  The structural connections between the Kochen-Specker (KS) theorem, pre- and
post-selection (PPS) paradoxes, and anomalous weak values are explored in
detail. All PPS paradoxes, such as the 3-box paradox, the Quantum Cheshire Cat,
and the Quantum Pigeonhole principle, construct a particular type of
ontological model that assigns an eigenvalue to each observable (independent of
context) of a system such that these assignments are consistent with the PPS.
It is shown that such an ontological model must be explicitly contextual in the
sense of the KS theorem, or otherwise implies either a restriction on free
random choice or explicitly retrocausal behavior. We call such models
PPS-contextual. The structure of each paradox is always such that there are
particular contexts of mutually commuting observables that violate the product
rule or sum rule, when the ontological model is extended to include observables
that are not measured during the experiment. These paradoxes are
counterfactual, in the sense that they are not directly observed, and also
because the product and sum rules are always obeyed by projective measurements
in actual experiments. It is shown that by adopting an alternate ontological
model, where all hidden variables are weak values (which are not always
eigenvalues, but obey the sum rule by definition), the same contexts that
presented the original paradox must also contain observables with anomalous
weak values. These anomalous weak values are not counterfactual because they
can be probed through weak measurements on an ensemble of identically pre- and
post-selected states, allowing this localized signature of KS contextuality to
be experimentally observed. The weak values of all observables of a system can
in principle be measured during an experiment, making this model a promising
candidate for describing PPS-contextual ontological `elements of reality.'
