Consider the statement
$(1)$ If I'm in Richmond, then I'm in Virginia.
This statement is an implication $p \implies q$, where $p$ is "I'm in Richmond" and $q$ is "I'm in Virginia." Incidentally, this statement is true.
Now consider the statement
$(2)$ If I'm in Springfield, then I'm in Virginia.
This implication is false, because there are Springfields in other states, for example, Illinois. So if we let $r$ be the statement "I'm in Springfield", what is the correct way of showing that statement $(2)$ is false?
I suppose it should be $(r \not\implies q)$ or rather $\lnot(r\implies q)$, but this is equivalent to $\lnot(\lnot r \lor q)$, which is equivalent to $(r \land \lnot q)$, which means "I'm in Springfield and I'm not in Virginia," which is clearly not the case either because there is definitely a Springfield in Virginia.
I think $a\not\implies b$ doesn't mean "$a$ doesn't necessarily imply $b$," I think it means "$a$ never implies $b$." Is this correct? If so, how should I represent the statement, "It is not the case that if I'm in Springfield then I'm in Virginia," in other words, "Being in Springfield doesn't necessarily imply that I'm in Virginia."?