The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

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Saturday, February 17, 2007

Don't Ask Alice - Part the Second

This is the second installment on Dodgson's unsolved sorites - for the first go here: Don't ask Alice, I don't think she'll know. This was the first of Dodgson's sorites I solved. I was unfamiliar with them until I ran across a reference to this sorites by Steven Den Beste, who cited it as an example of problems which were too complex or laborious to solve. Having done research on this topic spurred by Brown's book Laws of Form, and having found the proper form of his suggestion for inference and proven a theorem, I decided to attempt a solution. In 30 minutes I found one and e-mailed Den Beste asking if it was correct. To my surprise, he replied that he did not know because neither Dodgson nor anyone else he was aware of had ever provided a solution; he did however point me to an on-line version of Dodgson's Symbolic Logic, where I discovered all eight of his unsolved problems, and began work on them.

So, without further preamble, here is the second of Dodgson's [hitherto] unsolved sorites:

(1) A logician who eats pork-chops for supper, will probably lose money;
(2) A gambler, whose appetite is not ravenous, will probably lose money;
(3) A man who is depressed, having lost money and likely to lose more, always rises at 5 a.m.;
(4) A man, who neither gambles nor eats pork-chops for supper, is sure to have a ravenous appetite;
(5) A lively man, who goes to bed before 4 a.m., had better take to cab-driving;
(6) A man with a ravenous appetite, who has not lost money and does not rise at 5 a.m., always eats pork-chops for supper;
(7) A logician, who is in danger of losing money, had better take to cab-driving;
(8) An earnest gambler, who is depressed though he has not lost money, is in no danger of losing any;
(9) A man, who does not gamble and whose appetite is not ravenous, is always lively;
(10) A lively logician, who is really in earnest, is in no danger of losing money;
(11) A man with a ravenous appetite has no need to take to cab-driving, if he is really in earnest;
(12) A gambler, who is depressed though in no danger of losing money, sits up till 4 a.m.;
(13) A man, who has lost money and does not eat pork-chops for supper, had better take to cab-driving, unless he gets up at 5 a.m.;
(14) A gambler, who goes to bed before 4 a.m., need not take to cab-driving, unless he has a ravenous appetite;
(15) A man with a ravenous appetite, who is depressed though in no danger of losing money, is a gembler.
Univ. "men"; a = earnest; b = eating pork-chops for supper; c = gamblers; d=getting up at 5; e = having lost money; h = having a ravenous appetite; k = likely to lose money; l = lively; m = logicians; n = men who had better take to cab-driving; r = sitting up till 4.
The solution proceeds exactly as the first problem. Using the classes Dodgson suggested, write the premises as the conjunction:
ρ(M~B~K) · ρ(C~HK) · ρ(LE~K~D) · ρ(CBH)ρ(L~RN) · ρ(H~EDB) · ρ(M~K~N) · ρ(A~C~LEK~) · ρ(CHL) · ρ(L~M~A~K~) · ρ(A~H~N~) · ρ(C~LKR) · ρ(E~BDN) · ρ(C~RHN~) · ρ(H~LKC)
Applying the theorem, canceling those terms which appear both complemented and uncomplemented (shown underlined above), produces the conclusion:
ρ(M~DRA~),   or in conventional symbolic notation:
M·¬D·¬R ⇒ ¬A ,   with the interpretation:
"A logician who goes to bed before 4 a.m. and doesn't rise at 5 a.m. is not earnest." (there are, of course, other equivalent symbolic representations and interpretations of this expression). Q.E.D.

An exhaustive analysis of this sorites discovers that if premises can be used more than once (perfectly allowable logically) then these 15 premises are redundant; the set {1, 2, 4, 6, 7, 9, 10, 11, 13, 14} is sufficient for the conclusion. It appears Dodgson was at the limit of his compositional skills with this problem.

2 Comments:

Anonymous Anonymous said...

Sir, I am no logician, but I find this to be one of the coolest things I have seen in many moons.

Cheers to you!

12:36  
Anonymous Anonymous said...

Dear Mr. Watson

Arent't there 12 Boolean variables instead of only 11? I can't find the condition "being depressed" among your premises.

With best regards

Oly

12:13  

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