  We formalize various counting principles and compare their strengths over
$V^{0}$. In particular, we conjecture the following mutual independence
between: (1) a uniform version of modular counting principles and the
pigeonhole principle for injections, (2) a version of the oddtown theorem and
modular counting principles of modulus $p$, where $p$ is any natural number
which is not a power of $2$, (3) and a version of Fisher's inequality and
modular counting principles.
  Then, we give sufficient conditions to prove them. We give a variation of the
notion of $PHP$-tree and $k$-evaluation to show that any Frege proof of the
pigeonhole principle for injections admitting the uniform counting principle as
an axiom scheme cannot have $o(n)$-evaluations. As for the remaining two, we
utilize well-known notions of $p$-tree and $k$-evaluation and reduce the
problems to the existence of certain families of polynomials witnessing
violations of the corresponding combinatorial principles with low-degree
Nullstellensatz proofs from the violation of the modular counting principle in
concern.
