I am being asked if I can prove P, L, Q? (without using resolution-refutation).

Propositional clauses:
1. P -> V
2. !P -> (!V and F)
3. (V or F) -> Q
4. Q -> L

From here I was able, with resolution-refutation, to determine that we cannot prove P but we should be able to prove Q and L… After that I am completely stuck on how to proceed as I am not allowed to prove the question with that approach – only to help me see what answers I should get…

I assume I must either use Forward-Chaining or Backward-Chaining to solve the problems – but no knowledge is given, only implications – so how is one supposed to use either? In all my readings usually we would be given something like F=True (knowledge) or something similar and the chaining would come down to that – but with only implications I can’t see how anything can be proven…

All my attempts (and there have been many) have only added to my confusion.
Any help/hints would be greatly appreciated.
Thanks,

It seems that you are facing challenges in proving P, L, and Q using forward-chaining or backward-chaining, as you have only been provided with implication statements. I will try to explain how you can approach this problem using both forward-chaining and backward-chaining.

Forward-chaining is a method of reasoning from known facts towards a goal. In this case, you start with the given information and apply the implications to deduce new facts until you can reach a conclusion.

Backward-chaining, on the other hand, starts with the goal and works backward to find the facts that support that goal. You use the implications in reverse order to establish whether the statements you want to prove are logically supported.

Let's try both methods to see how they can be applied to your problem.

Forward-chaining:
1. Starting with the given clauses, you can first apply clause 1: P -> V. If we assume P to be true, then it implies V.
2. Clause 3 states that (V or F) -> Q. Since we have V from the previous step, we can conclude that Q is true.
3. Finally, clause 4 states that Q -> L. We know Q is true from the previous step, so L must also be true.

Therefore, using forward-chaining, you can prove that P is false, but you can prove L and Q.

Backward-chaining:
1. We start with the goal of proving P, L, and Q.
2. Considering clause 4: Q -> L, we see that if L is indeed true, then Q must also be true.
3. Now, we move to clause 3: (V or F) -> Q. By a similar argument, if Q is true, then (V or F) must be true.
4. Finally, let's focus on clause 1: P -> V. In order for (V or F) to be true, either V or F must be true. If we assume V is true, then P must also be true, based on the implication in clause 1.

From the backward-chaining analysis, we can conclude that P, L, and Q cannot all be proven true simultaneously. Proving P true leads to P and V both being true, while proving L true requires Q to be true. These two conclusions are not consistent, which means there is a contradiction.

In summary, using both forward-chaining and backward-chaining, we can deduce that P is false, and L and Q can both be proven true. However, it is not possible to prove all three statements simultaneously.

To prove statements without using resolution-refutation, you can use either backward-chaining or forward-chaining. Here's a step-by-step guide on how to proceed using backward-chaining:

1. Declare the goal you want to prove, which in this case is P, L, and Q.

2. Start with the given clauses and apply backward-chaining to determine if you can derive the goal.

3. Begin by considering the last implication, Q -> L. Since this is an implication, you can rewrite it as !Q or L.

4. Next, consider the third implication, (V or F) -> Q. Again, rewrite it as !(V or F) or Q. This implies (!V and !F) or Q.

5. From clause 4, we now have (!V and !F) or Q.

6. In clause 2, you have !P -> (!V and F). Using the negation of this clause, you can rewrite it as P or (!V and F).

7. Now, combining clauses 5 and 6, you have (P or (!V and F)) and ((!V and !F) or Q).

8. Simplify the above expression using distributive property. You can do this by noting that (P or (!V and F)) and ((!V and !F) or Q) is equivalent to (P or (!V and F) or ((!V and !F) or Q)).

9. Observe that the expression above has the form (A or B) or C. If you can prove any of A, B, or C, you have proven the entire statement.

10. In this case, you can prove (!V and F) from step 8, which satisfies B.

11. To prove (!V and F), observe that from clause 2, you have !P -> (!V and F). If you assume P, you can derive a contradiction, which means !P is true. Therefore, you can conclude that (!V and F) is true.

12. Since you have proven B (step 10), you can conclude that (P or (!V and F) or ((!V and !F) or Q)) is true.

13. Finally, you can conclude that P, (P or (!V and F) or ((!V and !F) or Q)), and (!V and F) are true.

In summary, using backward-chaining, you have proven that P, (!V and F), and Q are true. Therefore, you can assert that P, L, and Q are provable from the given clauses.