I wish I had time to write a longer reply because this sort of question has been on my mind lately and I have a lot to say about it. I teach logic at the university level. Given my experiences I've come to think that it's difficult to justify even at that level.
At the very least, I think we have to come up with more sophisticated arguments to convince students of its relevance (for those who aren't going on to study computer science). Year after year, I've seen our institution (and this applies to my own contribution) - fail to present that argument in a way that is convincing to students. I've come to believe that this is not a result of superficial engagement with the concepts by the students. Even my best students remain skeptical as to the relevance of logic. Rather I think they are reacting to a genuine concern about the usefulness of logic given it's actual nature.
To explain fully why would take some time - and I've got half a large blog post waiting to be finished on the matter. But for example - students get jaded when we logicians tell them that:
This apple is green.
Therefore, it is not red.
Is NOT a logically valid argument. But hang on they argue - you told us that a logically valid argument is one where the truth of the premises necessitates the truth of the conclusion. How does this argument fail to qualify?
They become even more jaded when we tell them that the logical form of English indicative conditionals such as:
If the moon is made of cheese, then Sarah Palin is the president of the United States.
are best represented by the material conditional. We present detailed and sophisticated arguments for this conclusion. But no matter how clearly we present the case, students remain frustrated over the fact that if that is a material conditional then it's true - even though we suppose that there is no connection between the moon being cheese and Sarah Palin being president.
There are countless other examples. The point though is that you can't just dismiss student frustration. Because no matter how tight your argument for the material conditional as being the logical form of that English sentence, you just can't deploy that knowledge in any normal context of dispute. If any argument in the real world hinged on the truth of an indicative conditional - trying to convince them that it is indeed true because it's a material conditional with a false antecedent is simply absurd - and the students know it.
As it happens, I do have an argument for the importance of understanding logic. But it's involved and subtle - and still it leaves logic as something that you can't really deploy in a day to day context. It's certainly not something that to which you'd want to subject elementary students. Other than that, the only clear use for the knowledge of logic is in the technology sector and philosophy. This is a path of specialisation that shouldn't be imposed on children at such a young age.
Just my two cents...