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I get asked English grammar questions sometimes.  They are usually difficult for the same reason that Feynman's differentiating under the integral worked: the person asking has already used all the tools to which they have access, and failed.  I don't have Feynman's method though; they're just tricky questions.

I also get asked to proof read things a lot.

The second of two verbs is not necessarily an infinitive

We tend to learn that the second of two verbs is in the infinitive; it’s not entirely true.  The second of two verbs is actually likely to be a noun phrase forming the object of a transitive verb.  There are (at least) three general ways to do this:

  1. I like to shop. (Infinitive)
  2. I like shopping. (Gerund)
  3. I like that I shop. (Explicit nominalisation using another phrase)

These are all grammatically correct, albeit with slightly different nuances.  However, consider this example:

  1. I recommend to shop.
  2. I recommend shopping.
  3. I recommend that you shop.

In this case the first is wrong, but it’s not obvious why.  The second and third are correct.  It’s the third that explains why the first is wrong: the subject of the second verb is different to that of the first (you instead of I); the subjects were the same in the “like” case.  The infinitive cannot express that, even though it’s clear from the use of “recommend”.  The gerund one is OK because gerunds don’t have subjects. 

This tends to happen with verbs like "recommend" and "allow" in a passive sense.  In these cases the right answer is often to use the verb in an explicit nominal form:

  1. I recommend to apply...  (wrong since the subject of apply is missing)
  2. I recommend applying... (OK but a bit ambiguous)
  3. I recommend that you apply... (OK, but it's not passive)
  4. I recommend application of... (OK and explicitly passive)