prompter: Can you present the derivation of the following logical statement?
((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), (Q ∨ ¬S) ⊢ (R → (P ∨ W))

assistant: To derive the statement (R → (P ∨ W)) from the premises ((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), and (Q ∨ ¬S), we can use a proof by contradiction. We will assume the opposite of what we are trying to prove (i.e., assume ¬(R → (P ∨ W))), and derive a contradiction.

Here's the derivation:

1. ¬(R → (P ∨ W)) (Assumption)
2. ¬(¬R ∨ (P ∨ W)) (Implication rule, equivalent to R ∧ ¬(P ∨ W))
3. R (Disjunctive syllogism, from 2 and ¬(P ∨ W))
4. ¬(P ∨ W) (From 2 and R)
5. ¬P ∧ ¬W (De Morgan's law, from 4)
6. Q ∨ ¬S (Premise)
7. (Q ∧ R) → P (Premise)
8. ¬(R → S) → (¬W → Q) (Premise)
9. ¬(R → S) (Assumption for a proof by contradiction)
10. ¬¬W → Q (From 8 and 9, by contraposition)
11. W ∨ Q (From 6 and 10, by disjunctive syllogism)
12. P ∨ W (From 7, 3, and modus ponens)
13. P (From 5 and 12, by disjunctive syllogism)
14. P ∨ W (From 13, by addition)
15. ¬P ∧ ¬W (From 5)
16. Contradiction: P ∧ ¬P (From 14 and 15)
17. Therefore, (R → (P ∨ W)) is true by contradiction (from 1).

Thus, we have shown that if ((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), and (Q ∨ ¬S) are true, then (R → (P ∨ W)) must also be true.

prompter: b) Use the resolution method of propositional logic to show that
the following set of clauses is unsatisfiable:
{{C, ¬C, A}, {C, ¬A}, {¬B, ¬A}, {¬B, C}, {¬C}, {A, B}}

assistant: To show that the set of clauses {{C, ¬C, A}, {C, ¬A}, {¬B, ¬A}, {¬B, C}, {¬C}, {A, B}} is unsatisfiable, we can use the resolution method.

First, we write out all possible pairs of clauses:

From {C, ¬C, A} and {C, ¬A}: We can resolve on C to obtain {¬A, A}, which can be simplified to the empty clause {}.
From {¬B, ¬A} and {¬B, C}: We can resolve on ¬B to obtain {¬A, C}, which can be simplified to {¬A, C}.
From {¬C} and {¬A, C}: We can resolve on C to obtain {¬A}, which can be simplified to {¬A}.
From {A, B} and {¬A}: We can resolve on A to obtain {B}, which cannot be further simplified.
At this point, we have obtained the empty clause {}, which indicates that the set of clauses is unsatisfiable. Therefore, there is no truth assignment that satisfies all the clauses, and the set of clauses is contradictory.

assistant: One possible derivation is:

1. (Q ∧ R) → P        Premise
2. ¬(R → S) → (¬W → Q) Premise
3. Q ∨ ¬S              Premise
4. | R                 Assumption
5. | | ¬W              Assumption
6. | | ¬(R → S)        Modus Tollens 4, 2
7. | | R ∧ ¬S          Conjunction Introduction 4, 6
8. | | Q               Disjunction Elimination 3, 7
9. | Q                 Negation Elimination 5-8
10.| Q ∧ R             Conjunction Introduction 4, 9
11.| P                 Modus Ponens 1, 10
12.| P ∨ W             Disjunction Introduction 11
13.R → (P ∨ W)         Conditional Proof 4-12

prompter: What do each of the operators mean?

assistant: - ((Q ∧ R) → P)  Premise
- (¬(R → S) → (¬W → Q))  Premise
- Q ∨ ¬S  Premise
- ¬R ∨ S  Material implication of premise 2
- ¬Q → ¬(¬W → Q)  Contrapositive of premise 2
- ¬Q → (W ∧ ¬Q)  De Morgan's law and double negation on premise 5
- W ∧ ¬Q  Modus ponens from premises 3 and 6
- W  Simplification from premise 7
- ¬Q  Simplification from premise 7
- R ∨ ¬R   Law of excluded middle
    - Case1: R 
        - Q ∧ R   Conjunction of premises 
        - Q   Simplification from subproof a 
        - P   Modus ponens from premises and subproof b 
        - P ∨ W   Addition with premise 8 
        - R → (P ∨ W)   Conditional introduction with subproofs a-d 
    - Case2: ¬R  
        - S   Disjunctive syllogism of premise4 and case2 
        - Contradiction: (Q ∧ R) contradicts with (¬S ∧ Q) in the case where S is true and both premises hold.
        
Therefore
(R→(P∨W))

prompter: Great, thank you!

assistant: I'm glad I was able to help, if there's anything else you want to ask, I'm here.

assistant: You're welcome! If you have any more questions, feel free to ask.

assistant: You're welcome! If you have any more questions or topics you'd like me to assist you with, feel free to ask!

prompter: That's good to hear, you're so kind. That's enough for now. See you later 😉