Logic - Modus Tollens

Modus tollens is modus ponens’s sharper sibling, and the logic behind every good experiment. The truth: if a prediction fails, the idea that made it is in trouble.

The core idea, in plain words

If you know “if $p$ then $q$ ” and you know $q$ is false, you may conclude $p$ is false.

\frac{p \to q \qquad \neg q}{\therefore\ \neg p}

The Latin is modus tollendo tollens, “the mode that denies by denying”. It is valid because $p \to q$ is logically equivalent to its contrapositive $\neg q \to \neg p$ , and modus tollens is just modus ponens applied to the contrapositive.

$p$	$q$	$p \to q$	$\neg q$
T	T	T	F
T	F	F	T
F	T	T	F
F	F	T	T ← both premises true, $p$ false

An everyday analogy

“If the dog had barked, the neighbour would have woken. The neighbour slept through. So the dog didn’t bark.” (Sherlock Holmes’s “dog that did nothing in the night-time” is this very argument.)

A small worked example

If $n$ is divisible by 4, then $n$ is even.
$30$ is not divisible by 4… careful, that’s the antecedent. Let’s use $n = 31$ : $31$ is not even.
∴ $31$ is not divisible by 4. ✔

Notice the trap I nearly walked into in step 2: knowing 30 isn’t divisible by 4 tells you nothing about whether 30 is even. That’s the next fallacy.

The impostor: denying the antecedent

\frac{p \to q \qquad \neg p}{\therefore\ \neg q} \quad \text{(INVALID)}

“If it’s raining, the pavement is wet. It’s not raining. So the pavement is dry.” The car-washer strikes again. Numerically: “if divisible by 4 then even; 30 is not divisible by 4; so 30 is not even.” False conclusion from true premises, so the form is invalid (row 3 of the table).

The four forms at a glance

Form	Premises	Conclusion	Valid?
Modus ponens	$p \to q$ , $p$	$q$	✔
Modus tollens	$p \to q$ , $\neg q$	$\neg p$	✔
Affirming the consequent	$p \to q$ , $q$	$p$	✘
Denying the antecedent	$p \to q$ , $\neg p$	$\neg q$	✘

A picture in words

Draw $p$ as a small circle inside a bigger circle $q$ (everything that is $p$ is $q$ ). If you are outside $q$ , you are certainly outside $p$ : modus tollens. If you’re outside $p$ , you could still be in the ring of $q$ : that’s why denying the antecedent fails.

graph LR
  NQ["not q"] -- "contrapositive: ¬q → ¬p" --> NP["so not p"]

Common misconceptions

“The contrapositive and the converse are the same.” The contrapositive $\neg q \to \neg p$ is equivalent to $p \to q$ . The converse $q \to p$ and the inverse $\neg p \to \neg q$ are not (they are equivalent to each other).
“A failed prediction disproves my particular hypothesis.” In practice $p$ is a bundle of assumptions (theory + instruments + conditions). Modus tollens tells you at least one of them is false, not which one (the Duhem–Quine problem).
“Tollens is a proof by contradiction.” Close cousins, not identical, but every tollens step can be rewritten as one.

How it connects

Modus Ponens is its twin; together they’re the two valid ways to use a conditional.
Karl Popper’s falsificationism is modus tollens as a philosophy of science: theories can be refuted by failed predictions but never proven by successful ones (that would be affirming the consequent). See Philosophy and Logic.

The truth revealed: you can’t prove an idea by its successes, but a single honest failure can bring it down.