Could you help me read (interpret) the truth tables of the two operators?
For the implication operator, the truth table is:
$$\begin{array}{c c | c} h& c& h \implies c \\ \hline T& T& T& \\ T& F& F& \\ F& T& T& \\ F& F& T& \\ \end{array}$$
Is the third column indicating when the operator holds, in other words, when a value of $s$ CAN imply a value in $t$?
For the "if and only if" conditional connective, the truth table is:
$$\begin{array}{c c | c} s& t& s \iff t \\ \hline T& T& T& \\ T& F& F& \\ F& T& F& \\ F& F& T& \\ \end{array}$$
2 Answers
Formally, $p \implies q$ means "(not $p$) or $q$." Thus if $p$ is false then not $p$ is true, so "(not $p$) or $q$" is also true.
Informally, $p \implies q$ means that if you know $p$ is true then you can conclude that $q$ is true. So if $p$ is false, then $p \implies q$ does not say anything about $q$. So the implication is always true. Whatever one wants is implied by a false statement.
The third column is showing you when the implication is false, an example is $$p \implies q$$
If p is true and q is false, the implication is false. "since truths cannot reach lies".
The "if and only if"(iff) is a doble implication, $p \iff q$ is equivalent to say $p \implies q \land q \implies p$
