Need help supplying proo f for the following problem using the rules of inference
1. (Z.Y)> X
2. ~O>Z
3. ~Y>O
4. ~O.T /S
To prove the conclusion S using the given premises using the rules of inference, we can follow these steps:
1. Use simplification (SIMP) rule on premises 2 and 3 to get ~O.
~O>Z and ~Y>O imply ~O.
2. Use modus tollens (MT) rule on ~O and premise 4 to get ~T.
~O and ~O.T imply ~T.
3. Use disjunction syllogism (DS) rule on the derived ~T and premise 1 to get (Z.Y).
~T and (Z.Y)>X imply (Z.Y).
4. Use simplification (SIMP) rule on (Z.Y) to get Y.
(Z.Y) implies Y.
5. Use conjunction elimination (CONJ) rule on Y and (Z.Y) to get Z.
(Z.Y) implies Z.
6. Use conjunction elimination (CONJ) rule on Y and (Z.Y) to get X.
(Z.Y) implies X.
7. Use simplification (SIMP) rule on ~O to get O.
~O implies O.
8. Use modus ponens (MP) rule on O and premise 2 to get Z.
O and ~O>Z imply Z.
9. Use modus ponens (MP) rule on O and premise 3 to get O.
O and ~Y>O imply O.
10. Use disjunctive syllogism (DS) rule on Z and O to get X.
Z and (Z.Y)>X imply X.
11. Finally, use conjunction introduction (CONJ) rule on Z and Y to get S.
Z and Y imply S.
Therefore, using these steps, we have derived conclusion S from the given premises using the rules of inference.
To derive the conclusion 'S', we can use the following rules of inference:
1. Modus Ponens (MP): If we have a conditional statement 'p → q' and also have 'p', then we can infer 'q'.
2. Modus Tollens (MT): If we have a conditional statement 'p → q' and also have '~q', then we can infer '~p'.
Now let's go through the steps:
1. (Z.Y) > X (Premise)
2. ~O > Z (Premise)
3. ~Y > O (Premise)
4. ~O . T (Premise)
5. (~Z . ~Y) > X (Implication elimination from premise 1)
6. ~(Z.Y) . ~(~Z . ~Y) (Contradiction introduction from step 5)
7. ~Z . ~Y (Simplification from step 6)
8. ~Y (Simplification from step 7)
9. O (Modus Tollens using premises 3 and 8)
10. ~O (Simplification from premise 4)
11. ⊥ (Contradiction introduction from steps 9 and 10)
12. S (Ex Falso Quodlibet - Principle of Explosion, since we derive a contradiction)
Therefore, we have derived the conclusion 'S' using the given premises and the rules of inference.