The phrase "narrow meaning" is grammatically correct and can be used in written English.
It is typically used to describe a specific or limited interpretation of a word or phrase. Example: In legal documents, every word must be scrutinized for its narrow meaning and potential implications.
"Design for Innovation" means considering that design cannot simply focus on a narrow meaning of "product use", because this could severely limit the diffusion of innovative products.
"Rural" has a very narrow meaning to the Census Bureau.
It shouldn't have a specialized, absurdly narrow meaning only for top executives, such as conviction of a felony.
The result was narrow, meaning that all political forces must respect the 48% who voted to remain.
The headphones jack is also very narrow meaning that most cables will not fit correctly forcing you to use the included headphones cable.
This meant that the second half of the celebrated slogan "tough on crime, tough on the causes of crime" could only have a very narrow meaning.
In spite of our different narrow meanings, I can communicate with a compatriot because our sentences like, say, "Pork chops are generally better in Memphis than in Portland" will have the same wide truth conditions in spite of their having different narrow meanings.
Of course it would also entail that the narrow meanings of "water" for oneself and one's Twin-Earth counterpart would not be similar at all (since we would endorse none of the same "wide" water inferences).
This would allow one to say that the narrow meanings of "cat" between my wife and I are very similar, since most of our "cat -beliefs have the same truth-cat -beliefs20] Of course, appeals to narrow content are not uncontroversial.
The "two factor" theory would also help support something like the similarity response mentioned above in that narrow meanings could be treated as similar if they embodied most of the "same" inferences where those inferences were themselves typed in terms of having the same wide content.
In much the same way, disagreement could be explained by the fact that a single sentence will have the same truth conditions for both me and my interlocutor (even though we each tie it to different narrow meanings), and so if I affirm it, and they deny it, then only one of us can be correct.
