NP-hardness of an extention of 2 sat
You can reduce from independent set. Given a graph, have a variable for each vertex, and for each edge, add the four clauses
$$ x \lor y, x \lor \lnot y, \lnot x \lor y, \lnot x \lor \lnot y, $$
where $x,y$ are the variables corresponding to the vertices connected by the edge. I'll let you complete the proof.