The Analytic Atavar

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Saturday, February 24, 2007

Don't Ask Alice - Part the Eighth

This is the eighth installment on Dodgson's unsolved sorites - for the previous seven 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]
Don't ask Alice - Part the Fourth [from Dodgson's Symbolic Logic, p. 190]
Don't ask Alice - Part the Fifth [from Dodgson's Symbolic Logic, p. 191]
Don't ask Alice - Part the Sixth [from Dodgson's Symbolic Logic, p. 192]
Don't ask Alice - Part the Seventh [from Dodgson's Symbolic Logic, p. 193]
Without further ado, here is the eighth of Dodgson's [hitherto] unsolved sorites:

(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 grandsom 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".]

Define the following two-place relations: e(X,Y) - X and Y are enemies, assumed symmetric, so that e(X,Y) = e(Y,X), and e(X,Y)~ means X and Y are friends; s(X,Y) - X is senior to Y, or Y is junior to X, assumed anti-symmetric so s(X,Y) = s(Y,X)~ and vice-versa; i(X,Y) - X is inferior to Y, or Y is superior to X, assumed anti-symmetric so i(X,Y) = i(Y,X)~ and vice-versa; c(X,Y) - X is a creditor of Y, Y is a debtor of X, assumed disjoint so c(X,Y)~c(Y,X)~ = U; f(X,Y) - X is the father of Y, or Y is the son of X, assumed disjoint so f(X,Y)~f(Y,X)~ = U; m(X,Y) - X is a master of Y, or Y is the servant of X, assumed disjoint so m(X,Y)~m(Y,X)~ = U; p(X,Y) - X is a persecutor of Y, or Y is a victim of X, assumed disjoint so p(X,Y)~p(Y,X)~ = U; and u(X,Y) - X is an uncle of Y, or Y is a nephew of X, assumed disjoint so u(X,Y)~u(Y,X)~ = U. We may write the premises as:
1) f(B,A) ⇒ m(A,B) or ρ(f(B,A)~m(A,B))
2) u(C,B) · i(A,C) ⇒ c(B,A) or ρ(c(B,A)i(A,C)~u(C,B)~)
3) e(C,D)~ · e(B,C) · f(A,B) ⇒ c(D,A)~ or ρ(c(D,A)~e(C,D)e(B,C)~f(A,B)~)
4) f(A,B) · c(C,B) ⇒ p(C,A) or ρ(c(C,B)~f(A,B)~p(C,A))
5) i(A,B) · m(B,D) · f(C,D) ⇒ s(A,C) or ρ(f(C,D)~i(A,B)~m(B,D)~s(A,C))
6) f(C,D) · f(D,A) · s(B,C) ⇒ u(B,A)~ or ρ(f(C,D)~f(D,A)~s(B,C)~u(B,A)~)
7) m(B,A) · i(B,C) · e(C,E)~ · e(D,E) ⇒ p(D,A)~ or ρ(e(C,E)e(D,E)~i(B,C)~m(B,A)~p(D,A)~)
8) e(A,B)~ · i(C,B) · m(C,E) · p(D,E) ⇒ e(A,D) or ρ(e(A,B)e(A,D)i(C,B)~m(C,E)~p(D,E)~)
9) e(A,B) ·p(B,C) · m(E,C) · f(E,D) ⇒ e(A,D)~ or ρ(e(A,D)~e(A,B)~f(E,D)~m(E,C)~p(B,C)~)

It must be kept in mind that the letters used in each premise are mere placeholders, indicating the relationships between the various parameters in the two-place functions but in general different from the same letter in a different premise. The problem is to find substitutions in each premise which will eliminate the most terms, leading to the minimal conclusion. To facilitate this process, construct a table with columns for each function and for each premise show the constituent functions with numbers as the parameter placeholders.

c(...)e(...)f(...)i(...)m(...)p(...)s(...)u(...)
1) f(1,2)~ m(2,1)
2)c(1,2) i(2,3)~ u(3,1)~
3)c(1,2)~e(3,1) e(4,3)~f(2,4)~
4)c(1,2)~ f(3,2)~ p(1,3)
5) f(1,2)~i(3,4)~m(4,2)~ s(3,1)
6) f(1,2)~ f(2,3)~ s(4,1)~u(4,3)~
7) e(1,2) e(3,2)~ i(4,1)~m(4,5)~p(3,5)~
8) e(1,2) e(1,3) i(4,2)~m(4,5)~p(3,5)~
9) e(1,2)~ e(1,3)~f(4,3)~ m(4,5)~p(2,5)~
Now we can begin eliminating terms by filling in parameters in a column, and filling in the same parameters in the remainder of the premise. We can assign the parameters to produce s(A,B) in lines 5 and 6, to produce u(A,C) in lines 2 and 6, to produce c(C,D) in lines 2, 3 and 4, to produce p(C,E) in lines 4, 7, 8, and 9, and to produce m(F,E) in lines 1, 5, 7, 8, and 9, using the theorem to eliminate terms. Further simplification can be obtained by selecting parameters to produce the same term in a column, and finally to elimination terms in the e(...) column, producing the final table:
c(...)e(...)f(...)i(...)m(...)p(...)s(...)u(...)
1) f(E,F)~ m(F,E)
2)c(C,D) i(D,A)~ u(A,C)~
3)c(C,D)~e(K,C) e(G,K)~f(D,G)~
4)c(C,D)~ f(E,D)~ p(C,E)
5) f(B,E)~i(A,F)~m(F,E)~ s(A,B)
6) f(B,I)~ f(I,C)~ s(A,B)~u(A,C)~
7) e(J,K) e(C,K)~ i(F,J)~m(F,E)~p(C,E)~
8) e(K,J) e(K,C) i(F,J)~m(F,E)~p(C,E)~
9) e(K,C)~ e(K,H)~f(F,H)~ m(F,E)~p(C,E)~
with the conclusion:

ρ(e(G,K)~e(I,K)e(H,K)~f(B,E)~f(E,F)~f(F,H)~f(E,D)~f(D,G)~f(B,I)~f(I,C)~i(D,A)~i(A,F)~i(F,I)~u(A,C)~) or
e(J,K)~ · f(B,E) · f(E,F) · f(F,H) · f(E,D) · f(D,G) · f(B,I) · f(I,C) · i(D,A) · i(A,F) · i(F,J) · u(A,C) ⇒ ¬[e(G,K) · e(H,K)]
Diagramming these relationships, we conclude that if the great-grandfather, B, had two sons E & I, and that if E had two sons F & D, who each had a son, H & G respectively, and that if I had a son C, and that if D is inferior to A who is inferior to F who is inferior to J - then if all these conditions are met, the conclusion Dodgson requested is:
The two great-grandsons (G & H) are not both enemies of their Father/Uncle's (F's) superior's (J's) friend (K).
We don't know where A fits in the family tree, only that he is an uncle of C (B's grandson); he could be either a son of B or a maternal uncle of C.  In completing the father column f(...), we assumed the existence of an individual I, a brother of E.  It is also possible that E had no brother, i.e., that E and I are the same individual.  There is not enough information from the premises to decide I's status.  None of this uncertainty affects the conclusion.
[Note: This article was corrected on 2/27/07 to fix a mistake in the table elimination and to show the new, proper conclusion.]

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