Someone living in a mansion has killed Aunt Anna.

1. Anna, Cal, and the butler live in the mansion, and are the only people who live therein.
2. A killer always hates his victim, and is never richer than his victim.
3. Cal hates no one that Aunt Anna hates.
4. Anna hates everyone except the butler.
5. The butler hates everyone not richer than Aunt Anna.
6. The butler hates everyone Anna hates.
7. No one in the mansion hates everyone else..

Who killed Aunt Anna?

Only the 'else' in the expression 'everyone else' of premise 7 is wrong, that is: (7) 'No one in the mansion hates everyone' would be correct. Now, there is no contradiction left.

From (4) follows

(8)     hates(anna,anna) & hates(anna,cal) & not hates(anna,butler)

From (3) follows

(9)     not hates(cal,anna) & not hates(cal,cal)

From (2) and (9) follows

(10)   not killed(cal,anna)

From (6) and (8) follows

(11)    hates(butler,anna) & hates(butler,cal)

From (7) and (11) follows

(12)    not hates(butler,butler)

From the Contraposition (Cp) of (5)

(5)     not richer_than(X,anna) --> hates(butler,X)
(Cp)   not hates(butler,X) --> richer_than(X,anna)

and (12) follows

(13)    richer_than(butler,anna)

From (2) and (13) follows

(14)    not killed(butler,anna)

From (1), (2), (10), and (14) follows

(15)    killed(anna,anna)

Edited on November 27, 2016, 12:20 pm

Edited on December 2, 2016, 3:47 am

Edited on December 2, 2016, 4:06 am

Posted by ollie on 2016-11-27 11:56:53