Some desultory logic

by , under thinking

(Or: what my brain does while my body swims.)

I have a pack of cards with letters on one side and numbers on the other. I lay out four of them on the table, like this:

E 6 J 11

I tell you “Each card with an even number has a vowel on the other side.”

How many cards, and which ones, do you need to turn over to prove me right or wrong? Is it even possible to prove me right or wrong from these four cards? For the cards on the table only, or for the whole pack?

Let’s consider E. Well, let’s not consider E because E isn’t worth consideration. The rule doesn’t actually say anything about the other side of a card with a vowel, except that some of them have even numbers. Whether E has an odd or an even number on the other side doesn’t fall under the scope of the rule.

Do you turn over 6? Yes, you do, because if it has a consonant on the other side you’ve proven me wrong. Conversely, you need to make sure that J doesn’t have an even number because even numbers need vowels. So you turn over J.

There’s no need to turn over 11 because the rule doesn’t cover odd numbers at all: whether there’s a vowel or a consonant on the other side of 11 doesn’t matter.

The inverse of my rule is not “Each card with an odd number has a consonant on the other side” but “Each card with a consonant has an odd number on the other side”. Cards with vowels can have odd or even numbers; cards with odd numbers can have vowels or consonants.

Interestingly, it’s possible that you’ll prove me wrong for the whole pack from these four cards: if 6 should have a consonant, or J an even number. You can’t prove me right for the whole pack; for that you need to check all even-numbered and all consonant cards in the pack. You can, of course, prove me right for only the cards on the table from the cards on the table.