The degree of falsity of an inequality in Boolean variables shows how many variables are enough to substitute in order to satisfy the inequality. Goerdt introduced a weakened version of the Cutting Plane (CP) proof system with a restriction on the degree of falsity of intermediate inequalities . He proved an exponential lower bound for CP proofs with degree of falsity bounded by , where n is the number of variables.
In this paper we strengthen this result by establishing a direct connection between CP and Resolution proofs. This result implies an exponential lower bound on the proof length of Tseitin-Urquhart tautologies when the degree of falsity is bounded by for some constant c. We also generalize the notion of degree of falsity for extensions of Lovász-Schrijver calculi (LS), namely for proof systems introduced by Grigoriev et al. . We show that any proof with bounded degree of falsity can be transformed into a proof. We also prove lower and upper bounds on the proof length of tautologies in with bounded degree of falsity.