Meaning of 'sense' in 'a man of eminent good sense'

Question

I would like to conclude the accurate meaning of sense in the following phrase.

a man of eminent good sense

That phrase is from one of example expressions in the entry for the adjective, 'eminent' in the Oxford Advanced Learner's Dictionary.

I assume the meaning of sense in that phrase is judgment, the 4th of the definitions defined by the Oxford Advanced Learner's Dictionary as follows. Why I assumed so is merely because eminent good sense is an uncountable noun and the 4th definition is the only one which is uncountable.

But meaning-wise, if it were a man of an eminent good sense or a man of eminent good senses, other definitions can be applied, wouldn't it be?

1. [countable] one of the five powers (sight, hearing, smell, taste and touch) that your body uses to get information about the world around you
2. [countable] a feeling about something important
3. [singular] an understanding about something; an ability to judge something
4. [uncountable] good understanding and judgement; knowledge of what is sensible or practical behaviour
5. senses [plural] a normal state of mind; the ability to think clearly
6. [countable] the meaning that a word or phrase has; a way of understanding something

Answer 1

You are correct in your understanding of sense in the example. There's nothing more to be said about that. So I will now address the other part of your question.

If a person is of something, used to describe a characteristic such as sense, intelligence, stupidity, good looks, and so on, then that characteristic will pretty much always be the characteristic in the abstract. It is the characteristic, not an instance of it. Thus, it will always be a mass (uncountable) noun. It can also be used in the plural for things that people actually have several of - "a man of good ideas".

If you were to use with instead of of, then you could use it for characteristics in the abstract (mass nouns) or specific instances (countable nouns). You then use a definite or indefinite article as appropriate by the usual 'rules'.

A man with an eminent good sense.

This could mean someone with one of the "five senses" that are particularly acute, but being so close to a familiar phrase means that people would usually take it the usual way instead. Part of the issue is that "good sense" is acting as a noun phrase, which we are used to meaning "sense" in the sense of "judgement". You can make people interpret "good sense" as less of a set phrase by making them think of good as a separate adjective, by shifting eminent to become the adverb form eminently:

A man with an eminently good sense.

This says that the level of 'goodness' is such as to make it eminent. However, the "good sense" being there will still make people think of judgement, so it's not going to shift meaning to allow any other sense of sense. So let's swap good for another word that only applies to the "forms of perception" sense of sense:

A man with an eminent acute sense.
A man with an eminently acute sense.

The first says that the sense, whichever it is, is both eminent and acute. The second says that it is so acute as to be eminent.

Finally, if you make it plural the shades of meaning change again:

A man with eminent good senses.
A man with eminently good senses.
A man with eminent acute senses.
A man with eminently acute senses.

All of these will tend to be taken as something other than "judgement", because that sense of sense doesn't become plural. The difference between eminent and eminently is the same as above.