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.)
The contraction of "is not" is "isn't." "Ain't" is a slang form that is attributed to "is not," "are not," and "am not," and is considered substandard English.
The two sentences are identical in meaning, just different in the way that the verb is placed; both are grammatically correct. In the first, the verb is contracted with "not," and in the second, the verb is contracted with the subject. Both forms are considered informal English, and the second is discouraged for use in standard English writing.
I would argue that there is a difference in meaning between the two sentences. The first simply states that the dog is not running; it could be walking or sleeping, or could have just stopped running.
The second carries a suggestion that the dog is specifically "not running" anymore or not running now. It carries a temporal component. The focus here is on the "not running" as the "not" is placed next to the "running". To me this suggests that the dog had been running or had been expected to be running, but is, in fact, not running. Using the second sentence to mean what the first sentence means would be unusual.
It’s a matter of emphasis. The OP’s two examples would be typically found in speech, not writing, and the way in which they were spoken would reinforce the meaning. For example, we might expect The dog ISN'T running to be in response to someone who said Hey, stop your dog running like that. On the other hand, The dog's not RUNning could be followed by There must be something wrong with him today. I recognize that these are not the best examples to illustrate the point, but I hope they will give some idea of the difference. The first emphasises that there’s something the dog isn’t doing, while the second emphasises what it is that the dog isn’t doing.
As for the propriety of using contractions in writing, Pam Peters, in ‘The Cambridge Guide to English Usage’ takes a typically pragmatic view:
The writers of formal documents may feel that [contractions] undermine the authority and dignity of their words. But the interactive quality that contractions lend to a style is these days often sought, in business and elsewhere. They facilitate reading by reducing the space taken up by predictable elements of the verb phrase, and help to establish the underlying rhythms of prose.
It’s perhaps worth adding that French uses contractions even in the most formal contexts. There are no non-contracted alternatives to je n’ai pas and so on. And of course there’s one quasi-contaraction that is always found in English, the singular possessive apostrophe -’s.
