THE GENERAL DUAL

D(atom) => atom

D(A /\ B) => D(A) \/ D(B)

D(A \/ B) => D(A) /\ D(B)

D(A == B) => D(A) XOR D(B)

D(A XOR B) => D(A) == D(B)

D(A -> B) => -(D(B) -> D(A))

D(forallx(A)) => thereexistsx(D(A))

D(thereexistsx(A) => forallx(D(A))

D(-A) => -(D(A))

D(TRUE) = FALSE

D(FALSE) = TRUE

IF A = B THEN D(A) = D(B)

other notes:
-(A==B) == A XOR B