The Froggy Problem

I’ve taught this as a lesson twice a year ever since I started working at the sixth form and I love it. Lewis Carroll had a maths degree from Oxford and filled all of his work with mathematical references particularly relating to branches of logic. The number of times that the empty set is the butt of the joke in the two Wonderland books is immense. He also wrote a lot of logic problems which we will have a look at after introducing a bit of notation.

We are going to use ⇒ to mean implies. So for example if A is the statement Lucy is a cat and B is the statement Lucy is a mammal, then A⇒B, but we can't necessarily say that B⇒A. All cats are mammals, but not all mammals are cats.

If we introduce the statement C: Lucy is an animal. Then we can see that if A⇒B and B⇒C then A⇒C. This the first of two universal rules that we will be employing.

The next symbol we will introduce is ¬ which is the pipe key from the the top left of your keyboard that you have probably never used before. We are going to use it to mean with converse of a statement. So if A is the statement that Lucy is a cat, then ¬A is the statement that Lucy is not a cat. The second rule is that if A⇒B then ¬B⇒¬A. So if Lucy is a cat then she is a mammal means the same thing as if Lucy is not a mammal then she is not a cat. This second statement is called the contrapositive of the first statement.

We are going to apply them to some problems set by Lewis Carroll. In each, we want to make the strongest possible statement that we can from a series of other statements. Here's the first:

(a)  All babies are illogical.
(b)  Nobody is despised who can manage a crocodile.
(c)  Illogical persons are despised.

Let's turn each into symbolic logic.

B⇒¬L

C⇒¬D

¬L⇒D

Where B is the property of the subject being a baby, L is being logical, C is the ability to manage crocodiles and D is being despised. We want to put all of these into one chain of implication. If we take the contrapositive of the second one we get D⇒¬C. So B⇒¬L⇒D⇒¬C so B⇒¬C, No baby can manage a crocodile. Or equivalently, by taking the contrapositive of this we get C⇒¬B which means: if you can manage a crocodile then you are not a baby.

Let's try another. Here are the statements if you want to have a go:

(a)  No interesting poems are unpopular among people of real taste.
(b)  No modern poetry is free from affectation.
(c)  All your poems are on the subject of soap-bubbles.
(d)  No affected poetry is popular among people of real taste.
(e)  No ancient poem is on the subject of soap-bubbles.

Most of my students find this quite difficult, but it is still a fun problem to have a go at. Let's get symbolic:

I⇒P

M⇒A

Y⇒B

P⇒¬A

¬M⇒¬B

Where (in short) I is interesting, P is popular with people of real taste, M is modern, A is full of affectation, Y are your poems and B is on the topic of (soap) bubbles.

Putting these together we can look for the letters which are only used once. Let's start with Y⇒B, which we need to connect to B which using a contrapositive ⇒M (I'm using the opposite of Modern as ancient). M⇒A directly which ⇒¬P using a contrapositive and finally ⇒¬I using another. So overall we get Y⇒B⇒M⇒A⇒¬P⇒¬I or Your poems are not interesting. A subtle insult! Alternatively we can take the contrapositive and get the equivalent statement that If the poem is interesting then it isn't yours.

I'm going to leave you with the epitome of a Lewis Carroll logic problems of this type. Having seen how much work goes into unravelling the verbal traps in the small problems above, behold the Froggy Problem. You will never see a more Victorian Puzzle full of waistcoat and cigars! Carroll died before publishing a solution but he claimed it contained a beautiful trap. There are various blogs publishing solutions online, but I leave it to you: what is the strongest stament you can make from these statements?

1. When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”
2. Whenever I let Froggy forget that 10 pounds he owes me, and he begins to strut about like a peacock, his mother declares “He shall not go out a-wooing!”
3. Now that Froggy’s hair is out of curl, he has put away his gorgeous waistcoat.
4. Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discover that my purse is empty.
5. When my tailor calls with his little bill, and I remind Froggy of that 10 pounds he owes me, he does not grin like a hyena.
6. When it is very hot, the thermometer is high.
7. When the day is fine, and I’m not in the humor for a cigar, and Froggy is grinning like a hyena, I never venture to hint that he’s quite the dandy.
8. When my tailor calls with his little bill and finds me with an empty pocket, I remind Froggy of that 10 pounds he owes me.
9. My railway shares are going up like anything!
10. When my purse is empty, and when, noticing that Froggy has got his gorgeous waistcoat on, I venture to remind him of that 10 pounds he owes me, things are apt to get rather warm.
11. Now that it looks like rain, and Froggy is grinning like a hyena, I can do without my cigar.
12. When the thermometer is high, you need not trouble yourself to take an umbrella.
13. When Froggy has his gorgeous waistcoat on, but is not strutting about like a peacock, I betake myself to a quiet cigar.
14. When I tell Froggy that he’s quite a dandy, he grins like a hyena.
15. When my purse is tolerably full, and Froggy’s hair is one mass of curls, and when he is not strutting about like a peacock, I go out on the roof.
16. When my railways shares are going up, and when it’s chilly and looks like rain, I have a quiet cigar.
17. When Froggy’s mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words.
18. When it is going to rain, and I am having a quiet cigar, and Froggy is not intending to go a-wooing, you had better take an umbrella.
19. When my railway shares are going up, and Froggy seems nearly mad with joy, that is the time my tailor always chooses for calling with his little bill.
20. When the day is cool and the thermometer low, and I say nothing to Froggy about his being quite the dandy, and there’s not the ghost of a grin on his face, I haven’t the heart for my cigar!