Exposure to antibiotics induces the expression of mutagenic bacterial stress–response pathways, but the evolutionary benefits of these responses remain unclear. One possibility is that stress–response pathways provide a short-term advantage by protecting bacteria against the toxic effects of antibiotics. Second, it is possible that stress-induced mutagenesis provides a long-term advantage by accelerating the evolution of resistance. Here, we directly measure the contribution of the Pseudomonas aeruginosa SOS pathway to bacterial fitness and evolvability in the presence of sublethal doses of ciprofloxacin. Using short-term competition experiments, we demonstrate that the SOS pathway increases competitive fitness in the presence of ciprofloxacin. Continued exposure to ciprofloxacin results in the rapid evolution of increased fitness and antibiotic resistance, but we find no evidence that SOS-induced mutagenesis accelerates the rate of adaptation to ciprofloxacin during a 200 generation selection experiment. Intriguingly, we find that the expression of the SOS pathway decreases during adaptation to ciprofloxacin, and this helps to explain why this pathway does not increase long-term evolvability. Furthermore, we argue that the SOS pathway fails to accelerate adaptation to ciprofloxacin because the modest increase in the mutation rate associated with SOS mutagenesis is offset by a decrease in the effective strength of selection for increased resistance at a population level. Our findings suggest that the primary evolutionary benefit of the SOS response is to increase bacterial competitive ability, and that stress-induced mutagenesis is an unwanted side effect, and not a selected attribute, of this pathway.