Post hoc ergo propter hoc

About

After this, therefore because of this. One event follows another, and the following is taken for consequence; the inference consults nothing but the order of arrival.

Taxonomy - Causal fallacy, temporal causation error. Genuine causal inference contrasts by demanding independent evidence; succession alone is never asked to carry it.

Informal Example - "The rooster crowed, then the sun rose; therefore the rooster caused the sunrise."

Minimal Symbolic Form

Canonical Form:

(A < B) ⊢ (A → B)

Where:

• A < B: A precedes B in time
• A → B: A causes B
• ⊢: the reasoner infers this

The entire fallacy resides in the turnstile: from "A earlier than B" the reasoner passes to "A caused B," and pays nothing on the way through.

Logical Analysis

Invalidity Statement:

(A < B) ⇏ (A → B)

Temporal succession does not entail causal implication.

Missing Necessary Condition

A valid causal inference would demand, at minimum:

¬∃C (C < A ∧ C → B)

No earlier variable C may be causing B. The fallacy does not establish this; it does not even inquire. Hence the gap.

Causal DAG Representation

Fallacious Assumed Model:

A → B

Correct Minimal Alternative Model:

A ← C → B

A hidden common cause C explains both A and B, and the assumed arrow was never needed. The fallacy is thus a model-selection error: a graph preferred for being the first to arrive.

Temporal Representation

Let:

• ◇B: B eventually occurs
• A U B: A holds continuously until B occurs

The fallacy corresponds to the invalid inference:

(A U B) ⊢ (A → B)

Or equivalently:

◇(A ∧ ◇B) ⇏ (A → B)

Even with the full apparatus of temporal logic, precedence purchases no causation.

Epistemic Representation

Epistemic Misstep

𝔼(A < B) ⇏ □_cause(A → B)

Where:

• 𝔼(A < B): the evidence that A precedes B
• □_cause: justified causal belief

Evidence of succession justifies belief in succession, and in nothing further. The failure is one of evidential standards, not of observation; the reasoner saw correctly and concluded anyway.

Countermodel

Coincidence Model

A < B ∧ ¬(A → B)

Reverse Causation Model

B → A ∧ A < B

Confounder Model

C → A ∧ C → B ∧ C < A

Three incompatible models, one temporal record. The data elects no winner; the inference simply declared one.

Historical Background

The underlying error of inferring causation from mere succession is discussed by Aristotle, particularly in his treatment of non causa pro causa ("the non-cause taken as cause"). The exact phrase post hoc ergo propter hoc does not appear in Aristotle, but the fallacy is recognized as a mistaken causal inference based on temporal order.

During the medieval Scholastic period, the Latin expression becomes standard in logical handbooks, where it is treated as a subtype of fallacia causae falsae. Scholastic authors distinguish it from related forms such as cum hoc ergo propter hoc.

In the early modern era (16th-18th centuries), the phrase appears consistently in logic manuals and university textbooks throughout Europe, solidifying its canonical form.

By the 19th and 20th centuries, with the rise of inductive logic and later probabilistic and causal analysis, the fallacy is reinterpreted as a formal error of causal identification: assuming an arrow A → B without eliminating confounding factors or alternative causal models.

Modern treatments in statistics and the philosophy of science describe the fallacy as a structural error in causal reasoning, independent of temporal logic.