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Theoretical calculations of the two-photon ionization cross section of argon and of the two-photon detachment cross section of the negative fluorine ion are presented for the energy region from threshold to the single-photon ionization or detachment threshold. Detailed analyses are presented of the effects of various kinds of electron correlations, whose contributions to the two-photon transition amplitudes are evaluated using variationally stable procedures to sum implicitly over intermediate states. These variationally stable procedures enable us to give much more reliable predictions in the region of the first few intermediate-state resonances in argon than were provided by previous calculations of one of us [A.F. Starace and T. F. Jiang, Phys. Rev. A 36, 1705 (1987)]. In addition, we show here for both argon and F- the large corrections to the p (1S) final-state channel cross sections provided by certain intermediate-state shake-up interactions, which we show to be essential for obtaining good agreement of dipole length and velocity cross sections. We provide detailed comparisons of our results with those of previous workers; in particular, our results for the negative fluorine ion are in excellent agreement with the recent absolute measurement of Kwon et al. [Phys. Rev. A 40, 676 (1989)].