The resolution performance of a two-dimensional detection system for PET, consisting of discrete scintillators read out by avalanche photodiodes, has been evaluated. The resolving capability of the system, as described by the FWHM, FWTM and spectral resolutions was determined throughout the camera field. For a 350 keV energy threshold, the in-plane resolution was 2.0 mm FWHM, 3.8 mm FWTM, and 3.6 mm spectral at the center, degrading to 5 mm FWHM, 8 mm FWTM, and 7.5 mm spectral at a relative radial distance of 0.5 from the center. Axially, the resolution was less than 3.5 mm FWHM, 5.7 mm FWTM throughout the useful field. The analysis of the various components of the response functions by theoretical and Monte-Carlo simulations has shown that the resolving power is limited by the crystal size everywhere except in the central region of the field. This clearly indicates that measurement of the depth of interaction within the detectors will be necessary in order to achieve a very high and uniform resolution throughout a large fraction of the field. In the central region, intercrystal scatter is the limiting factor, but the proposed detection system allows for an efficient discrimination against multiple interaction events by some shielding against forward scatter and by independent energy thresholding on every single detector.