Aristotelean contraposition consists of obverting an initial proposition, converting this obversion by switching the places of the first and second propositions (keeping any negation), and then obverting the conversion.

Obversion is often explained in this way:

  A-type proposition:
    "All S are P" becomes "No S are non-P".

  E-type proposition:
    "No S are P" becomes "All S are non-P".

  I-type proposition:
    "Some S are P" becomes "Some S are not non-P".

  O-type proposition:
    "Some S are not P" becomes "Some S are non-P".

I see two different unary operators "not" and "non-".  I am informed that they are not equivalent, as "not" defines a negative proposition type (E or O) while "non-" is considered to define a positive proposition type.

My question is whether there is a meaningful distinction between these two operators.  If there is a meaningful distinction, I see no explanation or interpretation of how "not non-P" differs from "P", and the list of examples (that seems to be repeated in multiple sources) seems woefully incomplete.  If there is not a meaningful distinction coming from Aristotle, is there a meaningful distinction for modern logics (particularly paraconstistent logics)?