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Abductive reasoning is ubiquitous in artificial intelligence and everyday thinking. However, formal theories that provide probabilistic guarantees for abductive inference are lacking. We present a quantitative formalization of abductive logic that combines Bayesian probability with the interpretation of abduction as a search process within the Algorithmic Search Framework (ASF). By incorporating uncertainty in background knowledge, we establish two novel sets of probabilistic bounds on the success of abduction when (1) selecting the single most likely cause while assuming noiseless observations, and (2) selecting any cause above some probability threshold while accounting for noisy observations. To our knowledge, no existing abductive or general inference bounds account for noisy observations. Furthermore, while most existing abductive frameworks assume exact underlying prior and likelihood distributions, we assume only percentile-based confidence intervals for such values. These milder assumptions result in greater flexibility and applicability of our framework. We also explore additional information-theoretic results from the ASF and provide mathematical justifications for everyday abductive intuitions.