The discussion is about why the statement $\bot \to \bot$ is considered "true" rather than "false". That is, why the truth table of the conditional connective is defined as it is.
An argument is considered valid if, it guarantees the conclusion is true when all the premises are true.
So if $\to$ is defined as it is, then the truth of both premises, $\{P\to Q, P\}$ are required to guarantee the truth of the conclusion $Q$. (Knowing only that one is true is not enough to ensure the conclusion is true.)
$$\begin{array}{cc|cc|cc}P & Q & P\to Q & P & Q
\\ \hline
\bot & \bot & \top & \bot & \bot
\\ \bot & \top & \top & \bot & \top
\\ \top & \bot & \bot & \top & \bot
\\ \top & \top & \top & \top & \top & \star
\end{array}$$
However if we defined $\dot\to$ so that $\bot \dot\to \bot = \bot$ we only need the one premise $P\dot\to Q$ to guarantee the conclusion. The conclusion is always true when that premise is, whatever $P$ may be.
$$\begin{array}{cc|c|cc}P & Q & P\dot\to Q & Q
\\ \hline
\bot & \bot & \bot & \bot
\\ \bot & \top & \top & \top & \star
\\ \top & \bot & \bot & \bot
\\ \top & \top & \top & \top & \star
\end{array}$$
An argument, as intended in the page you mentioned, consists of a collection of premises, used to establish the truth of one (or more) conclusion.
If you were to model this in, say, propositional logic, you would call the premises $p_1, \dotsc, p_n$ and the conclusion $c$.
Then, the argument would be encoded by the formula
$$
p_1 \land \dotsb \land p_n \implies c
$$
To attach a semantic meaning to this formula, i.e. if we want to establish if it is true or false, we need two ingredients:
- The truth values of $p_1,\dotsc,p_n$ and $c$ - you need to fix such values to obtain the truth value of the whole formula; the way you assign this truth values gives you an interpretation.
- A "meaning" for the logical connectives. This means, for example, that the truth value of the conjunction $\land$ can be computed by means of a function (and same goes for the implication).
If we call our interpretation $I$, we say that a formula is satisfied by $I$ (or true under that interpretation) if by assigning the truth values of all the variables as specified in $I$ and then computing the truth values of the logical connectives, the output is true.
As a mathematical convention - this is how implication is defined - a formula of the form $A \implies B$ is false when $A$ is true and $B$ is false; in all the other cases, it is true.
This means that, if the premise $A$ is false, the overall formula is true, no matter the value of $B$. But if $A$ is assumed to be true, then $B$ must be true for the argument to be true.
This means that for an argument to be valid you must be free to give any possible value to each of your variables and still obtain a true formula.
This can be generalized to arbitrary formulas (not only the one in argument form), and that is what the concept of tautology is about.
As an example, the formula $p \lor \neg p$ is a tautology: here, you only have two possible interpretations, one that makes $p$ true, the other makes $p$ false.
You can choose any, and the formula turns out to be true.
Another example of a valid argument is $p \implies p$: assume that something is true; then, that thing is true. Here, you can again choose between two interpretations and no matter what your choice is, the formula is true.
According to the language you are using, there are different ways of defining formula and truth values. You can distinguish between propositional formulas (the ones described above), first-order formulas (as an example, $\exists{x}. p(x) \implies q(x)$), modal formulas and many others. You can choose how many truth values are there: true and false, or true, false and unknown, or infinitely many.
Depending on the choices that you make here, the notion of truth and validity change. Above, I introduced the ones related to classical propositional logic.
Best Answer
You know contraposition. The contrapositive of " if A then B " is " if B is false, then A is false", or, more precisely, " if not-B, then not-A".
A proposition and its contrapositive sentence are equivalent, they mean exactly the same thing.
The definition of deductive validity says that a reasoning is valid just in case :
So
(1) if I know that a reasoning is valid
(2) and that its conclusion is actually false
(2) then , I can claim with certainty that at least one of its premises is false ( one or more, possibly all).