The 'A 4 B 7' Example


Suppose someone places 4 cards on a table in front of you.

Written on the visible face of the cards is the following(one character per card):

A     B     4     7
Suppose it is also true that each one of these cards has a letter on one side and a number on the other (and nothing else on either side).

Now suppose someone claims that the following rule holds for these cards:

If a card has an A on one side, then it has a four on the other.
Question: Which cards would you have to check in order to decide whether the rule really does hold for these cards?

The most popular answer: Most people say the card with the A and the card with the 4.

The right answer: The correct answer is the card with the A and the card with the 7.

Why? Think about this in terms of the arguments we could construct using this rule. The first premise of each argument will be the rule.

P1: If a card has an A on one side, then it has a four on the other.
Now, we will think about each card as supplying us with the second premise of the argument. Then, we'll assess the argument to see whether the rule tells us anything about the card.

Card #1: 'A'

P1: If a card has an A on one side, then it has a four on the other.
P2: The card has an A on one side.
What can we conclude from this?
C: The card has a 4 on the other side.
So, we'd better flip this card to check our conclusion. Why is this true? Because arguments with the form:
P1: If P then Q
P2: P
C: Q
are deductively valid. This argument form is called modus ponens or affirming the antecedent.
 

Card #2: 'B'

P1: If a card has an A on one side, then it has a four on the other.
P2: The card does not have an A on one side.
What can we conclude from this?

Nothing. The rule doesn't tell us what happens if there is no A on the card. So, we don't need to flip the card.

If you thought you could conclude:

C: The card does not have a 4 on the other side
you'd be commiting the fallacy of denying the antecedent.

Why? Because you'd be acting as though an argument with the form:

P1: If P then Q
P2: It is not the case that P
C: It is not the case that Q
was a deductively valid argument, but it's not.
 

Card #3: '4'

P1: If a card has an A on one side, then it has a four on the other.
P2: The card has a four on one side.
What can we conclude from this?

Nothing: The rule doesn't tell us that only cards with A's on them may have 4's on them. So again, we don't need to flip the card. If you thought you could conclude:

C: The card has an A on one side.
You'd be commiting the fallacy of affirming the consequent.

Why? Because you'd be acting as though an argument with the form:

P1: If P then Q
P2: Q
C: P
was deductively valid, but it's not.

Card #4: '7'

P1: If a card has an A on one side, then it has a four on the other.
P2: The card does not have a 4 on one side.
What can we conclude from this?
C: The card does not have an A on the other side.
Why? If it did, then the rule (i.e., P1) would be false because we would have here a card with an A on one side, but not a 4 on the other. As such, we have to flip this card to check the rule.

Arguments with the form:

P1: If P then Q.
P2: It is not the case that Q.
C: It is not the case that P.
are deductively valid. This argument form is called modus tollens or denying the consequent.

The Moral: Don't confuse valid argument forms like modus ponens and modus tollens with fallacious argument forms like affirming the consequent and denying the antecedent.

This is a very tricky moral to live up to.

[Philosophy 1200]