  Bell's theorem has long been considered to establish local realism as the
fundamental principle that contradicts quantum mechanics. It is therefore
surprising that the quantum pigeonhole effect points to the pigeonhole
principle as yet another source of contradiction [Aharonov et al., Proc. Natl.
Acad. Sci. USA, 113, 532-535 (2016)]. Here we construct two new forms of Bell's
inequality with the pigeonhole principle, then reconstruct Aharonov et al.'s
weak measurement on a bipartite system. We show that in both cases it is
counterfactual reasoning by the assumption of realism rather than the
pigeonhole principle being violated by quantum mechanics. We further show that
the quantum pigeonhole effect is in fact a new version of Bell's theorem
without inequality. With the pigeonhole principle as the same conduit, a
comparison between two versions of Bell's theorem becomes straightforward as it
only relies on the orthonormality of the Bell states.
