Counterfactual Causation

Origin. David Hume raised the problem of causation; regularity theories proved insufficient. David Lewis (1973) developed the counterfactual analysis: "C causes E" means if C had not occurred, E would not have occurred. Builds on possible worlds semantics for counterfactual conditionals. Refined through debates with preemption, overdetermination, and late preemption cases.

Models. Causation as counterfactual dependence. What would have happened if things had been different? C causes E if E counterfactually depends on C — in the closest possible worlds where C doesn't occur, E doesn't occur either. Captures singular causation (this C caused this E) rather than general causal laws.

Formalism.

Counterfactual conditionals: φ □→ ψ (if φ were the case, ψ would be the case).

Semantics (Lewis): Possible worlds ordered by similarity to the actual world. φ □→ ψ is true at world w iff:

There are no φ-worlds (vacuously true), or
Some φ-world where ψ holds is closer to w than any φ-world where ψ doesn't hold

Similarity ordering: Lewis proposed criteria (most to least important):

Avoid big, widespread miracles
Maximize region of perfect match
Avoid small, localized miracles
Maximize approximate match

Causal dependence: E causally depends on C iff:

C □→ E (if C occurred, E would occur) — typically trivially true if both actually occurred
¬C □→ ¬E (if C had not occurred, E would not have occurred)

Causation: C causes E iff there is a chain of causal dependence from C to E. (This handles cases where intermediate events exist.)

Symbols.

Symbol	Unicode	Name	Meaning
□→	—	Counterfactual	If...were...would; a two-character digraph
◇→	—	Might-counterfactual	If...were...might
□	U+25A1	White square / Box	The first half of the would-digraph
◇	U+25C7	White diamond	The first half of the might-digraph
≤w	—	Closer-than	World similarity ordering
→	U+2192	Rightwards arrow / Material conditional	Standard implication
¬	U+00AC	Not sign / Negation	Not
w	—	World	Point of evaluation, normally the actual world
C	—	Cause	Candidate cause event
E	—	Effect	Effect event
D	—	Backup	The preempted alternative cause
VC	—	System name	Lewis's logic of counterfactuals

Metatheory. The underlying conditional logic is settled: Lewis (1973) proved VC sound and complete for systems of spheres ordered by comparative similarity, without assuming the limit assumption, and satisfiability is decidable — PSPACE-complete for arbitrarily nested conditionals (Friedman and Halpern, 1994). The analysis of causation built on it is not a logic but a definition, and its structure is dictated by two failures. Counterfactual dependence is not transitive, so Lewis defined causation as its ancestral, a strictly stronger relation than dependence; the closure is what handles early preemption, where the backup is cut off before it acts and so the intermediate links each depend on their predecessor even though the effect does not depend on the cause. Late preemption defeats the chain analysis, since there is no intermediate to break, which is why Lewis (2000) replaced chains with influence. The structural-equations reconstruction of Halpern and Pearl (2001, 2005) makes the definition computational, and Eiter and Lukasiewicz (2002) fixed its cost: deciding actual causation is Σ₂ᴾ-complete in general and NP-complete for binary models.

Applies to. Philosophy of causation. Legal reasoning (causation in law). Historical counterfactuals. Explanation (explaining why by citing what would have been different). Moral responsibility (the agent's action made the difference).

Limitations. Dependence is sufficient but not necessary for causation, and the two standard counterexamples are structural rather than marginal. Under symmetric overdetermination — two assassins, one victim — removing either leaves the effect intact, so ¬C □→ ¬E is false for both and the test returns no cause at all. Under late preemption the backup is cut off only by the effect itself, so no intermediate event breaks the chain and the ancestral closure does not repair the verdict. The closure introduces its own defect: causation so defined is transitive by construction while dependence is not, which yields the wrong answer wherever an intermediate both raises and lowers the chance of the effect. The verdicts also turn on parameters the theory leaves open — how finely events are individuated, since a more fragile effect depends on more; whether backtracking readings are admitted; and how the similarity ordering weighs miracles against match, which Fine (1975) showed cannot be settled by overall resemblance. The analysis is of token causation, and gives no account of type-level causal claims.