Context is how I know whether “dog”(n) and “dog”(v) are two instances of the same word or not?
I don’t understand. What context?
(Is it possible you misread the first sentence of my post?)
The alphabetical sequence dog may represent either a noun or a verb. The word [dɒg] may represent either a noun or a verb. Which one is meant is understood by context. Whether they constitute the “same” word doesn’t really matter in the end.
It does if grammaticality is a property of a sequence of words rather than alphabetic symbols or phonemes.
Is it?
I’m having a lot of trouble understanding what it is you’re trying to get at, Frylock. I admit that might be my issue, as I’m pretty damn tired.
It’s possible someone could say “The dog can run” and mean dog[sub]verb[/sub] instead of dog[sub]noun[/sub], thus rendering it technically ungrammatical, but how could you tell?
I find it very silly to claim “Dog can run” can be grammatical on the grounds that “Dog” might be a name unrelated to the intended word “dog” or any such thing. I have no problem saying “Dog can run” is ungrammatical, and the point would be that this indicates that “dog” is in a different lexical class from, say, “Bob”. And in the OP’s variety of English, supposedly, “Dad” is in the lexical class of “Bob” rather than that of “dog”.
I also felt the OP’s question was, legitimately, a question of grammar; in their variety of English, there was supposedly a distinction in lexical class between “dad” and “father” which, in my own and in more standard lects, there was not, just as Frylock pointed out.
If the sentence is said while watching a dog run, you’re probably right, although I recall that some dialects do allow for omitting the article. But that’s assuming context again.
I suppose, but when the OP asks something like “Is one considered more grammatically correct?” I interpreted that not to be a question of what is grammatical or not in his dialect but rather a style or usage question in general. Otherwise, that question doesn’t really make sense to me. How is something more or less grammatical in the descriptive sense?
Here, incidentally, is a cite for considering such a sentence as “Door is open” ungrammatical.
Perhaps the confusion is that it may have seemed as though Frylock was referring to this as a distinction between “names” and "noun"s (although, looking back, I think he was careful to only ever correctly refer to this as a distinction between what he called “names” and other nouns). Both categories under discussion are subcategories of the nouns; it’s just a distinction between common singular count nouns, which require a determiner in a noun phrase, and other nouns (including proper nouns), which don’t.
The use of “the” is complicated.
I always use the form: “The FDA granted approval …”
But this form has become very common: “FDA granted approval …”
I imagine that’s the move from analyzing FDA/“Food and Drug Administration” as an ordinary phrase with the syntactic properties you would expect of that construction (in particular, with “administration” as a common singular count noun) to analyzing it as an opaque name with the syntactic properties of a proper noun.
Well I started out trying to be careful about, then stopped being careful a few posts back unfortunately. 
And that distinction (requiring or not requiring a determiner in the phrase) is a grammatical distinction, is what I was thinking.
It seems to me the OP can be fairly understood as asking whether “dad” can be used as a “singular count noun” (thanks for the clarifying terminology which I wish I’d used all along) or not in standard English. That strikes me as a “question about grammar” if anything is.
In case it’s not clear, I agree. As I said,
Ah, but as I think now as to how I’m spending my day, it’s a very piddling point to spend time arguing over, whether some guy in some thread somewhere on the Internet was or was not actually talking about “grammar” when he thought he was. I mean, you and I are correct, Frylock, but, eh, my enthusiasm for making an effort to argue that we are is waning in favor of “Whatever…”
In case it’s not clear, I know.
See now that’s the difference between you and me I’m afraid. When I am faced with actual responsibilities and see the opportunity to engage in deeply insignificant piddling problems on the SDMB, I gravitate toward the SDMB every time unless there’s an IMMEDIATE URGENT DEADLINE involved.
Yeah let’s not speculate about who’s going to get further using these respective strategies… :(
Similarly, people in the Navy tend to say something like, “USS Nimitz is sailing across the Pacific,” while civilians will add “The” to the beginning of the sentence.
An interesting discussion on Language Log about when acronyms and initialisms do and don’t take an initial article
Yep, it’s purely a matter of dialect and personal preference rather than grammar rule.
Mommy, mama, mom, and mother all have the same grammar.
There is such a distinction in my speech, although not the same as that described by the OP.
“Dad” can be a common noun or a term of address. “Father” can only be a common noun. I don’t call my father “Father”, anymore than I call my wife “Wife”. I associate using “Father” as a term of address with the speech of upper-class British people.
One thing you could claim is that if you heard someone you knew to be a native speaker of English distinctly say, “Dog can run,” it would elicit the response, “Who is Dog?” proving that the listener had been forced, by that structure, to interpret “Dog” as a proper noun in that case (since the common meaning of “dog” is a singular count noun that requires an article).