Let's say we've got the following model:
There is a set of people $A_1,...,A_n$ That is split into two groups: Those who always say the truth, and those who always lie.
Now, every person in this set makes a statement about some other people of the set, e.g. $A_1$ might say "$A_3$ and $A_4$ are liars, and $A_5$ is a truth-teller".
The theorem I've deduced (and of which I am asking whether it's correct):
No matter what statements we're given, for any person the guess whether he's a liar or truth-teller never will be better than pure chance.
Sketch: We'll model the statements as a formula in propositional logic, then deduce all models and show that there's equally many models in which a person $A$ is a truth-teller as there are models where $A$ is a liar.
Let's say a person $A$ says that $B_1,...,B_n$ are truth-tellers and $C_1,..,C_n$ are liars.
Then there's two possibilities:
- $A$ is a truth-teller. Then the statements are all true.
- $A$ is a liar. Then the statements are all false.
Let's say the predicate $T(\cdot)$ stands for being a truth-teller, so that $T(A)$ is true iff $A$ is a truth-teller. Then we can build up the formula
$$\,\bigg(T(A) \land T(B_1)\land ... \land T(B_n) \land\lnot T(C_1)\land ...\land\lnot T(C_n)\bigg) \lor\bigg(\lnot T(A) \land \lnot T(B_1)\land ... \land \lnot T(B_n) \land T(C_1)\land ...\land T(C_n)\bigg) $$ for person $A$, and for each other person another one that will look like this.
The whole formula then is the conjunction ($\land$) of all formulas like above for every person in the set.
Now let's say $v$ is a model of our formula. Then for every formula like the above one, either the left side of the disjunction or the right side of the disjunction has to be true.
Let's say wlog the left side is true.
Then, if we construct the interpretation $v'$ by inverting every assigment (i.e. $v'(A):= \lnot v(T(A)) $ ), in above formula now the right side is true.
Therefore, we've got for every model where $A$ is a truth-teller a dual one where $A$ is a liar, and thus, any deduction we make can never be better than pure chance.
My question is: Is my proof correct? Is my deduction correct? It feels wrong that all this information amounts to nothing.