1. I cannot but admire his courage.

2. I cannot help admiring his courage.
3. I cannot avoid admiring his courage.
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Are all the sentences grammatical?
What is the meaning of 'but' in Sentence 1?
Does 'help' in #2 mean 'avoid' in #3?

All are grammatically correct, but #1 is the best. #2 is OK, but #3 is not used.

Yes, all the sentences are grammatically correct.

In sentence 1, "but" is used as a conjunction that is used to connect two clauses, indicating contrast or disagreement. In this sentence, it is used to convey that the speaker cannot do anything else except admire his courage. It emphasizes that admiration is the only possible response.

In sentence 2, "help" means to be unable to prevent or stop something from happening. So, when the speaker says "I cannot help admiring his courage," it means that they are unable to stop themselves from admiring his courage. It implies a sense of involuntary admiration.

In sentence 3, "avoid" means to prevent from happening or to keep away from something. When the speaker says "I cannot avoid admiring his courage," it means that they are unable to prevent themselves from admiring his courage. It implies that admiration is inevitable, regardless of any effort to avoid it.

Thus, "help" in sentence 2 does not have the same meaning as "avoid" in sentence 3. "Help" in sentence 2 means being unable to prevent something, while "avoid" in sentence 3 means being unable to keep away from something.