Carroll’s Froggy Problem

I did not expect the post-Watson stir-craziness to be quite this intense.

The next few months are going to involve a lot of anxious uncertainty, so what better way to procrastinate my thesis work*** (on Leibniz, of course) than to putz around in the silly logic puzzles of Lewis Carroll? Apparently some people have considered this problem in light of Carroll’s writings on logic, and that gets really technical so I’m just going to dust off Formal Logic 101 for this. Which means my answer, all other error aside, might be totally off-base. Part of the reason I’m even ok posting this is because I suspect most people are too busy or too lazy to look through for my mistakes. But if you do look through this and find them, tell me!

What is the strongest conclusion that can be drawn from these premises? (See link above for source)

  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 doesnot 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!

Anyway, I decided to translate this as best I could into logic. Here’s what I came up with, and the strongest conclusion I drew:

First, the predicates.****

A – the day is fine (¬A – it looks like rain, it is going to rain)

B – I tell Froggy “You’re quite the dandy, old chap!” (¬B – I never venture to hint Froggy is quite the dandy//I say nothing to Froggy about his being quite the dandy)

C – I let Froggy forget the 10₤ he owes me (¬C – I remind Froggy of the 10₤ he owes me)

D – Froggy struts like a peacock (¬D – Froggy is not strutting around like a peacock)

E – Froggy’s mother declares “He shall no go a-wooing!” (¬E – Froggy’s mother lets him go a-wooing)

F – Froggy’s hair is out of curl (¬F – Froggy’s hair is one mass of curls)

G – Froggy has put away his gorgeous waistcoat (¬G – Froggy has a gorgeous waistcoat on)

H – I go to the roof

I – I enjoy a quiet cigar (¬I – I am not in the humor for a cigar)

J – I discover my purse is empty//my purse is empty// the tailor finds me with an empty pocket  (¬J – my purse is tolerably full)

K – my tailor calls with his little bill

L – Froggy does not grin like a hyena (¬L – Froggy is grinning like a hyena)

M – it is very hot (¬M – the day is cool/it is chilly)

N – the thermometer is high (¬N – the thermometer is low)

O – Froggy is not intending to go a-wooing

P – my railway shares are going up like anything

Q – you need not trouble yourself to take an umbrella (¬Q – you had better take an umbrella)

R – Froggy seems nearly mad with joy (¬L?)

So it should look something like this:

1. When A –> B

2. When C ∧ D –> E

3. Now that F –> G // therefore F ∧ G

4. When H ∧ I –> J

5. When K ∧ ¬C –> L

6. When M –> N

7. When A ∧ ¬I ∧ ¬L –> ¬B

8. When K ∧ J –> ¬C

9. P

10. When J ∧ (noticing) ¬G –> ¬C ∧ (things are apt) M

11. Now that ¬A ∧ ¬L –> ¬I // therefore ¬A ∧ ¬L ∧ ¬I

12. When N –> Q

13. When ¬G ∧ ¬D –> I

14. When B –> ¬L

15. When ¬J ∧ ¬F ∧ ¬D –> H

16. When P ∧ ¬A ∧ ¬M –> I

17. When ¬E –> R  ¬G

18. When (¬A ∧ I) ∧ O –> ¬R

19. When P ∧ R –> K

20. When ¬M ∧ ¬N ∧ ¬B ∧ ¬R –> ¬I

I suspected that R = ¬L, ¬R = L, and my work reflects this assumption. I am not certain it is safe for me to assume, though. I think it is necessary and safe to assume in the negation of 4 (shown below) isn’t a simple negation distribution where the ∧ becomes ∨, based on the way its phrased.

What do we know? Sorry this is a bit sloppy.

Here’s what I came up with:

A1. P (by 9)

A2. F ∧ G (by 3 – use of ‘now that’)

A3. ¬A ∧ ¬L ∧ ¬I (by 11 – use of ‘now that’)

A4. P ∧ ¬A ∧ ¬M –>I (by 16)

but ¬I (∧ ¬L ∧ ¬A by 11)

so ¬(P ∧ ¬A ∧ ¬M)

P ∧ ¬A (by A1 and A3)

∴M (by 16 and 11)

A5. M (by A4)

M–>N (by 6)

A6. N (by A5 and 6)

N–>Q (by 12)

A7. Q (by A6 and12)

A8. When K ∧ ¬C –> L (5)

but ¬L  (by A3) so ¬(K ∧ ¬C)

 so   ¬K∨ C

A9. When ¬E –> R ∧ ¬G (by 17)

but G (by A2)

so ¬(¬E)

∴E

A10. When P ∧ R –> K (by 19)

P (by 9)

R (if R=¬L by 11)

∴K

A11. ¬K∨ C (by A8)

K (by A10)

∴ C

A12. When (¬A ∧ I) ∧ O –> ¬R (by 18)

but R (if R=¬L by 11)

so ¬(¬A ∧ I ∧ O)

ok, the point of this one is that there is no point. we can’t say ¬O or O either way

A13. When K ∧ J –> ¬C (by 8)

but C (by A11)

so ¬(K ∧ J)

¬K ∨  ¬J

K (by A10)

∴ ¬J

This is what I think we can say, which isn’t the solution as far as I can tell: given[P, F, G, ¬A, ¬L (=R), ¬I]: M, N, Q, E, K, C, ¬J

Froggy’s hair is out of curl; Froggy put away his gorgeous waistcoat; it it very hot; the thermometer is high; the day is not fine (it looks like rain); Froggy is grinning like a hyena/Froggy seems nearly mad with joy; I am not in the humor for a cigar; Froggy’s mother declares, ‘He shall not go a-wooing!;’ my tailor calls with his little bill; I let Froggy forget the 10₤ he owes me; my purse is tolerably full.

My guess is that what we can say is that Froggy’s hair is out of curl and he is nearly mad with joy, though Froggy’s mother has declared he shall not go a-wooing; we can’t say whether or not Froggy actually intends to go a-wooing. If anyone is interested, I went through and verified my suspected answer through contradiction, but I’m not posting it here for the sake of length. I will happily include it or send to anyone that would like it.

Probably related to Carroll’s thought in creating this problem is an old English children’s verse:

A frog he would a-wooing go,
Whether his mother would let him or no.

Doesn’t help me much here, though. O stood for the first line of this and that was the one I couldn’t determine, except through a verification by contradiction (assuming the truth of not-O).

__________________

*There is an apostrophe in Master’s because it’s apparently some magical contraction of “of” – e.g. Master of Arts. Hopefully this will change with widespread use of Masters by the ignorant.

**Flirty Spinozist.

***And Kelly’s story. I haven’t forgotten!

****Apparently Carroll himself produced a dictionary for this problem:

  • Universe: “Cosmophases”;
  • E = this;
  • a = Froggy’s hair is out of curl;
  • b = Froggy intends to go a-wooing;
  • c = Froggy is grinning like a hyena;
  • d = Froggy’s mother permits him to go a-wooing;
  • e = Froggy seems nearly mad with joy;
  • h = Froggy is strutting about like a peacock;
  • k = Froggy is wearing a waistcoat that is gorgeous beyond words;
  • l = I go out on my roof;
  • m = I remind Froggy of the 10 pounds he owes me;
  • n = I take a quiet cigar;
  • r = I tell Froggy that he’s quite the dandy;
  • s = It is going to rain,
  • t = It is very hot;
  • v = My purse is empty;
  • w = My railway shares are going up;
  • z = My tailor calls with his little bill;
  • A = The thermometer is high;
  • B = You had better take an umbrella

(“Lewis Carroll’s Symbolic Logic” ed. William Warren Bartley, III, published by Harvester Press 1977)

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