The law of noncontradition states that (in a system $S$), for any proposition $p$, $\neg p$ and $p$ cannot be both true at the same time. Does it mean that $\neg p$ and $p$ cannot both be implied in $S$?
So does the validity of the law of noncontradiction imply consistency of $S$?
On
Consistency is a property of a formal system $S$ ... and it is a syntacticalproperty: it indeed says that $p$ and $\neg p$ cannot both be derived in system $S$
The Law of Non-Contradiction is something that is assumed to hold true for most logics. It is a semantical property in the sense that it is assumed that statements cannot be both true and false at the same time.
To see the difference, take classical propositional logic. In classical propositional logic we assume the Law of Non-contradiction. OK, but now we can try and define a formal system $S$ with axioms, rules, etc. And suppose that this system contains the rule:
Hokus Ponens
$\therefore \varphi$
(that is, from nothing, you can infer anything you want)
Now, clearly such a system $S$ is not consistent, since we can infer both $\varphi$ and $\neg \varphi$. And, as such, this sytem is not a good axiomatiation of classical propositional logic.
If the system $S$ is consistent, then it will not be possible to validly deduce the statements $p$ and $\neg p$ from the axioms using the rules of inference. If the system is inconsistent, then it will be possible to deduce such a contradiction from the axioms. In that event, we would be assured that at least one of the axioms of $S$ is false. Note that system $S$ I'm referring to is a classical system.