In the search for life on Earth-like planets around other stars, the first (and likely only) information will come from the spectroscopic characterization of the planet's atmosphere. Of the countless number of chemical species terrestrial life produces, only a few have the distinct spectral features and the necessary atmospheric abundance to be detectable. The easiest of these species to observe in Earth's atmosphere is O$_{2}$ (and its photochemical byproduct, O$_{3}$). But O$_{2}$ can also be produced abiotically by photolysis of CO$_{2}$, followed by recombination of O atoms with each other. CO is produced in stoichiometric proportions. Whether O$_{2}$ and CO can accumulate to appreciable concentrations depends on the ratio of far-UV to near-UV radiation coming from the planet's parent star and on what happens to these gases when they dissolve in a planet's oceans. Using a one-dimensional photochemical model, we demonstrate that O$_{2}$ derived from CO$_{2}$ photolysis should not accumulate to measurable concentrations on planets around F- and G-type stars. K-star, and especially M-star planets, however, may build up O$_{2}$ because of the low near-UV flux from their parent stars, in agreement with some previous studies. On such planets, a 'false positive' for life is possible if recombination of dissolved CO and O$_{2}$ in the oceans is slow and if other O$_{2}$ sinks (e.g., reduced volcanic gases or dissolved ferrous iron) are small. O$_{3}$, on the other hand, could be detectable at UV wavelengths ($\lambda$ < 300 nm) for a much broader range of boundary conditions and stellar types.