Cum hoc ergo propter hoc

About

With this, therefore because of this. Two things occur together, and the company is taken for causation. Where post hoc reads the order of arrival, cum hoc requires no order at all; mere coincidence suffices.

Taxonomy - Causal fallacy, correlational causation error. Legitimate causal inference contrasts by demanding intervention or structural justification; co-occurrence is only where it begins.

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

The conjunction is the whole of the evidence; the arrow is supplied gratis.

Logical Analysis

Invalidity Statement:

(A ∧ B) ⇏ (A → B)

Co-occurrence does not entail causal direction.

Missing Necessary Condition

A valid causal inference would demand, at minimum, ruling out:

∃C (C → A ∧ C → B)

Some C may be producing both A and B. The fallacy does not exclude this; it does not raise the question.

Causal DAG Representation

Fallacious Assumed Model:

A → B

Correct Minimal Alternative Model:

A ← C → B

A hidden common cause C explains the correlation, and no direct link survives its arrival. The fallacy is again a model-selection error, committed this time on company rather than order.

Probabilistic/Statistical Formulation

Observed Correlation

P(B | A) > P(B)

Fallacious Inference

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

Conditional probability is mistaken for interventional probability; seeing is mistaken for doing.

Epistemic Representation

𝔼(A ∧ B) ⇏ □_cause(A → B)

Where:

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

Correlation is evidence of correlation; the causal belief must be earned elsewhere.

Countermodels

Common Cause

C → A ∧ C → B

Reverse Causation

B → A ∧ (A ∧ B)

Pure Coincidence

(A ∧ B) ∧ ¬(A → B) ∧ ¬(B → A)

Three models fit the same correlation, and only one was considered. The data cannot say which is true; the inference did not ask.

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.