A first-order theory 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.
- 1 Preliminaries
- 2 Primitives
- 3 Definitions
- 4 Axioms
- 5 Theorems
- 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.
b is an individual constant with intended reading "Big Bang".
Ex is a one-place predicate with intended reading "x is an event".
xCy is a two-place predicate with intended reading "x is a cause of y".
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.
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
The Big Bang is an event
The Big Bang is a first cause
Events have effects
Causality is asymmetric
Causality is transitive
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
No event is a cause of itself
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
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