  Classical-realistic analysis of entangled systems have lead to retrodiction
paradoxes, which ordinarily have been dismissed on the grounds of
counter-factuality. Instead, we claim that such paradoxes point to a deeper
logical structure inherent to quantum mechanics, which is naturally described
in the language of weak values, and which is accessible experimentally via weak
measurements. Using as an illustration, a gedanken-experiment due to
Hardy\cite{hardy}, we show that there is in fact an exact numerical coincidence
between a) a pair of classically contradictory assertions about the locations
of an electron and a positron, and b) the results of weak measurements of their
location. The internal consistency of these results is due to the novel way by
which quantum mechanics "resolves" the paradox: first, by allowing for two
distinguishable manifestations of how the electron and positron can be at the
same location: either as single particles or as a pair; and secondly, by
allowing these properties to take either sign. In particular, we discuss the
experimental meaning of a {\em negative} number of electron-positron pairs.
