Mendelson's book, truth function of n argument Mendelson, Introduction to Mathematical Logic
How can you explain the highlighted sentence? I don't understand whether the statement letter are variables ranging over T or F, or the $x_i$ are the variables. Or if both why do we have two sets of variables ?

 A: A truth function of $n$ variables (arguments) is technically a function from $\{T,F\}^n$ to $\{T,F\}$. That is, it’s a function of the form $f(x_1,x_2,\ldots,x_n)$, where the allowed inputs for $v_1,\ldots,v_n$ are $T$ and $F$, and the output of the function is $T$ or $F$. The inputs are the truth values of the statement letters of interest, and the output is the truth value of some logical combination of them.
As an example, the truth function for the logical connective $\leftrightarrow$ (‘if and only if’) is a function of two variables defined as follows:
$$\begin{align*}
&f(T,T)=T\\
&f(T,F)=F\\
&f(F,T)=F\\
&f(F,F)=T\,.
\end{align*}$$
To find the truth value of an expression $A\leftrightarrow B$, for instance, we let $x$ be the truth value of $A$ and $y$ the truth value of $B$; then $f(x,y)$ is the truth value of $A\leftrightarrow B$.
The truth function for the connective $\neg$ (‘not’) is a function of one variable defined as follows:
$$\begin{align*}
&g(T)=F\\
&g(F)=T\,.
\end{align*}$$
With $x$ and $y$ as before, the truth value of $(\neg A)\leftrightarrow B$ is $f\big(g(x),y\big)$, because $g(x)$ is the truth value of $\neg A$.
