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2.3: Test your understanding of the conditional

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  • People sometimes find conditionals confusing.  In part, this seems to be because some people confuse them with another kind of truth-functional connective, which we will learn about later, called the “biconditional”.  Also, sometimes “if…then…” is used in English in a different way (see section 17.7 if you are curious about alternative possible meanings).  But from now on, we will understand the conditional as described above.  To test whether you have properly grasped the conditional, consider the following puzzle.[4]

    We have a set of four cards in figure 2.1.  Each card has the following property:  it has a shape on one side, and a letter on the other side.  We shuffle and mix the cards, flipping some over while we shuffle.  Then, we lay out the four cards:

    cards r square q star



    Figure 2.1


    Given our constraint that each card has a letter on one side and a shape on the other, we know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side; and so on.

    Consider now the following claim:

    For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card.

    Here is our puzzle:  what is the minimum number of cards that we must turn over to test whether this claim is true of all four cards; and which cards are they that we must turn over?  Of course we could turn them all over, but the puzzle asks you to identify all and only the cards that will test the claim.

    Stop reading now, and see if you can decide on the answer.  Be warned, people generally perform poorly on this puzzle.  Think about it for a while.  The answer is given below in problem 1.

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