Working backwards: modus tollens

You know P → Q. You know Q is false. ∴ P is false too.

TCube(a)→Small(a)

Premise one: if a is a cube, then a is small.

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How it works

Suppose P → Q is true and Q is false. If P were true, the promise would come due and Q would be true. It isn’t. So P is false. This step is modus tollens.

It is the engine behind “that can’t be, because then…”. Every time a puzzle rule tells you what would follow, and you can see it doesn’t, you have ruled something out.

P → Q and ¬Q → ¬P are true in exactly the same worlds. The second is called the contrapositive. Modus tollens is modus ponens run on the contrapositive.

Worth memorizing

From P → Q and ¬Q, conclude ¬P.
This is modus tollens.
P → Q and ¬Q → ¬P say the same thing.
The second is the contrapositive of the first.
Q → P is a different sentence.
Swapping the sides without negating them is a mistake. See affirming the consequent.

Practice

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Question 1

Premises: Large(c) → Cube(c). ¬Cube(c). Which conclusion follows?

Question 2

“If a is not small, then a is not a cube.”

Question 3

((Tet(c)→Small(c))∧¬Small(c))→¬Tet(c)

Question 4

¬Small(b)→¬Cube(b)

Question 5

A knight’s statements are true. A says something false. What follows?

Where you’ll use this

Knights and KnavesA false statement means the speaker is no knight.QueensA queen here would leave a row empty. So no queen here.Process of eliminationRule things out until one is left

Next: Process of eliminationIt’s P or Q. It isn’t P. ∴ Q. With more options, rule out all but one and the last is forced.