In the parlance of Talmud study and lomdus, what does the word "Hakira" mean and how should it be used? Why are Hakiras useful?
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Literally it means "investigation" or something similar (and is used in other contexts, like that of evidence in court), but it's used to mean "hair-splitting".
Well, almost really. A chakira is an investigation as to the exact nature of something, and is usually stated as a binary choice: is X an A or is X a B? (X can be an object, a state, an action, etc.) A practical difference is generally sought so that the distinction is clear and known to be real, and, where possible, proof is brought as to whether X is in fact an A or a B.
It's useful in that it clarifies the nature of X.
UPDATE: msh210 beat me to it by a few seconds, but I'll post anyhow so you see the similarity between our answers.
A hakira seeks to suggest two subtly-different explanations for something. If explanation A is correct, then X should apply in some other case; if explanation B is correct, then Y should apply instead.
It helps to think of different ways to view the subject. For instance: