Q:

Mathematical Logic : Implication

consider the statement, if today is Monday then tomorrow is Tuesday

how is the third condition true in this case?

A:

By the definition of the logical connective $\to$, $p\to q$ is defined to be true for every combination of truth values of $p$ and $q$ except $p$ being true and $q$ false.
This isn’t quite like normal English usage of if $p$ then $q$, which generally assumes some genuine connection between $p$ and $q$. The formal logical connective is defined without regard to any possible meaning of $p$ and $q$: its truth value is simply a function of the truth values of $p$ and $q$.
However, some intuitive justification of the definition is possible. What evidence would you require in order to be certain that an implication $p\to q$ was false? The only absolutely certain evidence would be a demonstration that $p$ is true and $q$ is not. Thus, we say in that case that $p\to q$ is false, and since we must (in classical logic) assign to each of the other three possibilities either the value True or the value False, we give truth the benefit of the doubt when $p$ is false.
However, your specific example muddies the issue because for the particular $p$ and $q$ that you’ve chosen, it is not possible for one to be true and the other to be false: it’s actually the case that $p\leftrightarrow q$, and only the top and bottom lines of your table are actually possible states of the universe.

