I had not heard the term abductive reasoning before, but recognised it as my own the moment it was explained. It is inferring the most likely answer or solution, rather than relying on deductive, mathematical-style proofs for everything. Some things do not have clear answers, or more exactly, whatever absolute answer they might have is not accessible to us. God either exists or he doesn't, but we are unable to prove this either way. People make their choice based on observing reality, thinking about it, and deciding which is the most likely.
Much interpretation of Scripture is like this, and it is an approach CS Lewis uses often. When a formal proof is available, it should win the day, and sometimes one should be attempted just to see how far one can go. We don't know what Jesus meant by the Beatitudes, but the words have meanings and taken as a batch they seem to point in a direction. Also, we can see some things that we can solidly conclude he did not mean. We at a much more distant point in both the first and last books of the Bible. People who try to reduce either book to proofs are the most likely to lead us astray.
In the wiki entry diagnosis is mentioned, another good instance of a problem that might have a right answer, but we do not have perfect knowledge about the body nor of all possible conditions. Clinicians therefore look at the data and choose what they believe is the most likely explanation for the data.