Can the boolean operator IMPLY be derived from the Aristotalian operator IMPLY if I treat the Aristotalian predicates as boolean objects?
The logical axioms are uproved assumptions of both formal systems. To my knowledge there is no derivation step possible, except a silly tautology using the principle of identity.
In theory, you could derive predicate calculus from Aristotalian logic, in that aristotalian logic supports a non-monotonic set of operators. (NOT And IMPLY). Which ought to make the rest of the logical operators derivable as theorems, where your domain of discourse is set theory itself. At least, that was the received wisdom I got as an undergrad. I've not tried it personally.
Chances are, I'd guess, that it's never been done, except maybe as an undergraduate exercise at some university. I'd bet equivalent things have been done in circuit design, without it occuring to anyone to equate the two domains. It's extremely easy to fold all of aristotalian logic into a tidy corner of predicate calculus (three line sorites fold into one line predicate calculus statements). Producing predicate calculus from aristotalian logic would be, by comparison, an onerous and pointless exercise.