Cum hoc ergo propter hoc

About

With this, therefore because of this. It is the fallacy of inferring that because two events or variables occur together, one must cause the other. Correlation is mistaken for causation. Unlike post hoc, no temporal order is required.

Taxonomy - Causal fallacy, correlational causation error. Contrasts with legitimate causal inference requiring intervention or structural justification.

Informal Example - "People who carry lighters are more likely to get lung cancer. Therefore, carrying a lighter causes cancer."

Minimal Symbolic Form

Canonical Form:

(A ∧ B) ⊢ (A → B)

Where:

• A ∧ B: A and B occur together
• A → B: A causes B
• ⊢: the reasoner infers

Logical Analysis

Invalidity Statement:

(A ∧ B) ⇏ (A → B)

Co-occurrence does not entail causal direction.

Missing Necessary Condition

A valid causal inference would require ruling out alternatives such as:

∃C (C → A ∧ C → B)

There may exist a common cause C producing both A and B.

Causal DAG Representation

Fallacious Assumed Model:

A → B

Correct Minimal Alternative Model:

A ← C → B

A hidden common cause C can explain the observed correlation between A and B without any direct causal link. This demonstrates the fallacy as a model-selection error based on correlation rather than temporal order.

Probabilistic/Statistical Formulation

Observed Correlation

P(B | A) > P(B)

Fallacious Inference

P(B | do(A)) > P(B)

The reasoner mistakes conditional probability for interventional probability.

Epistemic Representation

𝔼(A ∧ B) ⇏ □_cause(A → B)

Where:

• 𝔼(A ∧ B) = evidence of correlation
• □_cause = justified causal belief

Evidence of co-occurence alone does not justify causal certainty.

Countermodels

Common Cause

C → A ∧ C → B

Reverse Causation

B → A ∧ (A ∧ B)

Pure Coincidence

(A ∧ B) ∧ ¬(A → B) ∧ ¬(B → A)

Each model fits the data but contradicts the causal inference.

Historical Background

The error of inferring causation from coincidence is discussed implicitly by Aristotle under non causa pro causa, though not named in this form.

The phrase cum hoc ergo propter hoc appears in medieval and early modern logical manuals as a companion to post hoc ergo propter hoc, distinguishing false cause from simultaneity rather than succession.

In modern logic and statistics, the fallacy corresponds directly to confusing correlation with causation, a central concern in probability theory, experimental design, and causal inference.