Eddington's Liars Problem

If A, B, C and D each speak the truth once in three times (independently), and A affirms that B denies that C declares that D is a liar, what is the probability that D was speaking the truth?

There is some evidence that the problem was first proposed as a joke, but a few have taken it seriously.

In order to satisfy the conditions of the problem the possible statements of A, B, C, and D must be in the following form: 

	D tells the truth (T), or
	D lies (F)

	C declares that D is a liar (T/F), or
	C declares that D told the truth (T/F)

	B denies that C declares that D is a liar (T/F)
	= B affirms that C declares that D told the truth, or
	B affirms that C declares that D is a liar (T/F)
	= B denies that C declares that D told the truth


	A affirms that B denies that C declares that D is a liar (T/F)
	= A affirms that B affirms that C declares that D told the truth
The statement of A must be of a fixed form but B, C and D each have two possible statements, independently of each other. We have given some alternate equivalent statements to relate more easily to the previous statements. There are eight possible combinations of statements: 
1. D tells the truth p = 1/3
   C declares that D is a liar (F) p = 2/3
   B denies that C declares that D is a liar (F) p = 2/3
   A affirms that B denies that C declares that D is a liar (T) p = 1/3
   p = 4/81

2. D tells the truth p = 1/3
   C declares that D told the truth (T) p = 1/3
   B denies that C declares that D is a liar =
   B affirms that C declares that D told the truth (T) p = 1/3
   A affirms that B denies that C declares that D is a liar =
   A affirms that B affirms that C declares that D told the truth (T) p = 1/3
   p = 1/81

3. D tells the truth p = 1/3
   C declares that D is a liar (F) p = 2/3
   B affirms that C declares that D is a liar (T) p = 1/3
   A affirms that B denies that C declares that D is a liar (F) p = 2/3
   p = 4/81

4. D tells the truth p = 1/3
   C declares that D told the truth (T) p = 1/3
   B affirms that C declares that D is a liar (F) p = 2/3
   A affirms that B denies that C declares that D is a liar (F) p = 2/3
   p = 4/81
 

5. D lies p = 2/3
   C declares that D is a liar (T) p = 1/3
   B denies that C declares that D is a liar (F) p = 2/3
   A affirms that B denies that C declares that D is a liar (T) p = 1/3
   p = 4/81


6. D lies p = 2/3
   C declares that D told the truth (F) p = 2/3
   B denies that C declares that D is a liar =
   B affirms that C declares that D told the truth (T) p = 1/3
   A affirms that B denies that C declares that D is a liar (T) p = 1/3
   p = 4/81


7. D lies p = 2/3
   C declares that D is a liar (T) p = 1/3
   B affirms that C declares that D is a liar (T) p = 1/3
   A affirms that B denies that C declares that D is a liar (F) p = 2/3
   p = 4/81


8. D lies p = 2/3
   C declares that D told the truth (F) p = 2/3
   B affirms that C declares that D is a liar (F) p = 2/3
   A affirms that B denies that C declares that D is a liar (F) p = 2/3
   p = 16/81
Cases 1 through 4 are favorable to D telling the truth. The sum of the numerators of these cases is 13 and the sum of the numerators of all cases is 41, so that the probability that D told the truth is 13/41. The above assumes that C has heard D's statement and knows whether it is true or false. If he does not know, the probability of his lying is 1/2.

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