5.5. Proof of Validity of Resolution

So to begin with, let us try to understand what exactly is the resolution rule. It is a very important inference rule and it is used extensively in this programming language called PROLOG.

The resolution rule is a fundamental inference rule in propositional logic. It is integral to various programming languages, especially in artificial intelligence contexts like PROLOG. The rule provides a method to infer conclusions from given premises, much by resolving contradictions. This means that if we have a statement and its negation, we can simplify our logical statements by using their joint truth to reach further conclusions.

Imagine two people discussing whether it will rain. One states, 'If it is cloudy, it will rain,' and another counters, 'If it is not raining, it is not cloudy.' Both can reach a conclusion about the weather by resolving their statements, much like how the resolution rule helps simplify and conclude logical arguments.

The important property here is that I have a literal L which is present in positive form in C₁ and negative form in C₂.

A clause is essentially a disjunction of literals, and a literal can be a variable or a boolean constant (True/False). The resolution rule tells us that if we have two clauses, C₁ containing a literal in its positive form (L) and C₂ containing the same literal in its negative form (¬L), we can resolve these clauses to simplify our logical expressions. The remaining parts of the clauses after canceling out L and ¬L become the conclusion.

Think of canceling a subscription service. If one person states, 'I will keep the subscription if I am happy,' and another says, 'I will not keep the subscription if I am unhappy,' their joint statements can lead to a resolution: if they determine their happiness, they can either choose to continue or cancel together, based on that conclusion.

So in argument form the resolution rule can be stated as follows. It says that if you are given the clauses C₁ and C₂ where, C₁ is C' ∨ L and C₂ is C' ∨ ¬L. Then based on these two premises, you can conclude the conclusion C' ∨ C''.

The conclusion derived from applying the resolution rule shows that if both C₁ and C₂ are true, then at least one part of C' or C'' must also be true. This translates the resolution of the clauses into a sound logical conclusion, exemplifying the power of the resolution as a valid inference mechanism. The combination of C' and C'' arises from extracting the disjunctions of the remaining parts of both clauses.

Consider two friends debating whether they will go out for dinner: one states, 'We can go if I finish my work,' and the other adds, 'We can go if it does not rain.' By resolving their statements, if one is true (they finish work), it's inferred they will go to dinner, symbolically presenting how resolution aids in deriving conclusions.

And remember as per our definition of argument forms since we are saying that our resolution is a valid argument form.

To showcase the validity of the resolution rule, it is established that the conjunction of the premises implies a conclusion that is always true, known as a tautology. By assuming the premises are true and showing the conclusion follows logically without exception, we reinforce that resolution is indeed a valid inference method in logical deduction.

Think of a legal argument where premises present evidence that collectively leads to an undeniable conclusion (e.g., 'If all evidence points towards guilt, the accused must be guilty'). Here, the logical validity mirrors resolution by inferring conclusions that must logically follow from accepted premises.

The claim here is that a constant false will belong to the resolvent of set of these clauses if and only if this, the set of clauses in S is unsatisfiable.

Unsatisfiability refers to a situation where no truth assignment to the variables makes all clauses true. If the resolution process yields a constant False (or the empty clause), it indicates that there are contradictions within the set of premises, thus rendering them unsatisfiable. This is critical in validating arguments by revealing inherent contradictions in their assumptions.

Imagine a group of friends planning an outing: if one insists on leaving tomorrow while another insists on leaving next week, a conclusion of inaction might emerge (an 'impossible condition'), demonstrating unsatisfiability. This analogy relates closely to recognizing contradictions in arguments, validating the unsatisfactory nature of certain premises.

The goal is the following you are given an argument form and you have to verify whether this argument form is valid or invalid.

The strategy involves transforming premises and conclusions into clausal form and then adding the negation of the conclusion to the premises. If the resulting set of clauses is unsatisfiable (e.g., leads to a constant False), the original argument form is valid. Thus, verifying validity through resolution refutation reinforces logical consistency in argumentation.

Imagine trying to prove a scientific theory: if experiments consistently yield results contradicting the theory, researchers must refute the theory's validity. This is akin to the proof by resolution, where the aim is to expose conflicts that determine whether a premise holds or fails logically.

--