The motto of the institution where I work is:
To explore the potential of nature to improve the quality of life
Is this (double use of the to-infinitive) grammatically correct? And if so, is it common to set up a sentence like this?
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There are two ways to interpret this.
To improve the quality of life is an infinitive clause. It could be modifying explore:
[Our purpose is] to explore... to improve the quality of life.
In this interpretation, to improve the quality of life is the purpose for your exploration.
To improve the quality of life could also be modifying potential:
[Our purpose is] to explore the potential [to improve the quality of life] of nature.
This may sound clunky, but I simply switched the locations of the two modifiers applied to potential (to improve... and of nature) in order to illustrate my point.
These two methods of understanding yield very similar results, and the slogan is probably intentionally ambiguous. For more meaningless slogans and a few good laughs, read this Cracked.com article.
Hope this helps!