negating connectives

["And Rosta" <a.rosta@ntlworld.com>]

As per standard logic, negating a connective reverses its truth table. E.g.

 E           na E
 T T : T     F
 T F : F     T
 F T : F     T
 F F : F     T

My question is, firstly:
How do we negate a connective so as to mean "this connective yields a
false/wrong truth table, but its truth-reversal does not necessarily yield a
true/correct truth table"?
For example, if I know that p iff q, I would like to be able to somehow say
that I know that it is false/wrong that p and q.

And secondly:
In asking the first question, am I falling victim to the fallacy of
construing connectives as possible-worlds operators, so that the answer
to my question needs to be sought amid the logic of possible-world
operators rather than the logic of connectives?

--And.