NAND with Equal Arguments/Proof 1
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Theorem
- $p \uparrow p \dashv \vdash \neg p$
That is, the NAND of a proposition with itself corresponds to the negation operation.
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \uparrow p$ | Premise | (None) | ||
2 | 1 | $\neg \paren {p \land p}$ | Sequent Introduction | 1 | Definition of Logical NAND | |
3 | 1 | $\neg p$ | Sequent Introduction | 2 | Rule of Idempotence: Conjunction |
$\Box$
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\neg p$ | Premise | (None) | ||
2 | 1 | $\neg \paren {p \land p}$ | Sequent Introduction | 1 | Rule of Idempotence: Conjunction | |
3 | 1 | $p \uparrow p$ | Sequent Introduction | 2 | Definition of Logical NAND |
$\blacksquare$