I'm not interested in what Aristotle or the stoics wrote; how did "common" people understand practical logic, especially as it relates to rhetoric.
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You can see by G.E.R. Lloyd, Magic, Reason and Experience: Studies in the Origin and Development of Greek Science with Chapter 2 dedicated to Dialectic and demonstration, including : Early philosophical argumentation, The development of rethoric, The criticism of rethoric.
See also, by the same author :
But I suppose that you will never find investigations about the logic of "common people" in ancient Greece. Men without instruction scarcely write books; and very few books of ancient Greek philosophy and science survived (see Stoic books on logic).
Not specifically about Grekk-Roman world, you can see the studies of french anthropologist Lucien Lévy-Bruhl, namely :
Rhetoric was in fact understood as an everyday, practical skill for common Athenian male citizens. They were expected to use it to defend themselves in court and to participate in the democratic process. The ability to deftly defend oneself against a lawsuit was a major reason for the popularity of formal schools of rhetoric.
Rome is actually a bit different, though. Philosophy and rhetoric split into different fields during the Roman Empire, known as the Second Sophistic
The Romans did still use formal logic in rhetoric but not at all to the same degree as the Greeks had. They no longer believed it could uncover the truth through debate, so this emphasis was lost.