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Post by jdwatson on Apr 18, 2009 17:24:53 GMT -5
Following my initial post on Carroll's unsolved sorites, I was contacted by Mr. Amirouche Moktefi, who informed me of the book Symbolic Logic by Lewis Carroll, Part I - Elementary, 1896 - Fifth Edition, Part II - Advanced, never previously published, Edited by William Warren Bartley, III, where he discusses these eight problems based on galley proofs and other documents he discovered. The book appears to be out of print so I ordered a used copy of the paperback edition and reviewed Bartley's discussion of these eight problems.
Following my review, I revised my original paper (still at the same URL) to reflect this new information. As far as I could determine, Bartley only provides solutions to 3 (#1, 2, & 4) of the 8 problems. In summary:
Problem 1 : The Problem of the School-Boys – The solution obtained by my method is precisely the one Dodgson gives as the solution. Problem 2 : The Pork-Chop Problem – Again, the solution obtained here is identical with the symbolic solution Dodgson gives, however, his interpretation of the formula is in error. Problem 3 : Froggy's Problem – No solution to this problem is given by Bartley, so my result is original. Problem 4 : Members of Parliament Problem – Here, I discovered an error I had made in formulating premise 12 – when this was amended, the correct solution, matching Dodgson's, is obtained. He however apparently intentionally introduced 4 redundant premises. When these are removed, an even stronger conclusion is possible. These redundant premises would have been revealed by a thorough analysis like that done on Problem 2 in my companion paper Aristotelian Logic and Syllogism. Problem 5 : The Problem of Six Friends and Their Wives – No solution to this problem is given by Bartley, so my result is original. Problem 6 : The Salt and Mustard Problem – No solution to this problem is given by Bartley, so my result is original. Indeed, Bartley seems to imply that there is no solution to this problem. Problem 7 : The Self-Conscious Brothers Problem – This problem is not even discussed in Book XIII, but is instead an unsolved problem at the end of the next Book, so my result is original. Problem 8 : The Great-Grandson Problem – No solution to this problem is given by Bartley, so my result is original – in fact, Dodgson apparently believed it was so difficult that no one would be able to solve it.
The fact that I could reproduce the 3 solutions Bartley provides gives me greater confidence in my solutions for the remaining 5.
I would appreciate any comments members of this forum might have and would be happy to answer any questions.
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Post by bobbixler on May 9, 2014 17:24:30 GMT -5
For completeness I believe that there are 3 conclusions that can be made from the 20 statements that make up the problem. (See www.connectedglobe.com/froggy/index.html). They are: 1. It is very hot. (The thermometer is high). (From statements #9,#11,#16) 2. Froggy's mother does not permit him to go a-wooing. (From statements #3,#17) 3. You need not take an umbrella. (From statements #9,#11,#16,#6,#12) Conclusion #2 above is relatively trivial as it's proof only requires statements #3 and #17. However it is considered a retinend. Conclusion #3 is not considered a retinend at first glance but can be considered one after realizing that statement #18 is in direct contradiction to statement #11. So statement #18 should be removed before working the problem. When this is done conclusion #3 above becomes a retinend. I think that this is what Carroll was referring to when he spoke of a little trap in this problem. If one is interested in all valid logical conclusions derived from the premises then I believe these 3 conclusions are required although conclusion #1 is not considered a retinend. I know of no other source giving result #3 above. This riddle is a series of SIMULTANEOUS if-then statements so sequence of process should not make a difference. The "if" portion of each statement would be the premise and the "then" portion would be the conclusion. Sometimes the conclusion of one statement is the premise of another statement. This would be called an eliminand. Eliminands glue two statements into a meaningful basis for inference. They are not passed on to the conclusion if you follow what Carroll appears to ask for which is for the retinends only in the conclusion.
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Post by GoetzKluge on May 9, 2014 18:52:29 GMT -5
I know of no other source giving the result that I obtained. This riddle is a series of SIMULTANEOUS if-then statements so sequence of process should not make a difference. The "if" portion of each statement would be the premise and the "then" portion would be the conclusion. Sometimes the conclusion of one statement is linked to the premise of another statement. This would be called an eliminand. Eliminands glue two premises into a meaningful basis for inference. They are not passed on to the conclusion. There are many correct statements that can be inferred from this problem but the answer desired is the "complete" solution. This is what can be inferred after all the eliminands are eliminated. I find only one conclusion when I do this. With "sequence" I do not mean a sequence in time. There are statements, which only yield a conclusion once previous statements yielded the required parameters. With the pure logic I have no issue. But there may be statements which are a matter of personal preference, e.g. with regard to statements containing the terms "cool", "warm", "hot", "very hot"="thermometer is high" and also "fine". To Carroll (but not necessarily to me) no rain and not cool may be a requirement for "fine" (and rain or cool may be not fine). "Fine day", as an example, is not explicitely defined as true or false. And a very hot day isn't necessarily a fine day to everybody. The solution could be that some interpretations don't yield any result and therefore could be excluded. Alternatively, different preferences could lead to different solutions. Besides your solution related to the Umbrella the riddle also could provide an answer to whether to Carroll a very hot day is or isn't a fine day. Or did you find out, that some "fuzzy" terms do not contribute to the result anyway? How did you deal with that?
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Post by bobbixler on May 10, 2014 11:08:00 GMT -5
Gotz, Just went through your coding list but then you deleted it. Is that correct? Anyway here is what you need to change:
WARMTHINGS==HOT/THERMOMETER
10. {WAISTCOAT,PAYBACK},{PURSE},{WARMTHINGS},{ }
change 3,9,11 to reflect always TRUE: {RAILWAY SHARES,RAIN,GRIN} always FALSE: {CURL,WAISTCOAT,CIGAR}
Bob
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Post by GoetzKluge on May 10, 2014 12:05:58 GMT -5
Gotz, Just went through your coding list but then you deleted it. Is that correct? Anyway here is what you need to change: WARMTHINGS==HOT/THERMOMETER 10. {WAISTCOAT,PAYBACK},{PURSE},{WARMTHINGS},{ } change 3,9,11 to reflect always TRUE: {RAILWAY SHARES,RAIN,GRIN} always FALSE: {CURL,WAISTCOAT,CIGAR} Bob Sorry. I am not so sure about #10. Let me think about it. By the way, if CIGAR is always false, how can UmBRELLA in statement #8 become true? The coding is back (syntax changes for processing reasons). The code below already contains your proposals: #!/usr/bin/lua
--[[ 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! --]]
value={True=true,False=false,Nil=nil}
-- Format: {index, true conditions, false conditions, true results, false results} statements = { {_,{'True'},{'False'},{'RAILWAY','RAIN','GRIN'},{'CURL','WAISTCOAT','CIGAR'}}, -- Bob (statement 3,9,11) {_,{'HOT'},{'False'},{'WARMTHINGS'},{'CHILLY','COOLDAY'}}, -- Bob (WARMTHINGS) {_,{'THERMOMETER'},{'False'},{'VERYHOT','WARMTHINGS'},{'CHILLY','COOLDAY'}}, -- Bob (WARMTHINGS) {_,{'WARMTHINGS'},{'False'},{'HOT','THERMOMETER'},{'_'}}, -- Bob (WARMTHINGS) {_,{'CHILLY'},{'False'},{'_'},{'HOT','VERYHOT','THERMOMETER'}}, {_,{'COOLDAY'},{'False'},{'_'},{'HOT','VERYHOT','THERMOMETER'}}, {_,{'VERYHOT'},{'False'},{'THERMOMETER'},{'CHILLY','COOLDAY'}}, {_,{'FINEDAY'},{'False'},{'_'},{'CHILLY','RAIN'}}, {_,{'RAIN'},{'False'},{'_'},{'FINEDAY'}}, {_,{'FROGGYOBEYSMOTHER'},{'False'},{'_'},{'_'}}, -- no assignment yet {1,{'FINEDAY'},{'False'},{'DANDY'},{'_'}}, {2,{'PEACOCK'},{'PAYBACK'},{'_'},{'WOOING'}}, {3,{'True'},{'CURL'},{'_'},{'WAISTCOAT'}}, {4,{'ROOF','CIGAR'},{'False'},{'_'},{'PURSE'}}, {5,{'TAILOR','PAYBACK'},{'False'},{'_'},{'GRIN'}}, {6,{'VERYHOT'},{'False'},{'THERMOMETER'},{'_'}}, {7,{'FINEDAY','GRIN'},{'CIGAR'},{'_'},{'DANDY'}}, {8,{'TAILOR'},{'PURSE'},{'PAYBACK'},{'_'}}, {9,{'True'},{'False'},{'RAILWAY'},{'_'}}, {10,{'WAISTCOAT','PAYBACK'},{'PURSE'},{'WARMTHINGS'},{'_'}} , --Bob (I had {10,{'WAISTCOAT'},{'PURSE'},{'PAYBACK','WARMTHINGS'},{'_'}}) {11,{'RAIN','GRIN'},{'False'},{'_'},{'CIGAR'}}, {12,{'THERMOMETER'},{'False'},{'_'},{'UMBRELLA'}}, {13,{'WAISTCOAT'},{'PEACOCK'},{'CIGAR'},{'_'}}, {14,{'DANDY'},{'False'},{'GRIN'},{'_'}}, {15,{'PURSE','CURLS'},{'PEACOCK'},{'ROOF'},{'_'}}, {16,{'RAILWAY','CHILLY','RAIN'},{'CIGAR'},{'_'},{'_'}}, {17,{'WOOING'},{'False'},{'JOY','WAISTCOAT'},{'_'}}, {18,{'RAIN','CIGAR'},{'WOOING'},{'UMBRELLA'},{'_'}}, {19,{'RAILWAY','JOY'},{'False'},{'TAYLOR'},{'_'}}, {20,{'COOLDAY'},{'THERMOMETER','DANDY','GRIN'},{'_'},{'CIGAR'}}, }
function conclude(idx,trues,falses,setTrue,setFalse) conclusion=true defined=true for _,valueIndex in ipairs(trues) do if value[valueIndex]==nil then defined=false else conclusion = conclusion and value[valueIndex] end end for _,valueIndex in ipairs(falses) do if value[valueIndex]==nil then defined=false else conclusion = conclusion and not(value[valueIndex]) end end if (defined and conclusion) then for _,valueIndex in ipairs(setTrue) do value[valueIndex]=true end for _,valueIndex in ipairs(setFalse) do value[valueIndex]=false end end return defined end
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Post by GoetzKluge on May 12, 2014 15:19:24 GMT -5
Hi Bob, Thank you for your comments. Perhaps my code can help to test different interpretations. But for processing that table, I would have to build a decision tree based on it. I do not feel like doing that yet. It would be just too close to my engineering work, that is, too "calculatable". That is why I chose a hobby which is the contrary of that: I am interested in how people interpret uncertain information (especially "nonsense") while also having to consider how other individuals and groups respond to these interpretations. Also, as far as taking risks arising from deviant interpretations is concerned, there may be an interesting difference between discussing the interpretations of textual "nonsense" on one side and pictorial "nonsense" on the other side. In short: I like to hunt the Snark. Best regards from Munich Götz www.snrk.de"Only those questions that are in principle undecidable, we can decide." (Heinz von Foerster: Ethics and Second-Order Cybernetics, 1990-10-04, Système et thérapie familiale, Paris)
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Post by bobbixler on Apr 7, 2015 10:07:28 GMT -5
I've also worked problem #7 using my own methods and got the same result as the original poster (jdwatson). That conclusion is: "If John has a brother then they are of different heights." Here is that problem:
(1) Brothers, who are much admired, are apt to be self-conscious; (2) When two men of the same height are on opposite sides in Politics, if one of them has his admirers, so also has the other; (3) Brothers, who avoid general Society, look well when walking together; (4) Whenever you find two men, who differ in Politics and in their views of Society, and who are not both of them ugly, you may be sure that they look well when walking together; (5) Ugly men, who look well when walking together, are not both of them free from self-consciousness; (6) Brothers, who differs in Politics, and are not both of them handsome, never give themselves airs; (7) John declines to go into Society, but never gives himself airs; (8) Brothers, who are apt to be self-conscious, though not both of them handsome, usually dislike Society; (9) Men of the same height, who do not give them- selves airs, are free from self-consciousness; (10) Men, who agree on questions of Art, though they differ in Politics, and who are not both of them ugly, are always admired; (11) Men, who hold opposite views about Art and are not admired, always give themselves airs; (12) Brothers of the same height always differ in Politics; (13) Two handsome men, who are neither both of them admired nor both of them self-conscious, are no doubt of different heights; (14) Brothers, who are self-conscious, and do not both of them like Society, never look well when walking together.
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Post by bobbixler on Jun 13, 2015 10:45:11 GMT -5
I've been working on Carroll's (last?) and most difficult problem, the problem of the great-grandsons (#8):
(1) A man can always master his father; (2) An inferior of a man's uncle owes that man money; (3) The father of an enemy of a friend of a man owes that man nothing; (4) A man is always persecuted by his son's creditors; (5) An inferior of the master of a man's son is senior to that man; (6) A grandson of a man's junior is not his nephew; (7) A servant of an inferior of a friend of a man's enemy is never persecuted by that man; (8) A friend of a superior of the master of a man's victim is that man's enemy; (9) An enemy of a persecutor of a servant of a man's father is that man's friend.
The Problem is to deduce some fact about great-grandsons.
[N.B. In this Problem, it is assumed that all the men, here referred to, live in the same town, and that every pair of them are either "friends" or "enemies," that every pair are related as "senior and junior", "superior and inferior", and that certain pairs are related as "creditor and debtor", "father and son", "master and servant", "persecutor and victim", "uncle and nephew".]
In statement #6 above does "his" refer to "man's" or "junior"? I've worked this both ways using my own methods and am unable to find any combination of people and their attributes which satisfy all conditions listed although I have come very close.
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