This is a false comparison and as such, doesn't really do anything to counter my point.
It is logically identical to your leash --> dogs psychological example. If it's a false comparison, it would be useful for you to point out how they're actually different.
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We used leashes because no matter how well trained a dog may be, there is still an element of unpredictability to their behavior and you never know what might provoke them to act aggressively toward a person or another animal. Children tend to not have that element of unpredictability to their behavior.
Children are less capable of harming people than adult dogs (usually, though truly unstable children have killed people a leash is not the answer there), but given that blocking aggressive behavior isn't the primary reasoning for using a leash in either case...
Except leashes don't really make the kid any safer, all it does is ensure they don't run off.
I'm not saying a kid is necessarily safe while leashed, but in principle a kid that has run off in a public place is less safe at greater distance from the parent(s).
when they hit the limit of their range, they are violently snapped back and probably slam their head on the ground as a result. How is that safer?
We're not arguing about cartoon physics.
The whole point of being a parent is to be the guardian of your child. To do that, you have to actually pay attention to your child, whether they are on a leash or not.
This is not in dispute. The idea that using leash is abuse is in dispute. Nobody claimed that a leash is a substitute for actual attention that I saw. What it will do is block a sudden run-off at a moment's distraction. It won't make breakfast for you, educate children, cure cancer, or ensure psychological well being. That it won't do those things does not mean using it is abuse.
Then that parent isn't doing a good enough job. Disciplining your kids isn't going to work no matter how hard you try if you just plain suck at it. It's all about setting boundaries early and having real consequences that you follow through on when the child crosses those boundaries.
There are some children where there is nothing you can do in terms of consequences to prevent them suddenly attempting to run off. It doesn't matter what boundaries you set or what consequences you give them. The mental/medical issue is an example, but it isn't the only example. You can't predict what an immature (or even some mature people) might do in a split second when given x unknown stimuli to 100% certainty.
And this is also ignoring the possibility of the child being taken against its will. If you're talking about some place like a theme park with lots of crowding, even 95% of your attention on something is insufficient in principle; it's not human nature to never break focus for 5 seconds across a day. Toss in Tim's example of multiple children and this gets unrealistic fast.
Surprisingly, there does seem to be a lively debate on the matter among parents and child psychologists, yet no one has done an actual study to see what the effects of leashing a child might have on the child. Seems like since the issue is a pretty big topic among parents that someone would try to provide a definitive answer to the question.
That's...not ideal

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Well if they want their children to survive, perhaps they should start by actually paying attention to them instead of just strapping a leash on them and then going back to checking their Twitter.
Find me a person who will never get distracted for 5 seconds for any reason and you'll be presenting me a new sentient species.
Personally I think parents are way wayyy too overprotective these days. Leashes are not necessary! If you can't ensure that your kid doesn't make it to adulthood without a leash then you are probably just a crappy parent. (Not you personally)
I have no children, so my stake in this is logical/argumentative rather than personal. It's a reach to claim leashes are "necessary" since many get by just fine without them. However, it's also a reach to claim they're "abuse", especially when (somehow) we clearly lack evidence to make that conclusion one way or the other. The motivation for using them can be reasonable.
Leashes are nothing but a quick solution for lazy parents who don't have time to parent properly and effectively.
I find this comment to be inane unless we're talking about parents who just go around leashing their kids all the time (I have yet to see this, but it's a big world so some idiot's probably done it). The most typical use would be in very crowded areas, as an extra measure against getting separated. I don't see how this makes a parent "lazy" (one could simply not take their child into such areas ever, which uses less effort by a margin), nor how it reflects on how the child is taught or handled in other situations.
If I see a child leashed at Disney or something I don't think much of it.
If I see a child leashed by a parent walking him/her into school one-on-one my thought process doesn't go to "lazy". Rather, the more likely explanations when witnessing something like that is a) child has medical condition like Commodore alluded to or b) the parent is overprotective to a fault/is using a thought process that I don't understand.
I expect parents that routinely put children on a leash when doing simple outings are not the typical group that would be doing it just to look at twitter or something. Most parents will simply opt to take their children to places where they don't require undivided attention and then skip the leash entirely, so TYPICALLY seeing the leash implies something else.