On a characterization of the material conditional $\to$ I understand the usual motivation behind the truth table for the logical connective $\to$.
However, I would like to know if there is a more fundamental reason for that truth table. Something that would have to do with arguments and validity.
A.G.Hamilton writes in Logic for Mathematicians that "the significance of the conditional statement $A\to B$ is that its truth enables the truth of $B$ to be inferred from the truth of $A$, and nothing in particular to be inferred from the falsity of $A$".

Question: What does Hamilton mean precisely? Something like $\to$ is the only binary truth function such that

*

*The argument form $(p\to q),p;\therefore q$ is valid


*The argument form $(p\to q),\sim p;\mathcal{A}$ is invalid unless $\mathcal{A}$ is a statement form logically equivalent to $\sim p$ ?


From the statements of $\{A{\to}B, A\}$ we can infer something new; that $B$ is true.
However, from the statements of $\{A{\to}B, \neg A\}$ we cannot infer anything new.
That is all.
