The Analytic Atavar

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Monday, February 19, 2007

Don't Ask Alice - Part the Fourth

This is the fourth installment on Dodgson's unsolved sorites - for the first three go here:
Don't ask Alice, I don't think she'll know [from Dodgson's Symbolic Logic, p. 186]
Don't ask Alice - Part the Second [from Dodgson's Symbolic Logic, p. 187]
Don't ask Alice - Part the Third [from Dodgson's Symbolic Logic, p. 188]
Without further ado, here is the fourth of Dodgson's [hitherto] unsolved sorites:

(1) Any one, fit to be an M.P., who is not always speaking, is a public benefactor;
(2) Clear-headed people, who express themselves well have had a good education;
(3) A woman, who deserves praise, is one who can keep a secret;
(4) People, who benefit the public, but do not use their influence for good purpose, are not fit to go into Parliament;
(5) People, who are worth their weight in gold and who deserve praise, are always unassuming;
(6) Public benefactors, who use their influence for good objects, deserve praise;
(7) People, who are unpopular and not worth their weight in gold, never can keep a secret;
(8) People, who can talk for ever and are fit to be Members of Parliament, deserve praise;
(9) Any one, who can keep a secret and who is unassuming, is a never-to-be-forgotten public benefactor;
(10) A woman, who benefits the public, is always popular;
(11) People, who are worth their weight in gold, who never leave off talking, and whom it is impossible to forget, are just the people whose photographs are in all the shop-windows;
(12) An ill-educated woman, who is not clear-headed, is not fit to go into Parliament;
(13) Any one, who can keep a secret and is not for ever talking, is sure to be unpopular;
(14) A clear-headed person, who has influence and uses it for good objects, is a public benefactor;
(15) A public benefactor, who is unassuming, is not the sort of person whose photograph is in every shop-window;
(16) People, who can keep a secret and who use their influence for good purposes, are worth their weight in gold;
(17) A person, who has no power of expression and who cannot influence others, is certainly not a woman;
(18) People, who are popular and worthy of praise, either are public benefactors or else are unassuming.
Univ. "persons"; a = able to keep a secret; b = clear-headed; c = constantly talking; d = deserving praise; e = exhibited in shop-windows; h = expressing oneself well; k = fit to be an M.P.; l = influential; m = never-to-be-forgotten; n = popular; r = public benefactors; s = unassuming; t = using one's influence for good objects; v = well-educated; w = women; z = worth one's weight in gold.

The solution proceeds exactly as the first three problems. It will be seen in what follows that premise 12 is redundant; premise 9 can be split into two simpler premises and writing the premises except 12 as a conjunction:
ρ(CK~R) · ρ(B~H~V) · ρ(AD~W~) · ρ(K~R~T) · ρ(D~SZ~) · ρ(DR~T~) · ρ(A~NZ) · ρ(C~DK~) · ρ(A~MS~) · ρ(A~RS~) · ρ(NR~W~) · ρ(C~EM~Z~) · ρ(A~CN~) · ρ(B~L~RT~) · ρ(E~R~S~) · ρ(A~T~Z) · ρ(HLW~) · ρ(D~N~RS)
Applying the theorem, canceling those terms which appear both complemented and uncomplemented (shown underlined above), produces the conclusion:

ρ(B~K~VW~),    or 'Clear-headed women fit to be M.P.s are well-educated.' Q.E.D.
It is clear that if premise (12), ρ(K~VW~), is included, it becomes the minimal conclusion, with the conclusion above derivable from it, rendering all the other premises irrelevant. Dodgson apparently erred by including this premise in the group - he was guilty of "begging-the-question".

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