Answer the following question using truth tables Does implic
Answer the following question using truth tables.
Does implication distribute over disjunction from the right, left, both, or neither?
This is my modern class. Don\'t understand what\'s it asking. I think it\'s to do with propositional arguments.
Solution
Implication is Left Distributive over Disjunction
.
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | p(qr) | Assumption | (None) | ||
| 2 | 2 | p | Assumption | (None) | ||
| 3 | 1, 2 | qr | Modus Ponendo Ponens: E | 1, 2 | ||
| 4 | 2 | p | Law of Identity | 2 | ||
| 5 | 5 | q | Assumption | (None) | ||
| 6 | 5 | pq | Rule of Implication: I | 4 – 5 | Assumption 4 has been discharged | |
| 7 | 5 | (pq)(pr) | Rule of Addition: I1 | 6 | ||
| 8 | 8 | r | Assumption | (None) | ||
| 9 | 8 | pr | Sequent Introduction | 8 | True Statement is implied by Every Statement | |
| 10 | 8 | (pq)(pr) | Rule of Addition: I2 | 9 | ||
| 11 | 1 | (pq)(pr) | Proof by Cases: PBC | 3, 2 – 7, 8 – 10 | Assumptions 2 and 8 have been discharged |
