Resolution method and counterexample

In summary, the resolution method can be used to show that a proposition is not true. With a counterexample, it is possible to show that a proposition is not true even if you don't use all parts of the conclusion.
  • #1
joonteee
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Hi!
Am new to this forum, but I have looked around here for some time now, since am studying a course of logic in the context of computer science. I have a very important exam in a few days, and while I thought I got it, I got shocked when I was looking on previous graded exams to see what I could work on in the final days. And when I came to the part about resolution method and counterexamples for predicate logic on one exam I realized that either there is something I don't understand, or they actually gave me a bad exam (It's supposed to be highest grade).

The picture look kinda bad since I changed a few words on my native language manually and had to refit it for the upload. So just ask if there is something I can clarify.
The question on the exam is as follows:
"Is the following true? If it is, then show that with the resolution method, if it is not true then show that with a counterexample."
You can see the sub-questions A and B and under that the answers to each sub-question in the picture.
One thing I don't get if this is actually correct is that when doing resolution method you don't have to use all parts of the conclusion? In this case only \lnot q(a) is used and not \lnot s(a).
Other than that I get sub-question A. With B I first of all can't see why it wouldn't be possible to use only {a} to make a counterexample? And second it looks to me as q(b)=1 kind of negates the premise?

View attachment 8708

I would be extremely grateful if someone could help me understand. I really want to get this and it is also hugely important for my studies.

With kind regards

Joonteee
 

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  • #2
Hi Joonteee,It looks like you have an interesting question on your exam. First, let's address the resolution method. When using the resolution method, you don't need to use all parts of the conclusion. You can use only parts of the conclusion that are relevant to the problem. In this case, the only part of the conclusion that is relevant is \lnot q(a). As for sub-question B, it is possible to use only {a} to make a counterexample. This would be a counterexample for the premise, since it shows that for some value of a, the premise is not true. However, q(b)=1 does not negate the premise. The premise states that for all a, q(a) = 0, so q(b)=1 does not contradict the premise since it is not claiming anything about q(b).I hope this helps! Good luck on your exam!
 

1. What is the resolution method in logic?

The resolution method is a logical technique used to determine the validity of an argument by showing that it is impossible for the premises to be true while the conclusion is false. It involves creating a new statement, known as the resolvent, by combining two given statements using a logical operator such as "and" or "or". This process is repeated until a contradiction is reached, which means the argument is invalid.

2. How does the resolution method work?

The resolution method works by identifying the premises and conclusion of an argument and converting them into logical statements. These statements are then combined using the resolution rule, which states that if two statements contain complementary terms (one is the negation of the other), they can be resolved to create a new statement. This process is repeated until a contradiction is reached, or all possible combinations have been exhausted.

3. What is a counterexample in logic?

A counterexample in logic is an example that disproves a statement or argument by showing that it is not universally true. It is a specific case where the premises are true, but the conclusion is false, thus invalidating the argument. Counterexamples are important in logic as they help identify the limitations of a statement or theory and can lead to the refinement or rejection of a hypothesis.

4. How is the resolution method used to find counterexamples?

The resolution method can be used to find counterexamples by attempting to prove the negation of a statement or argument. If a counterexample exists, the resolution process will eventually reach a contradiction, proving the argument to be invalid. If no contradiction is reached, then the original statement is considered to be universally true. By using the resolution method, we can systematically search for counterexamples and disprove arguments.

5. What are the limitations of the resolution method and counterexamples?

Although the resolution method and counterexamples are powerful tools in logic, they also have their limitations. The resolution method can only be used to evaluate arguments that can be written in propositional or first-order logic. It is not applicable to arguments involving modal logic or higher-order logic. Additionally, finding counterexamples can be a time-consuming process and may not always be possible, especially for complex arguments. It also relies on the accuracy of the premises, which may not always be true in real-world scenarios.

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