P1: Butler is telling truth
P2: Cook is telling truth
P3: Gardener is telling truth
P4: HandyMan is telling the truth
- if the butler is telling the truth then so is the cook;
- P1 => P2 i.e ~P1 or P2 is true -(1)
- the cook and the gardener cannot both be telling the truth;
- the gardener and the handyman are not both lying;
- if the handyman is telling the truth then the cook is lying
- P4 => ~P2 i.e ~p4 or ~p2 is true i.e ~(P4.P2) is true - (4)
When P1 is true:
- P2 must be true (By 1) -(5)
- P3 must be false (By 2) -(6)
- P4 must be false (By 4 and 5)
When P1 is false
- P2 can be true or false (By 1)
- P1 is false P2 is true
- P3 must be false - (By 2)
- P4 must be false - (By 4)
- But this means p3 or p4 is false. This will contradict (3). So this is not possible.
- P1 is false P2 is false.
- P3 or P4 must be true. (P3,P4 = TT,TF,FT)
So there are the following possible solutions
P1,P2,P3,P4 = {T,T,F,F} and {F,F,T,F}, {F,F,T,T}, {F,F,F,T}
Hence we can't uniquely determine the values of P1,P2,P3,P4