tautologies and contradictions with $r$ I'm really struggling to understand tautologies and contradictions.
I've been able to do $(p \rightarrow q) \leftrightarrow (\lnot q \rightarrow \lnot p)$ and I understand why it is a tautology,  however I don't understand when they involve the character r. For example,
$$p \rightarrow (q \rightarrow r)$$
How do I construct a truth table from this? What even is r? I'm so confused. If someone could actually explain the concept or knows of a website that explains it well I'm open to anything!
 A: $r$ is simply another variable, just like $p$ and $q$ are variables. That means the truth value of the statement is a function of the truth values of $p, q, r$. 
So you need $2^3 = 8$ rows in your truth table in order to consider all $8$ possible distinct truth-value assignments to the variables $p, q, r$. 
The number of rows you need for a truth-table for any statement is a function of the number of distinct variables in the statement: if there are $n$ variables, then you need to consider $\,2^n\,$ possible distinct truth-value assignments to those variables, since each of the $n$ variables can take on two possible truth-values: true or false.
Wolfram Alpha does a nice job with truth-tables: Note the $8$ rows needed.

Note that the only possible truth value assignment that makes the entire statement false is when $p$ and $q$ are both true, but $r$ is false. Then, and only then, we have that $$\;\underbrace{\underbrace{p}_{T} \rightarrow (\underbrace{\underbrace{q}_{T} \rightarrow \underbrace{r}_{F}}_{F}}_{F})\;\text{ is false.}$$ 
The statement, however, is not a tautology nor a contradiction, as your title suggests. The truth value of the statement is contingent: it depends on the truth-value assignments of the variables. 


*

*In a tautology, a statement is true under every possible truth-value assignments of its variables, and 

*In a contradiction, the statement is false under every possible
truth-value assignments of its variables.

