Binary Resolution:
We looked at unit resolution (a propositional inference law) in the last lecture:
A∨B, ¬B /A
We may have this a bit further to propositional binary resolution:
A ∨ B, ¬ B∨ C /A∨C
Binary resolution have its name from the truth that each sentence is a disjunction of accurately two literals. We say the two disparate literals B and ¬B are resolved - they are detached when the disjunctions are combined.
The binary resolution principal may be looked to be sound because if both C and A were not true then at least one of the sentences on the peak line would be untrue. As this is an inference principal, we are pretending that the peak line is true. Hence we can't have both C and A being untrue, which means either C or A should be true. So we can infer the base line.
So far we've only looked at propositional version of resolution. In first-order logic we have to also deal with variables and quantifiers. As we'll look under, we don't
Have to worry for quantifiers: we are going to work with sentences that just contain less variables. Remind that we treat these variables as implicitly unique quantified, and that they may have any value. This allows us to state a much common first-order binary resolution inference law:
A∨B, ¬C∨D
Subset (θ, B) = Subset (&theta, C)
Subset (θ, A ∨ D)
This law has the side condition Subset (θ, B) = Subset(&theta, C), which demands there to be a substitution θ which forms B and C the similar before we may use the law. (Note θ may replace fresh variables while forming B and C similar. It doesn't have to be a ground substitution!) If we may search such a θ, then we may create the resolution step and apply θ to the outcome really, the first-order binary law is simply equal to applying the substitution to the real sentences, and then applying the propositional binary law.