Lesson 4.4 — Modus Tollens

Modus Ponens's twin. Instead of confirming the IF part, you deny the THEN part.

The Shape

You have an IF-THEN sentence. You confirm that the THEN part is FALSE. You conclude that the IF part must also be FALSE.

The Latin name means "the method of taking away." We took away the THEN, and the IF goes with it.

Why It Works

Look at the IF truth table:

Suppose you know two things:

• IF X THEN Y is TRUE (so we're in rows 1, 3, or 4 — not row 2)
• Y is FALSE (so we're in rows 2 or 4)

Both must hold, so we're in row 4. And in row 4, X is FALSE.

Modus Tollens in one sentence: when the consequent is false (and the IF holds), the antecedent must be false too.

Apply It Yourself

Same premise as 4.3. This time, instead of confirming the antecedent, you'll set the value of the consequent and see what falls out.

When you set the consequent to FALSE, only one row of the truth table is consistent — and in that row, the antecedent is also FALSE. That's the Modus Tollens move. When you set the consequent to TRUE, two rows fit; the antecedent could be either. No conclusion follows from that direction.

Everyday Examples

"If I had eaten breakfast, I wouldn't be hungry. But I am hungry. So I didn't eat breakfast."

"If the alarm had gone off, the cat would be hiding. The cat isn't hiding. So the alarm didn't go off."

"If she had said yes, we'd be celebrating. We're not celebrating. So she didn't say yes."

Modus Tollens is detective work. You know what should have happened. The evidence isn't there. So the trigger didn't happen.

Modus Ponens vs. Modus Tollens

Both are valid. Both let you draw conclusions from an IF-THEN. The two moves cover the two informative directions; the other two combinations (confirming the THEN, or denying the IF) tell you nothing — those are the fallacies in Lessons 5.1 and 5.2.

A Real-World Trap

Same as Modus Ponens: the IF-THEN itself needs to be true. Modus Tollens is valid logic, but if the original IF is wrong, the conclusion can still be off.

Valid shape. But Premise 1 might be false — some happy people don't smile much. The conclusion is unreliable.

Practice

For each, decide: is this Modus Tollens? Is it valid?

1. "If I had studied, I would have passed. I didn't pass. So I didn't study."
2. "If I had studied, I would have passed. I didn't study. So I didn't pass."
3. "If it had rained, the street would be wet. The street is dry. So it didn't rain."
4. "If she's lying, her face turns red. Her face isn't red. So she's not lying."

Answers:

1. Yes — Modus Tollens. Valid.
2. NO — this is "denying the antecedent," a fallacy we'll cover in Lesson 5.2. Invalid.
3. Yes — Modus Tollens. Valid.
4. Yes — Modus Tollens. Valid, provided the premise about face-turning-red is actually reliable.

Recap

• Modus Tollens: IF X THEN Y, plus NOT Y, gives you NOT X.
• The "detective" move — the expected result didn't happen, so the trigger didn't either.
• Valid whenever the IF-THEN itself is true.
• Often confused with "denying the antecedent" (Lesson 5.2). Don't.