Goerdt (1991) considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form $\sum a_ix_i+\sum b_i(1-x_i)\ge c,\ a_i,b_i\ge0$ is its constant term $c$.) He proved a superpolynomial lower bound on the proof length of Tseitin tautologies when the degree of falsity is bounded by $\frac{n}{\log^2n+1}$ ($n$ is the number of variables).
In this short note we show that if the degree of falsity of a length $l$ proof is bounded by $b(n)=o(n)$, this proof can be easily transformed into a resolution proof of length $O(l\cdot{n\choose{b(n)-1}}64^{b(n)})$. Therefore, a superpolynomial bound on the proof length of Tseitin tautologies in this system for $b(n)=o(\frac{n}{\log n})$ follows immediately from Urquhart's (1987) lower bound for resolution proofs.
We show that the main theorem actually yields an exponential
(rather than superpolynomial) lower bound for Cutting Plane proofs
with degree of falsity bounded by a linear function of a number
of variables.
