Strange weak values

We develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter-factual processes. We present the condition for the choice of the post-selected state to give a negative weak value of a given projection operator and strange values of an observable in general. The general framework is applied to Hardy's paradox and the spin-1/2 system to explicitly address the issues of counter-factuality and strange weak values. The counter-factual arguments which characterize the paradox specify the pre-selected state, and a complete set of the post-selected states clarifies how the strange weak values emerge.