
Introduction to Logic

Tools for Thought

Reasoning with the Resolution Principle is analogous to reasoning with the Propositional Resolution Principle. We start with premises; we apply the Resolution Principle to those premises; we apply the rule to the results of those applications; and so forth until we get to our desired conclusion or until we run out of things to do.
As with Propositional Resolution, we define a resolution derivation of a conclusion from a set of premises to be a finite sequence of clauses terminating in the conclusion in which each clause is either a premise or the result of applying the Resolution Principle to earlier members of the sequence. And, as with Propositional Resolution, we do not use the word proof, because we reserve that word for a slightly different concept, which is discussed in the next section.

As an example, consider a problem in the area kinship relations. Suppose we know that Art is the parent of Bob and Bud; suppose that Bob is the parent of Cal; and suppose that Bud is the parent of Coe. Suppose we also know that grandparents are parents of parents. Starting with these premises, we can use resolution to conclude that Art is the grandparent of Coe.

The derivation is shown below. We start with our five premises  four simple clauses for the four facts about the parent relation p and one more complex clause capturing the definition of the grandparent relation g. We start by resolving the clause on line 1 with the clause on line 5 to produce the clause on line 6. We then resolve the clause on line 3 with this result to derive that conclusion that Art is the grandparent of Cal. Interesting but not what we set out to prove; so we continue the process. We next resolve the clause on line 2 with the clause on line 5 to produce the clause on line 8. Then we resolve the clause on line 5 with this result to produce the clause on line 9, which is exactly what we set out to prove.
1.  {p(art, bob)}  Premise 
2.  {p(art, bud)}  Premise 
3.  {p(bob, cal)}  Premise 
4.  {p(bud, coe)}  Premise 
5.  {¬p(x, y), ¬p(y, z), g(x, z)}  Premise 
6.  {¬p(bob, z), g(art, z)}  1, 5 
7.  {g(art, cal)}  3, 6 
8.  {¬p(bud, z), g(art, z)}  2, 5 
9.  {g(art, coe)}  4, 8 

One thing to notice about this derivation is that there are some deadends. We first tried resolving the fact about Art and Bob before getting around to trying the fact about Art and Bud. Resolution does not eliminate all search. However, at no time did we ever have to make an arbitrary assumption or an arbitrary choice of a binding for a variable. The absence of such arbitrary choices is why Resolution is so much more focussed than natural deduction systems like Fitch.

Another worthwhile observation about Resolution is that, unlike Fitch, Resolution frequently terminates even when there is no derivation of the desired result. Suppose, for example, we were interested in deriving the clause {g(cal,art)} from the premises in this case. This sentence, of course, does not follow from the premises. And resolution is sound, so we would never generate this result. The interesting thing is that, in this case, the attempt to derive this result would eventually terminate. With the premises given, there are a few more resolutions we could do, e.g. resolving the clause on line 1 with the second literal in the clause on line 5. However, having done these additional resolutions, we would find ourselves with nothing left to do; and, unlike Fitch, the process would terminate.

Unfortunately, like Propositional Resolution, Resolution is not generatively complete, i.e. it is not possible to find resolution derivations for all clauses that are logically entailed by a set of premise clauses. For example, the clause {p(a), ¬p(a)} is always true, and so it is logically entailed by any set of premises, including the empty set of premises. Resolution requires some premises to have any effect. Given an empty set of premises, we would not be able to derive any conclusions, including this valid clause.
Although Resolution is not generatively complete, problems like this one are solved by negating the goal and demonstrating that the resulting set of sentences is unsatisfiable.

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