I'm having trouble combining/forming a single propositional formula from 3 recursively expressed statements. The original setting of the problem is something like:
There are three people $A$, $B$ and $C$.
$A$ said: $B$ and $C$ told the truth if and only if $C$ told the truth.
$B$ said: If $A$ and $C$ told the truth, then it is not the case that: if $B$ and $C$ told the truth, then $A$ told the truth.
$C$ said: $B$ did not tell the truth if and only if $A$ or $C$ told the truth.
From the statements I was able to formulate 3 propositional formulas for each person. I used $T_A$, $T_B$ and $T_C$ to represent the variables for the statements of each candidate, if they were true. That is,
for $A$: $T_A \Leftrightarrow ((T_B \land T_C) \Leftrightarrow T_C)$
for $B$: $T_B \Leftrightarrow ((T_A \land T_C) \Rightarrow \lnot ((T_B \land T_C) \Rightarrow T_A))$
for $C$: $T_C \Leftrightarrow ((T_A \lor T_C) \Leftrightarrow \lnot T_B)$
Now I am supposed to combine the 3 formulas above to a single one. My thought is after doing this, we could enumerate what would the value of the whole formula be for all the possible values of $T_A$, $T_B$ and $T_C$. If the whole formula turned out to be $true$ somewhen, then we could tell who actually lies judging from the value of $T_A$, $T_B$ and $T_C$.
I'm stuck here because the statements are recursive. I assume the 3 formulas I wrote above are correct. (Please correct me if I was wrong!) I was kind of always chasing my tail and didn't come up with a solution.
How should I combine the above 3 formulas correctly?
Thanks in advance.