Lewis Carroll's Logic Problem #1, page 386

Question

This is one of Lewis Carroll's logic problems for which no solution was given. I have reached a solution and am looking for someone to confirm it. This problem is taken from William Bartley's book "Lewis Carroll's Symbolic Logic" and is problem #1 found on page 386 of the first edition. It is important to note that the conclusion(s) intended to be found are to be formed from what are called retinends and not from eliminands. This is explained by the statements: 1) If A then B, and 2) If B then C. In these two statements B would be an eliminand because its only purpose is to link the two statements. Here A and C would be retinends and would be found in the conclusion. Retinends and eliminands are formed from the program statements. Also, include the contrapositives of program statements when identifying the eliminands and retinends. After that is done only consider the true attributes (variables) and not the false ones. Here is that problem:

1. I don't like walking with any London-friend of mine, unless he wears a tall hat;
2. The Colonel is ready to 'Play billiards with any man who is not the subject of general conversation, unless he happens to be fat;
3. A man, who knows what o'clock it is and who never yields to me in argument, is attractive;
4. No old, tight-rope-dancer ever rouses me to enthusiasm;
5. Any novelist, whom I take pleasure in cutting, always cuts me dead;
6. A man, who goes about in kid-gloves, but without his coat, is a humbug;
7. All my intimate friends in London are young men;
8. No man, who is the subject of general conversation, ever rouses me to enthusiasm, unless he is a horsey man;
9. A man, who has his wits about him and does not choose his own wines, is always "at home" when I call;
10. A man, who is a good shot and never tells pointless anecdotes, is sure to have a good temper;
11. All humbugs, who write novels, are intimate friends of mine;
12. I like to walk with a good-tempered man, unless he goes about in his shirt-sleeves;
13. A man, who never loses his umbrella, and is not easily taken in, is sure to be an early riser;
14. Fat men, who do not dance on the tight-rope, are universally respected;
15. I regard with contemptuous pity a man who fails in life, and who runs across the street;
16. A man, who does not stick to business, is not likely to be elected Mayor, unless he has bushy whiskers;
17. An elephant hunter always rouses me to enthusiasm, unless he happens to be a farmer;
18. Any London ftiend of mine, who tells pointless anecdotes, is a humbug;
19. I never invite an old man to dinner, unless he has lent me money;
20. A man, who does not stick to business, does not run across the street, and has bushy hair, is in no danger of getting a bad fall;
21. A man, who gets up late and sometimes loses his umbrella, has little chance of marrying an heiress;
22. An old man, who cares for appearances, always wears kid-gloves;
23. A good-tempered man never cuts me dead, unless he is a humbug;
24. A man, who never tells pointless anecdotes and has never lent me money, has his wits about him;
25. A man, who chooses his own wines and always yields to me in argument, is the sort that I invite to dine with me;
26. I always try to be civil to a man who fails in life, unless he has bushy whiskers;
27. All farmers are horsey men;
28. A novelist is a dull companion, unless he rouses me to enthusiasm;
29. All men, who get up early, and stick to business and win universal respect, are rich;
30. Any London friend of mine, to whom I try to be civil, will probably be elected Mayor;
31. Any good-tempered man, who has lent me money and does not care for appearances, is willing to shake hands with me when I am in rags;
32. The only men, with whom the Colonel will play billiards, are either horsey men or farmers;
33. I always invite an attractive man to dine with me, provided he is rich;
34. A man, who is apt to walk on tip-toe and whom I regard with contemptuous pity, is sure to be one who sticks to business;
35. The only men, who are always "at home" to me, but whom I never invite to dinner, are magistrates;
36. I always make an intimate friend of a man who will shake hands with me when I am in rags and will give up his umbrella to me when it is raining;
37. Any London friend of mine, who understands horses, is universally respected;
38. An unattractive man, who chooses his own wines, is easily taken in;
39. I have sufficient courage to insult any novelist, unless he happens to be a good shot;
40. An old man, who is apt to walk on tip-toe, will probably get a bad fall;
41. A man, who never knows what o'clock it is, and who has never lent me money, will probably marry an heiress;
42. No London-friend of mine, who has his wits about him, is easily taken in;
43. I never forget any old man who is willing to shake hands with me when I am in rags;
44. A novelist, who does not stick to business, is sure to fail in life;
45. I do not dare to insult an ill-tempered man, unless he happens to be an intimate friend of mine;
46. Those magistrates, who will not shake hands with me when I am in rags, always choose their own wine;
47. All dull companions are either horsey men or elephant-hunters;
48. Men, who wear tall hats and kid gloves, always cut me dead;
49. A man, who has bushy whiskers and is universally respected, is apt to walk on tip-toe;
50. I delight in cutting a man, whom I perfectly remember, but who will not give up his umbrella to me when it is raining.

Following is a list of attributes supplied by Carroll: Univ. "men"; a = apt to tell pointless anecdotes; b = apt to walk on tip-toe; c = "at home" to me; d = attractive; e = caring for appearances; f = choosing his own wines; g = dull companions; h = early risers; j = easily taken in; k = elephant-hunters; 1 = farmers; m = fat; n = good shots; p = good-tempered; q = having bushy whiskers; r = having his wits about him; s = horsey; t = humbugs; u = intimate friends of mine; v = invited to dine with me; w = knowing what o'clock it is; x = likely to be elected Mayor; y = likely to get a bad fall; z = likely to marry an heiress; A = London-friends of mine; B = magis- trates; C = men to whom I try to be civil; D = men who cut me; E = men who have lent me money; F = men whom I dare insult; G = men whom I delight to cut; H = men with whom I like to walk; J= men with whom the Colonel is willing to play billiards; K = novel- ists; L = old; M = regarded by me with contemptuous pity; N = re- membered by me; P = rich; Q = rousing me to enthusiasm; R = running across a street; S = sometimes losing his umbrella; T = sticking to business; U = subjects of general conversation; V = successful in life; W = tight-rope dancers; X = universally respected; Y = wearing a coat; Z = wearing a tall hat; BB = wearing kid gloves; GG = willing to give up his umbrella to me when it is raining; EE = willing to shake hands with me when I am in rags; LL = yielding to me in argument. `

Answer 1

First I convert each statement to an implication:

 1. A & ~Z -> ~H
 2. ~U & ~m -> J
 3. w & ~LL -> d
 4. L & W -> ~Q
 5. K & G -> D
 6. BB & ~Y -> t
 7. u & A -> ~L
 8. U & ~s -> ~Q
 9. r & ~f -> c
10. n & ~a -> p
11. t & K -> u
12. p & Y -> H
13. ~S & ~j -> h
14. m & ~W -> X
15. ~V & R -> M
16. ~T & ~q -> ~x
17. k & ~l -> Q
18. A & a -> t
19. L & ~E -> ~v
20. ~T & ~R & q -> ~y
21. ~h & S -> ~z
22. L & e -> BB
23. p & ~t -> ~D
24. ~a & ~E -> r
25. f & LL -> v
26. ~V & ~q -> C
27. l -> s
28. K & ~Q -> g
29. h & T & X -> P
30. A & C -> x
31. p & E & ~e -> EE
32. ~s & ~l -> ~J
33. d & P -> v
34. b & M -> T
35. c & ~v -> B
36. EE & GG -> u
37. A & s -> X
38. ~d & f -> j
39. K & ~n -> F
40. L & b -> y
41. ~w & ~E -> z
42. A & r -> ~j
43. L & EE -> N
44. K & ~T -> ~V
45. ~p & ~u -> ~F
46. B & ~EE -> f
47. ~s & ~k -> ~g
48. Z & BB -> D
49. q & X -> b
50. N & ~GG -> G

We can then convert the implications to disjunctions; since $p\implies q$ is equivalent to $\lnot\ p \lor q$, and $\lnot(p \land q)$ is equivalent to $\lnot\ p \lor \lnot\ q$, we can convert an implication like $r \land s \implies t$ into $\lnot\ r \lor \lnot\ s \lor t$.

1. ~A | Z | ~H
2. U | m | J
3. ~w | LL | d
4. ~L | ~W | ~Q
5. ~K | ~G | D
6. ~BB | Y | t
7. ~u | ~A | ~L
8. ~U | s | ~Q
9. ~r | f | c
10. ~n | a | p
11. ~t | ~K | u
12. ~p | ~Y | H
13. S | j | h
14. ~m | W | X
15. V | ~R | M
16. T | q | ~x
17. ~k | l | Q
18. ~A | ~a | t
19. ~L | E | ~v
20. T | R | ~q | ~y
21. h | ~S | ~z
22. ~L | ~e | BB
23. ~p | t | ~D
24. a | E | r
25. ~f | ~LL | v
26. V | q | C
27. ~l | s
28. ~K | Q | g
29. ~h | ~T | ~X | P
30. ~A | ~C | x
31. ~p | ~E | e | EE
32. s | l | ~J
33. ~d | ~P | v
34. ~b | ~M | T
35. ~c | v | B
36. ~EE | ~GG | u
37. ~A | ~s | X
38. d | ~f | j
39. ~K | n | F
40. ~L | ~b | y
41. w | E | z
42. ~A | ~r | ~j
43. ~L | ~EE | N
44. ~K | T | ~V
45. p | u | ~F
46. ~B | EE | f
47. s | k | ~g
48. ~Z | ~BB | D
49. ~q | ~X | b
50. ~N | GG | G

We then collect all variables that appear either only as themselves or only as their negations: ~A, ~K, ~L; these are the "retinends" (as far as I can figure). However, I can't (yet) find a combination of these that are implied by the premises, so it is likely I have another mistake somewhere.