The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

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Friday, February 16, 2007

Don't Ask Alice, I Don't Think She'll Know

Charles Dodgson (a.k.a. Lewis Carroll) is today remembered for his books Alice in Wonderland and Through the Looking Glass. Few are aware that he was a logician and prolific author of many logic puzzles, and was the master composer of sorites. He explained his elementary methods in his book, Symbolic Logic, of which I have a copy of the 4th (last) Edition. There are hundreds of exercises in the book for which he helpfully provided the answers.

He was planning on writing a sequel demonstrating more advanced methods, and as a teaser gave eight problems in the appendix for which he provided no solutions, dying before the new book was published. He commented:

"I will conclude with eight Problems, as a taste of what is coming ... I shall be very glad to receive, from any Reader, who thinks he has solved any one of them ..., what he conceives to be its complete Conclusion."
Unfortunately, his solutions are lost, and I am unaware of anyone solving these puzzles - they are generally considered too complex or not worth the labor required.

This belief is misfounded. Using a simple boolean notation and a remarkable theorem first suggested by G. Spencer Brown, later corrected and proven by me, I will present the solutions to these problems. Beginning at the beginning with the first problem:

All the boys, in a certain School, sit together in one large room every evening. They are of no less the five nationalities - English, Scotch [sic], Welsh, Irish, and German. One of the Monitors (who is a great reader of Wilkie Collins' novels) is very observant, and takes MS. notes of almost everything that happens, with the view of being a good sensational witness, in case any conspiracy to commit murder should be on foot. The following are some of his notes: -
(1) Whenever fome of the English boys are singing "Rule Britannia", and some not, some of the Monitors are wide-awake;
(2) Whenever some of the Scotch [sic] are dancing reels, and some of the Irish fighting, some of the Welsh are eating toasted cheese;
(3) Whenever all the Germans are playing chess, some of the Eleven are not oiling their bats;
(4) Whenever some of the Monitors are asleep, and some not, some of the Irish are fighting;
(5) Whenever some of the Germans are playing chess, and none of the Scotch [sic] are dancing reels, some of the Welsh are not eating toasted cheese;
(6) Whenever some of the Scotch [sic] are not dancing reels, and some of the Irish not fighting, some of the Germans are playing chess;
(7) Whenever some of the Monitors are awake, and some of the Welsh are eating toasted cheese, none of the Scotch [sic] are dancing reels;
(8) Whenever some of the Germans are not playing chess, and some of the Welsh are not eating toasted cheese, none of the Irish are fighting;
(9) Whenever all the English are singing "Rule Britannia", and some of the Scotch [sic] are not dancing reels, none of the Germans are playing chess;
(10) Whenever some of the English are singing "Rule Britannia", and some of the Monitors are asleep, some of the Irish are not fighting;
(11) Whenever some of the Monitors are awake, and some of the Eleven are not oiling their bats, some of the Scotch [sic] are dancing reels;
(12) Whenever some of the English are singing "Rule Britannia", and some of the Scotch [sic] are not dancing reels, ****
Here the MS. breaks off suddenly. The Problem is to complete the sentence, if possible.
The solution proceeds as follows: recognize the following classes: E - English, S - Scotch, W - Welsh, I - Irish, G - German, M - monitors, L - the team of eleven, A - those awake, R - those singing 'Rule Britannia', D - those dancing reels, T - those eating toasted cheese, C - those playing chess, O - those oiling bats, and F - those fighting, and write the 11 premises as the conjunction:
ρ(ρ(E~R~)ρ(E~R)ρ(M~A~)~) · ρ(ρ(S~D~)ρ(I~F~)ρ(W~T~)~) · ρ(ρ(G~C)~ρ(L~O)~) · ρ(ρ(M~A)ρ(M~A~)ρ(I~F~)~) · ρ(ρ(G~C~)ρ(S~D~)~ρ(W~T)~)) · ρ(ρ(S~D)ρ(I~F)ρ(G~C~)~) · ρ(ρ(M~A~)ρ(W~T~)ρ(S~D~)) · ρ(ρ(G~C)ρ(W~T)ρ(I~F~)) · ρ(ρ(E~R)~ρ(S~D)ρ(G~C~)) · ρ(ρ(E~R~)ρ(M~A)ρ(I~F)~) · ρ(ρ(M~A~)ρ(L~O)ρ(S~D~)~)
Applying the theorem, canceling those terms which appear both complemented and uncomplemented (shown underlined), produces the conclusion :
ρ(ρ(E~R~)ρ(S~D)ρ(M~A))
so the complete sentence would be :
'Whenever some of the English are singing "Rule Britannica", and some of the Scotch [sic] are not dancing reels, all of the Monitors are awake.' Q.E.D.

I will not go into the details of the notation or the transcription of the premises, for that would require far too much space and time. The purpose of this calculation is to show that the solution to such a seemingly complex problem is actually quite simple if the proper methods are available.

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