The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

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Sunday, February 18, 2007

Don't Ask Alice - Part the Third

This is the third installment on Dodgson's unsolved sorites - for the first two go here:
Don't ask Alice, I don't think she'll know
Don't ask Alice - Part the Second
Without further ado, here is the third of Dodgson's [hitherto] unsolved sorites:

(1) When the day is fine, I tell Froggy "You're quite the dandy, old chap!";
(2) Whenever I let Froggy forget that £10 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 he owes me, he does not grin like a hyæna;
(6) When it is very hot, the thermometer is high;
(7) When the day is fine, and I'm not in the humour for a cigar, and Froggy is grinning like a hyæna, 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 he owes me;
(9) My railway-shares are going up like anythin!
(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 he ower me, things are apt to get rather warm;
(11) Now that it looks like rain, and Froggy is grinning like a hyæna, 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 the dandy, he grins like a hyæna;
(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 railway-shares are going up, and when it is 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 solution proceeds exactly as the first two problems. Define the following propositional statements: R – 'I go out on the roof.', Q – 'I enjoy a quiet cigar.', P – 'My purse is empty.', T – 'My tailor calls with his little bill.', O – 'I remind Froggy of the £10 he owes me.', D – 'I tell Froggy "You're quite the dandy, old chap."', S – 'Froggy struts about like a peacock.', A – 'Froggy goes out a-wooing.', C – 'Froggy's hair is curly.', W – 'Froggy wears his gorgeous waistcoat.', G – 'Froggy grins like a hyena.', J – 'Froggy seems nearly mad with joy.', C – 'The day is cool.', H – 'The day is hot.', M – 'The thermometer is high.', L – 'It looks like rain.', and B – 'You had better take an umbrella.'. Interpret the statement 'The day is fine.' as meaning it is neither cool nor hot, or (CH)~. Since premise (9) tells us that railroad-shares are going up, it can be omitted and this condition in statements (16) and (19) can be ignored. Finally, premise 17 can be expanded into two simpler statements. Using these propositional statements, write the premises as the conjunction:
ρ(CDH) · ρ(A~OS~) · ρ(CW~) · ρ(PQ~R~) · ρ(G~O~T~) · ρ(H~M) · ρ(CG~HO~Q) · ρ(OP~T~) · ρ(HO~P~W~) · ρ(G~L~Q~) · ρ(B~M~) · ρ(GQW~) · ρ(D~G) · ρ(C~PRS) · ρ(C~L~Q) · ρ(A~J) · ρ(A~W) · ρ(ABL~Q~) · ρ(J~T) · ρ(C~DGMQ~)
Applying the theorem, canceling those terms which appear both complemented and uncomplemented (shown underlined above), produces the conclusion:

ρ(L~),    or 'It doesn't look like it will rain.' Q.E.D.
Again, what seems to be a very complex, seemingly impossible problem turns out to have a simple solution which is easily found.

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