Resolution Method:
For a minor miracle occurred in 1965 where Alan Robinson published his resolution method as uses a method to generalised version of the resolution rule of inference we saw in the previous lecture. So there it has been mathematically proven to be refutation-complete over first order logic. Because if you write any set of sentences in first order logic that are unsatisfiable so that means taken together they are false however in that they have no models so the resolution method will eventually derive the False symbol as indicating that the sentences somehow contradict each other.
Now we see in particularly if the set of first order sentences comprises a set of axioms and the negation of a theorem you want to prove then the resolution method can be used in a proof-by-contradiction approach. Because if your first order theorem is true such as proof by contradiction using the resolution method is guaranteed to find the proof to a theorem eventually.