The act or process of using apostrophe, a literary technique where the speaker directly addresses an absent person or personifies an object.
From apostrophe plus the noun-forming suffix -ation (the act or process of). Emerged in rhetoric and literary criticism to name the specific practice of making apostrophic addresses.
The fancy name 'apostrophation' for what's really just 'talking to someone who isn't there' shows how English loves taking simple concepts and dressing them up in Greek and Latin clothes!
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