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We can not exaggerate the value of literature as a medium by which to cultivate our minds.

Could you tell me if the sentence is grammatical? If that is grammatical, could it be that the infinitive clause modifies which? However hard I think of it, the sentence should be: We can not exaggerate the value of the literature as a medium by which we cultivate our minds.

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anonymousWe can not exaggerate the value of the literature as a medium by which we (can) cultivate our minds.

The paraphrase of an infinitive construction such as you have here usually contains a modal verb.

The analysis of the infinitive clause takes its cue from your paraphrase without the infinitive.

anonymousCould you tell me if the sentence is grammatical?

It is, but "can not" should be "cannot".

anonymouscould it be that the infinitive clause modifies which?

No. "which" is the relative word that connects the infinitival relative clause (by which to cultivate our minds) to the noun "medium". Thus, the relative clause modifies "(a) medium".

Here are a few similar constructions. I've underlined the noun which is modified by the infinitival relative clause.

There was a lack of raw metal [with which to produce the coins].
The students at the academy have many blessings [for which to be thankful].
There's no better city [in which to mount a show about recovery] than New Orleans.

CJ