We propose misogyny detection as an Argumentative Reasoning task and we investigate the capacity of large language models (LLMs) to understand the implicit reasoning used to convey misogyny in both Italian and English. The central aim is to generate the missing reasoning link between a message and the implied meanings encoding the misogyny. Our study uses argumentation theory as a foundation to form a collection of prompts in both zero-shot and few-shot settings. These prompts integrate different techniques, including chain-of-thought reasoning and augmented knowledge. Our findings show that LLMs fall short on reasoning capabilities about misogynistic comments and that they mostly rely on their implicit knowledge derived from internalized common stereotypes about women to generate implied assumptions, rather than on inductive reasoning.