Abductive reasoning is a process whereby one reasons to the truth of an explanation from its ability to account for what is observed. It is therefore sometimes also known as inference to the best explanation.
: All beans in that bag are white. : These beans are from that bag. : Therefore, these beans are white.
: These beans are from that bag. : These beans are white. : Therefore, all beans in that bag are white.
: These beans are white. : All beans in that bag are white. : Therefore, these beans are from that bag.
It is not completely clear what Peirce meant by abduction. Clearly the inference cannot be to just any explanation, e.g., there might have been many other bags full of white beans. But before we decide what constitutes the best explanation, we had been inquire into the nature of explanation itself.
There is an extensive literature about explanation in the Philosophy of Science, for example, (FourDecades). Obviously it is not just a matter of devising a statement which has the thing to be explained (explicandum) as a consequence. You want it to ‘give the reason for’ the observation. A thorough account of what constitutes the ‘reason’ for something is notoriously difficult to formulate. For some, it’s a matter of subsuming the observation under a general covering law, while for others, it’s a matter of giving a causal or mechanistic story with the observation as the outcome. Note also that there is a growing literature now on the concept of ‘explanatory proofs’ in mathematics, it being felt that one may have proved a mathematical fact without understanding ‘why’ it is true.
For some, abduction for Peirce is the process of coming up with a good explanation. Otherwise, if it is merely a case of assessing a range of existing rival hypotheses as explanations, it may be possible to employ Bayesian reasoning, generally taken to be a form of inductive reasoning.
Wesley Salmon, Four Decades of Scientific Explanation.