5.1. Lecture -05

Hello everyone, welcome to this lecture on resolution just a quick recap. In the last lecture we discussed about valid arguments, argument form when exactly we say an argument form to be valid and so on and we also saw various rules of inferences. The plan for this lecture is as follows in this lecture we will discuss about resolution which is an important influence rule and based on resolution we will see a proof strategy which is called as proof by resolution refutation.

In this introduction, the lecturer provides a brief overview of the previous lecture and outlines the current lecture's objectives. The previous discussion focused on valid arguments and rules of inference, fundamental concepts in logic and reasoning. The current lecture aims to explain the resolution rule in logical reasoning, which will be further applied to a proof strategy called proof by resolution refutation. Understanding these concepts is crucial for students as they build their foundation in discrete mathematics and logic, particularly in relation to validity in arguments.

Imagine you are solving a puzzle, where each piece must fit together logically. In the last lecture, you learnt the edges of the puzzle - how to create a valid picture using different pieces. This lecture is like learning how to fit in the middle pieces by using specific rules (the resolution rule) to stabilize your puzzle as you work through it. Each step you take is based on the previous pieces you've connected.

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. So recall I said that PROLOG is an important programming language, which is used in AI applications. So what exactly is this resolution rule? So it says the following, imagine you are given two clauses. So C is the clause and C is another clause. The important property here is that I have a literal L which is present in positive form in C1 and negative form in C2. So you can imagine that C1 is a huge clause consisting of one or more literals, one of the literals is L. So just to recall a literal is propositional variable or the constants True or False.

The resolution rule is a key concept in logical reasoning, commonly used in programming languages like PROLOG, particularly valuable in artificial intelligence applications. The rule hinges on the idea that you have two clauses, C1 and C2, which contain a literal L. For the resolution to work, L must appear positively in one clause (C1) and negatively in the other clause (C2). This relationship allows us to 'cancel out' the literal, leading us to a conclusion formed by the remaining parts of the clauses, denoted as C' and C''. This process simplifies our reasoning about the statements.

Think of two friends debating. One insists on a positive statement (like 'It is sunny today') while the other argues the opposite (like 'It is not rainy today'). If they agree on the weather not being rainy and one of them claims it’s sunny, you can conclude that some clarity about the weather—perhaps a good day out—is coming together from their debate. The resolution rule acts like your understanding as they clarify their points.

The resolution rule says that if it is given that the clause C1 and C2 are true, then based on the truth of these 2 clauses we can conclude, conclusion C' ˅ C''. So in some sense you can imagine that resolution rule is something equivalent to cancellation rule. That means you can cancel out the literal L if it is available in positive form in C1 and negative form in C2 and whatever is left in C1 and C2 you take the disjunction of that will be the conclusion of C.

The resolution rule can be viewed as a cancellation method within logical statements. If both clauses (C1 and C2) are deemed true, it allows us to assert a new conclusion: the disjunction of the remaining parts of both clauses after L (the literal) has been removed. This simplification is critical in logical deductions and proofs, providing clarity and efficiency in resolving logical arguments.

Imagine two friends who both agree on certain facts but disagree on one detail. If you eliminate the disagreement (like comparing whether it is sunny or not) by accepting the truth of what both agree on, you can simplify the situation down to what ultimately remains true about the day - allowing you to decide what to do next based on the most straightforward information.

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''. I stress that to apply the resolution rule you need C1 and C2 to be clauses. That means C1 and C2 have to be compound propositions which are available in the form of clause. It should not be available in a different form.

In formal argumentation, the resolution rule is articulated as follows: given two clauses, where C1 is of the form C' disjunction L and C2 as C' disjunction ¬L, the conclusion is derived through the resolution pathway as C' disjunction C''. This highlights the necessity that both C1 and C2 must exist in a structured clause form to effectively apply the rule. Proper application leads us to valid conclusions within the framework of logical deduction.

Consider a legal case; the prosecutor and defense might take positions based on certain evidence (the clauses). For a judgment to be reached, all evidence (the clauses) presented must follow a specific format (like logical statements) to ensure a fair verdict can be concluded from the facts (the resolution rule's application). This legal analogy helps illustrate the structure and importance of proper argument forms.

And since we are saying that resolution as a valid inference rule we will prove that, assume for the moment resolution is a valid inference rule, it means that we can say that the conjunction of clauses C1 and C2 where C1 and C2 have the common literal L available in positive as well as in negative form in C1 and C2 respectively implies the disjunction of C' and C'' is a tautology. It will always be a true statement we will prove that very soon.

The text discusses proving that the resolution rule acts as a valid inference rule. This assertion hinges on establishing that when C1 and C2 contain a common literal L (both in positive and negative forms), the outcome generated through the resolution—namely C' disjunction C''—is indeed a tautology. A tautology is a formula that remains true in all possible interpretations, ensuring that the rules applied yield reliable deductions.

Imagine a universal truth, like 'All humans are mortal.' No matter what situation you apply this statement to, it holds true across all contexts. Proving the tautology for resolution is akin to showing that, under all circumstances, the conclusions drawn from C1 and C2 following resolution will always lead you to correct and trustworthy outcomes—much like how mortality is a truth that stands unchallenged across different individual lives.

So now we want to prove that indeed the resolution is the, indeed resolution in principle that we are stating here is a valid argument form. So what we have to prove is we want to prove this statement that indeed this implication is a tautology. So for that we assume that a left hand side of this implication namely the conjunction of C1 and C2 is true. Why we are assuming it to be true because remember we want to show that this implication is a tautology and this implication is true for all other cases.

In the proof of resolution validity, we assume that the conjunction of the clauses C1 and C2 is true to establish that the implication culminating from the resolution rule remains a tautology. By doing so, it sets a premise from which we can explore various scenarios—particularly focusing on how this implication holds across all contexts. Essentially, it is mapping a path through logical reasoning to show that our initial assumptions about the conjunction lead consistently to true outcomes.

Think of testing a hypothesis—the process of scientific discovery often requires that you assume a hypothesis is true in order to explore its implications. Just as a scientist assumes their hypothesis to derive observations and conclusions, this exercise in proving the resolution rule moves methodically from a solid base of truth (the conjunction) to ensure their implications logically follow under scrutiny.

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