SOLUTION: Setting up a prove in Fitch notation Premises: PvQ ~QvR Conclusion; PvR

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Question 1167696: Setting up a prove in Fitch notation
Premises:
PvQ
~QvR



Conclusion; PvR
Found 2 solutions by solver91311, CPhill:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!

Prove by exhaustive cases.

Assume P, then P v R follows directly

Assume ~P, then Q follows from P v Q, but if Q, then ~Q v R implies R and therefore P v R.

John

My calculator said it, I believe it, that settles it


I > Ø

Answer by CPhill(1987) About Me  (Show Source):
You can put this solution on YOUR website!
## Fitch Proof Setup for $P \lor R$
Here is the setup for a proof of $P \lor R$ from the premises $P \lor Q$ and $\neg Q \lor R$ using Fitch notation. This proof requires a **proof by cases** (Disjunction Elimination).
### Goal: $\mathbf{P \lor R}$
| Line | Formula | Justification |
| :--- | :--- | :--- |
| 1 | $P \lor Q$ | Premise |
| 2 | $\neg Q \lor R$ | Premise |
| 3 | $\qquad \mathbf{P}$ | **Assumption** ($\lor$ Elim) |
| 4 | $\qquad P \lor R$ | $\lor$ Intro: 3 |
| 5 | $\qquad$ **Q** | **Assumption** ($\lor$ Elim) |
| 6 | $\qquad \neg Q \lor R$ | Repetition (or simply refer to 2) |
| 7 | $\qquad \qquad \mathbf{\neg Q}$ | **Assumption** ($\lor$ Elim) |
| 8 | $\qquad \qquad \perp$ | $\neg$ Elim: 5, 7 |
| 9 | $\qquad \qquad \mathbf{R}$ | **Assumption** ($\lor$ Elim) |
| 10 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 9 |
| 11 | $\qquad P \lor R$ | $\lor$ Elim: 6, 7-8, 9-10 |
| 12 | $P \lor R$ | $\lor$ Elim: 1, 3-4, 5-11 |
***
## Completed Proof
The key insight is that $\neg Q \lor R$ serves as a crucial intermediate step within the second case of the main proof.
| Line | Formula | Justification |
| :--- | :--- | :--- |
| 1 | $P \lor Q$ | Premise |
| 2 | $\neg Q \lor R$ | Premise |
| 3 | $\qquad \mathbf{P}$ | **Assumption (Case 1)** |
| 4 | $\qquad P \lor R$ | $\lor$ Intro: 3 |
| 5 | $\qquad \mathbf{Q}$ | **Assumption (Case 2)** |
| 6 | $\qquad \neg Q \lor R$ | Repetition: 2 |
| 7 | $\qquad \qquad \mathbf{\neg Q}$ | **Assumption (Subcase 2a)** |
| 8 | $\qquad \qquad \perp$ | $\neg$ Elim: 5, 7 |
| 9 | $\qquad \qquad R$ | $\perp$ Elim: 8 |
| 10 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 9 |
| 11 | $\qquad \qquad \mathbf{R}$ | **Assumption (Subcase 2b)** |
| 12 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 11 |
| 13 | $\qquad P \lor R$ | $\lor$ Elim: 6, 7-10, 11-12 |
| 14 | $P \lor R$ | $\lor$ Elim: 1, 3-4, 5-13 |