Activity 2-5: What are you implying?

You are given these four cards: Each of these cards has a letter on one side and 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 turn to 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. a 2 is even 3. a can be written as 3n 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 = 2 B: x 2 = 4 C: 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 even B: b is less than a C: ab is odd

Carom is written by Jonny Griffiths, 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.