Use the 17 rules of inference to prove the following arguments valid?
1)(Z>X).(X>~Z)
2)Z v X
3)~X/A.B
To prove the given arguments valid using the 17 rules of inference, we will proceed step by step:
1) (Z > X) . (X > ~Z)
2) Z v X
3) ~X / A . B
Let's analyze each statement and determine how to prove the argument valid.
To make it easier to follow, we'll use shorthand notation for the rules of inference:
- Premise: P
- Assumption: {A}
- Conjunction Introduction: A . B
- Conjunction Elimination: A or B
- Disjunction Introduction: A v B
- Disjunction Elimination: {A, {A} ⊢ C, {B} ⊢ C} ⊢ C
- Implication Introduction: {A} ⊢ B
- Implication Elimination: {A, A > B} ⊢ B
- Modus Tollens: {A > B, ~B} ⊢ ~A
- Modus Ponens: {A, A > B} ⊢ B
- Double Negation Elimination: ~~A ⊢ A
- Transposition: {A > B} ⊢ (~B > ~A)
- Simplification: A . B ⊢ A
- Addition: A ⊢ A v B
- Conjunction Elimination: A . B ⊢ A
- Conjunction Introduction: {A} ⊢ A . B
Now, let's use these rules to prove the argument valid:
1) (Z > X) . (X > ~Z)
- Premise
2) Z v X
- Premise
3) ~X
- Premise
4) {Z > X} ⊢ Z > ~X
- Transposition on (1)
5) {~X} ⊢ ~Z
- Modus Ponens on (4) and (3)
6) {Z > ~X, ~X} ⊢ ~X
- Modus Ponens on (5)
7) {Z > ~X, ~X} ⊢ Z
- Modus Ponens on (4) and (6)
8) {Z > ~X, ~X} ⊢ Z v X
- Disjunction Introduction on (7)
9) {Z > X, X > ~Z} ⊢ Z v X
- Conjunction Elimination on (1) and (2)
10) {Z > ~X, Z v X} ⊢ A . B
- Conjunction Introduction on (8) and (9)
11) {Z > ~X, Z v X, ~X} ⊢ B
- Simplification on (10)
12) {Z > ~X, Z v X} ⊢ A
- Simplification on (10)
13) {Z > ~X, Z v X, ~X} ⊢ A . B
- Conjunction Introduction on (12) and (11)
14) {Z > ~X, Z v X, ~X} ⊢ {~X ⊢ A . B}
- Assumption on (13)
15) {Z > ~X, Z v X} ⊢ ~X ⊢ A . B
- Conjunction Introduction on (14)
16) {Z > ~X, Z v X} ⊢ A . B
- Disjunction Elimination on (3), (15), and (10)
17) Z v X ⊢ ~X ⊢ A . B
- Modus Ponens on (2) and (16)
Based on these steps, we have successfully proven the validity of the argument using the 17 rules of inference.