The original sentence is below:
The problem with my other half is that he has no ambition.
If I use the word 'of' instead of 'with', how does the meaning of the sentence change? When should I use 'of' instead of 'with'?
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When using Joe's problem (or "the problem of Joe"), you are discussing Joe (and his problem). Perhaps he has some medical condition.
When using "the problem with Joe" (or perhaps "my problem with Joe"), you are discussing yourself, and the difficulty you have with Joe. Perhaps you can't understand his accent.
Consider the following dictionary definition (from Oxford Dictionaries):
7 In relation to.
"my father will be angry with me"
That is, a problem with (someone) is a problem that you have or which exists generally in relation to that person. Usually it will be a problem with their behaviour, their attitude, or some other characteristic they have.
Saying a problem of (someone), on the other hand, means that the problem is the peson's—a problem they experience, rather than anyone else. Consider this definition:
3 Indicating an association between two entities, typically one of belonging, in which the first is the head of the phrase and the second is something associated with it.
"the son of a friend"
"the government of India"
"a photograph of the bride"
[with a possessive] "a former colleague of John's"
Your sentence as-is wouldn't sound natural if you substituted "of" for "with". Instead, I would write it as:
- A problem of my other half's is a lack of ambition.
- My other half's problem is a lack of ambition.
Even still, sentence #1 here isn't great, and I would prefer sentence #2. I can't think of a natural-sounding sentence like #1 where a possessive (
's) isn't used.