What else but sail to save a daughter's life,
and pray she still drew breath?
It seems to me that the article in bold is redundant. Why is it used in this text? Is it grammatically correct?
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It's the life of a daughter.
The countably singular noun "daughter" needs some determiner, even with the genitive inflection. The indefinite article "a" is the determiner for "daughter", and in turn the genitive phrase "a daughter's" is the determiner for the countably singular "life".
Nothing is redundant. Everything that needs counting is counted.
Assuming the daughter is the writer's own daughter, the "a" would be incorrect in a strictly mundane grammar lesson on sentence structure. In that case, the sentence would be given as, ".. to save my daughter's life", even if the writer has more than one daughter.
However in this case, by using the indefinite article "a", the author speaks of his own daughter in a general sense, as if she were the daughter of any person in that situation. He moves the context into a more philosophical, universally significant realm to emphasize the emotion which is shared by humanity in general. It makes the story more poignant and moving to the reader.