In the following sentence, should "in" or "with" be used?
"They are all involved ____ the program."
Assuming there is one correct way to say it, is there ever a situation where the other is used, or is always only one way?
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Kate is involved in a romantic relationship, with Jack.
Kate is involved with Jack, in a romantic relationship.
Depends upon the tense, and the way we form sentences.
In your case, involved in is more suitable, and if you need to use with (maybe to prevent repetition within the paragraph), the correct verb would be associated as in "They are all associated with the program".
Technically, the two sentences have different meanings. However, for all intents and purposes (i.e. every day usage) they can be viewed as having the same meaning. If I were writing, I would use "in."
If you do want to choose, this is my thinking on the differences in meaning:
When we use the preposition "in:" "They are all involved in the project." we are using "in" to talk about a state or condition. It is possible many native speakers will subconsciously feel that "they" think about the project all the time, work on it to the exclusion of other things, and in general are immersed in the project.
When we use the preposition "with:" "They are involved with the project." we are using "with" to show a relationship or when people work together. I feel that using "with" is less immersive than "in," however the word "involved" has connotations of full immersion or demanding one's full attention, so this seems to complicate the meaning.