The empirical evidence tends to confirm the quantum - mechanically (QM) predicted violation of Bell's - type inequalities. The latter are widely accepted to be correct representation of the objective reality (OR) underlying EPR's gedankenexperiment. Contrariwise, we argue here that the starting premises leading to Bell's and CHSH's inequalities are not appropriate formulations of OR in the EPR context. Indeed, the essence of EPR argument is the correlation between pair of particles travelling with some well defined orientations. To measure such correlation, one requires detectors with high resolution, and some appropriately defined relative orientations. On the contrary, experimental tests of Bell's inequalities typically involve counting the number of particles with some value of spin (or polarisation) using low-resolution detectors and completely arbitrary orientations. Depending upon the relative orientation of good-resolution detectors, we obtain two families of CHSH-type inequalities. One of these families is the most often used version of CHSH inequality, that was derived by Clauser and Horne with the supplementary no - enhancement assumption: here, it is obtained without special provisions. The other family is completely new and refers to coincidence counts from multiple events. Experimental test to distinguish between the QM and OR descriptions of nature should take into account these new constraints.