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1) “According to X, this crisis first arose in the religious sphere, and then spread as to cover philosophy, the sciences and the pseudo-sciences.”

Is “spread as to cover” right?

2) “These theories purport to prove there is indubitable knowledge”.

Is it better to add “that” after prove?

3) “It is important to consider what he understands as ‘fideism’”.

Understand “as” or perhaps “by”?

4) “The evil demon hypothesis has a similar target as the dream argument.”

This doesn’t sound correct; maybe “has a target similar to that of the dream argument”?

Sextus
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Hello Sextus

1) I think you're right to doubt 'spread to cover'. '...and then spread to philosophy etc' would be fine.

2) I would add 'that'; or perhaps say 'prove the existence of'.

3) Yes, 'understands by', or perhaps 'means by'.

4) I would say: “The 'evil demon' hypothesis has a similar target to the 'dream' argument.”

See you
MrP
Comments  
1. No, neither is comma followed by and. I'm not sure the two commas are necessary either. The following is not unclear:

According to X this crisis first arose in the religious sphere then spread to cover philosophy, the sciences and the peudo-sciences.

2. No, I think it is optional.

3. Depends on the meaning. As written, we are considering a number of things, and trying to decide which of those things he understands to be "fideism." Change as to by, and we know he is using the word "fideism" but we don't know what he means when he uses that word. So the question cannot be answered without additional information.
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Thanks. And what about the example number 4?

Sextus
 MrPedantic's reply was promoted to an answer.