[edit]

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

Theorems[edit]

Simple equivalences[edit]

Let c and d 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.

Events are infinite[edit]

No event is a cause of itself[edit]

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

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[edit]

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

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[edit]

The Big Bang is in every full set of causes.