Notes: p→q; p→r; ¬q∨¬r; ∴¬p please prove using natural deduction.

Thursday, December 29, 2011

1. basic assumption

2. $\displaystyle{p}\to{r}$ basic assumption

3. $\displaystyle\neg{q}\vee\neg{r}$ basic assumption(goal: $\displaystyle\neg{p}$ )

In order to proof this we will need to divide it into two mini proofs:

Mini proof 1:

4. $\displaystyle\neg{q}$ (Assume $\displaystyle\vee{E}$ to prove $\displaystyle{p}$ )

5 $\displaystyle\neg{p}$ ( $\displaystyle\to{E}$ lines 1,4)

Mini proof 2:

6. $\displaystyle\neg{r}$ (Assume $\displaystyle\vee{E}$ to prove $\displaystyle{p}$ )

7. $\displaystyle\neg{p}$ ( $\displaystyle\to{E}$ lines 2,6)

CONCLUSION: (now we can show the proof)

8. $\displaystyle\neg{p}$ ( $\displaystyle\vee{E}$ lines3, 4-5, 6-7)

This is how we proof by natural deduction.

Notes