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I don't want people thinking I can cure their upset stomachs and stomachaches.

In this sentence, shall we replace 'thinking' by 'to think'
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In this sentence, shall we replace 'thinking' by 'to think'
We can replace it, or we can leave it as 'thinking'. Either one makes sense.
The form 'thinking' simply emphasizes the progressive, continuous, habitual, or on-going aspect.
The form 'to think' would create a punctual (point-in-time) aspect, which the author obviously did not want in this situation.

I don't want my daughter to go to that movie. (one-time, punctual aspect)
I don't want my daughter going to that sort of movie. (in general, habitual aspect)

CJ
Comments  
RishonlyI don't want people thinking I can cure their upset stomachs and stomachaches.

In this sentence, shall we replace 'thinking' by 'to think'

I'd go with TO THINK as you have the verb WANT. Like I WANT somebody TO do something.
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 CalifJim's reply was promoted to an answer.
Got the key. Thanks CalifJim and EyeSeeYou.