Activity 2-5: What are you implying?

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You are given these four cards:

Each of these cards has a letter on one sideand a digit on the other.

This rule may or may not be true:if a card has a vowel on one side,

then it has an even number on the other.

Which cards must you turnto check the rule for these cards?

This is a famous question that has tested people’s grasp of logic for years.

Most people pick Card with A, correctly.

Many pick Card with 2 in addition.

The right answer is to pick Card with 7 in addition to the A.

But… if the other side of the Card with 2 is a vowel, that’s fine.

And if the other side of the Card with 2 is a consonant, that’s fine too!

If the other side of the 7 is a vowel, there IS a problem.

In the statements below, a, n and m are positive integers

1. a is even

2. a2 is even

3. a can be written as 3n + 1

4. a can be written as 6m + 1

Consider these four assertions:

If we make a card with one statement on the front and one on the back, there are six possible cards we could make.

Define A I B to signify ‘A implies B’, Define A RO B to signify ‘A rules out B’, and

A NINRO B to mean ‘A neither implies nor rules out B’.

Task: how many different types of card do we have with these definitions?

We can see that the statements for Card 1 mean that this is of type (I, I): each side implies the other.

We conclude we have four different types of card:

1 is (I, I),3 and 5 are (RO, RO),

2 and 4 are (NINRO, NINRO), while 6 is (I, NINRO).

Are these the only possible types of card?

Yes, since if A RO B, then B RO A, so (RO,I) and RO, NINRO)

are impossible cards to produce.

Can we be sure that if A RO B, then B RO A?

We can use a logical tautology called MODUS TOLLENS:

if A implies B, then (not B) implies (not A).

Now A RO B means A I (not B).If A I (not B), then by Modus Tollens,

not (not B) implies (not A).Thus B implies (not A), that is B RO A.

What mistake do people make with the four-card problem?

They assume that A I B can be reversed to B I A.

So Vowel I Even reverses to (Not Even) I (Not Vowel),which shows we need to pick the 7 card.

if A implies B, then (not B) implies (not A).

In fact, A I B CAN be reversed, but to (not B) I (not A) – Modus Tollens.

Let’s try a variation on our initial four-card problem.

You are given four cards below. Each has a plant on one side and an animal on the other.

Task: you are given the rule:if one side shows a tree, the other side is not a panda.

Which cards do you need to turn over to check the rule?

This time the obvious reversal works, for ‘Tree RO Panda’ is the same as ‘Panda RO Tree’:

you DO need to turn over the two cards named in the question.

Given two circles, there are four ways that they can lie in relation to each other.

Task:if you had to assign

(I, I),(RO, RO),

(NINRO, NINRO), and (I, NINRO)

to these, how would you do it?

Let’s invent a new word, DONRO, standing for DOes Not Rule Out.

Task: is it true that if A DONRO B and B DONRO C, then A DONRO C?

Pick an example to illustrate your answer.

A: x = 2B: x2 = 4C: x = 2

A: The shape S is a red quadrilateral

B: The shape S is a rectangle

C: The shape S is a blue quadrilateral

A: ab is evenB: b is less than a

C: ab is odd

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net

With thanks to:The Open University and my teachers on the Researching Mathematical Learning course.

Mathematics In School for publishing my original article on this subject.