How exactly can you contract the phrase “is not”?
More specifically, what’s the difference between the sentences, “The dog isn’t running.” and “The dog’s not running.”?
They both sound correct to my ear, but I’m not sure if the latter is grammatically correct.
Both versions are correct, and both mean “The dog is not running.”
Any difference in connotation is more likely to be expressed via context or tone than the placement of the contraction, in my opinion.
(For what it’s worth, the dog’s not doesn’t technically contain a contraction of is not – it contains a contraction of dog is. But I don’t think that’s what you’re asking.)