The search for life on planets outside our solar system will use spectroscopic identification of atmospheric biosignatures. The most robust remotely detectable potential biosignature is considered to be the detection of oxygen (O₂) or ozone (O₃) simultaneous to methane (CH₄) at levels indicating fluxes from the planetary surface in excess of those that could be produced abiotically. Here we use an altitude-dependent photochemical model with the enhanced lower boundary conditions necessary to carefully explore abiotic O₂ and O₃ production on lifeless planets with a wide variety of volcanic gas fluxes and stellar energy distributions. On some of these worlds, we predict limited O2 and O₃ buildup, caused by fast chemical production of these gases. This results in detectable abiotic O₃ and CH₄ features in the UV-visible, but no detectable abiotic O₂ features. Thus, simultaneous detection of O₃ and CH₄ by a UV-visible mission is not a strong biosignature without proper contextual information. Discrimination between biological and abiotic sources of O₂ and O₃ is possible through analysis of the stellar and atmospheric context—particularly redox state and O atom inventory—of the planet in question. Specifically, understanding the spectral characteristics of the star and obtaining a broad wavelength range for planetary spectra should allow more robust identification of false positives for life. This highlights the importance of wide spectral coverage for future exoplanet characterization missions. Specifically, discrimination between true and false positives may require spectral observations that extend into infrared wavelengths and provide contextual information on the planet's atmospheric chemistry.