Prehistoric Logic

Modern logic often has an axiom that an entity A cannot also be not-A.

Prehistoric logic might not have thought this so self evident. What did antiquity see as self evident? Coming-into-Being and Passing-out-of-Being were probably so, reflected in myriads of stories about Creation, Birth, and Death.
We might state this as:  "If A, then possibly not-A". We could re-state this as "not-A or not-A" which is a tautology and really does not do the trick. So we are left with "A and not-A", since "A or not-A" is also tautologous.

Of course, prehistoric logic was never so discrete and "punctuated" as is our logic. It was a flow logic that was made up from the grammar of live images. Imagine, if you will, creating a system of logic from the films of Francois Truffaut using only the images as symbols. Then you would get close to the experience.