Dogs not allowed?

I went walking in the veld behind our house today, and kept seeing these signs. I took it to mean that one is not allowed to walk ones dog there, and i thought it was strange since there were no houses or tarred roads, just grass and lava rocks and it looked to be ideally suited for dog walking. When I got home I looked up the sentence and saw I got it all wrong, it meant ´only on a leash´.

In Reykjavik one is not allowed to let your dog run around freely, except in one place that is almost like an island. And it seems that it is the same here in the country. I am glad I do not have a dog, if it was well behaved I would have loved to have it run around in the veld. On the other hand, lots of people ignore the sign and the rules.