Logical park life

This is one for the computer science geeks to argue over. At a park the other day I strode past a sign proclaiming:

No dogs or skateboards allowed

I felt it was awfully nice of them to let me bring them in. “What?” you proclaim. Didn’t the sign forbid such actions? Mais non; and should anyone challenge me, here’s my proof (I’ve shown my working so the kid next to me can copy)

"No dogs" is equivalent to "!dog"

Since my, rather decrepit, dog clearly exists:

!dog = !true = false

Substituting false for No dogs, gives us:

false OR skateboards

And since my skateboard clearly exists:

false OR true

equates true. Therefore, as long as I have both a dog and a skateboard — which I can guarantee if I bring them into the park together — I’m sorted. It also works if I leave the dog at home and just bring my skateboard, which is great because the dog’s rubbish at skateboarding anyway: it can’t even master simple things like Ollies. Guess you can’t teach an old dog new tricks after all.