Lewis Carroll's Froggy Problem

This is Lewis Carroll's Froggy problem from the book "Symbolic Logic Part I" by Lewis Carroll. At the end of the book (in Appendix, Addressed to Teachers) he writes:
"I will conclude with eight Problems, as a taste of what is coming in Part II.
I shall be very glad to receive, from any Reader, who thinks he has solved any one of them
(more especially if he has done so without using any Method of Symbols),
what he conceives to be its complete Conclusion." The Froggy Problem was the third of these eight problems.

Regrettably Lewis Carroll died before publishing Part II. However, after a great deal of detective work
a considerable portion of the unpublished proofs and notes on which Lewis Carroll was working have been assembled
by William Warren Bartley III. The result is Lewis Carroll's Symbolic Logic"Lewis Carroll's Symbolic Logic" Edited by William Warren Bartley, III Published by Harvester Press 1977
ISBN 0-85527-984-2.

I have produced a symbolic representation of the problem using the Dictionary provided in Bartley's book.

I have also produced a representation using the subscript notation, as explained by Carroll in his book.

From that I have produced the Register of Attributes according to the rules set out in Chapter III of Book XII of "Symbolic Logic". This process reveals that there are four Retinends, these being E, a, b, d. It remains then to solve the problem using the Method of Trees starting with the root Ea'b'd.

My tree can be found in this PDF. As far as I am aware this is the first time a solution to this sorites has been published using Carroll's Method of Subscripts and Method of Trees. In fact I have been unable to locate any previous valid solution to this sorites by any method at all, which, 114 years after the problem was first published, is surprising.

I have also produced a representation using the subscript notation, as explained by Carroll in his book.

From that I have produced the Register of Attributes according to the rules set out in Chapter III of Book XII of "Symbolic Logic". This process reveals that there are four Retinends, these being E, a, b, d. It remains then to solve the problem using the Method of Trees starting with the root Ea'b'd.

My tree can be found in this PDF. As far as I am aware this is the first time a solution to this sorites has been published using Carroll's Method of Subscripts and Method of Trees. In fact I have been unable to locate any previous valid solution to this sorites by any method at all, which, 114 years after the problem was first published, is surprising.

I have tried to apply the methods described by Lewis Carroll to this problem as far as I am able. I confess I am not *totally* happy with the result and I am sure I am missing at least *something* (but hopefully not *everything*). The results of my efforts are contained here on these pages. The solution which I obtain is, in subscript notation, which can be written as **No Ea'd are b'** or in English "If Froggy's hair was a mass of curls and he had his mother's permission he would go a-wooing today". I would be interested to hear from anyone who arrives at a *different* solution using the methods descrribed by Lewis Carroll. I have also produced what seems to me a validation of the conclusion. The alternative method referred to by Carroll is his method of underscores plus the concept of "barred premisses". The only guide to how this method works is contained in a manuscript dated 7/2/93, reprinted in Bartley as part of Book XIII Chapter XII. I have derived the same conclusion as that given above, using this alternative method and you can find the full derivation on page underlines_and_bars.html.

Lewis Carroll stated that this problem contains a "beautiful 'trap' ". It seems to me that this trap relates to the form of propositions 3, 9 and 11. Consider proposition 9, "My railway shares are going up like anything!" I think the trap to which Carroll refers is the danger of treating "My railway shares are going up" in the same way as one would treat "Socrates is a man". The clue to this is in the dictionary for the problem which Carroll produced and which is presented in Bartley's book. The universe of discourse is stated to be "cosmophases" and the first item is given as **ε=this**. The fact that Carroll used epsilon, whereas in all other cases he uses the english alphabet, seems to me significant and suggests that this may well have been the trap to which he refers. His definition of "cosmophases" is "the state of the Universe at some particular moment" and his use of this concept is described in his essay, dated March 15, 1897 on DeMorgan's Problem (page 480 of Bartely's book).

Since producing this solution I have analysed more thoroughly the use of language in Carroll's formulation of the problem and arrived at a slightly different interpretation of the meaning of some of the premises, which removes the "mass of curls" condition from the solution. This can be found here and a re-working of the new formulation using Carroll's methods is in preparation (as of 25 January 2013)

Carroll requires the
Complete Conclusion which he explains (Book XIII Chapter I) means stating
"*all* the relations, among the Retinends only, which can be deduced from the Premisses."

Problem: To achieve the complete conclusion.

- 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 purse, 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!

The dictionary is the only fragment that remains in the documents so far found and published in "Lewis Carroll's Symbolic Logic" Edited by William Warren Bartley, III Published by Harvester Press 1977
ISBN 0-85527-984-2:

Dictionary for Froggy's 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;

Copyright (C) Graham Hawker 31 March 2010