The truth is that get was first a Sumerian word for document (then written on clay.) It made its way into Rabbinic Hebrew via Aramaic and Akkadian. Shtar, the Rabbinic Hebrew word for contract, is likewise an Akkadian loan word.
Even after it entered Hebrew, the word get meant document. It became our word for one specific type of document, i.e. the divorce document, via specialization, a not-very-unusual type of semantic change. In our own language deer (originally any animal) and girl (any child) also changed via specialization.
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