NOR is not Associative
Theorem
Let $\downarrow$ signify the NOR operation.
Then there exist propositions $p,q,r$ such that:
- $p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$
That is, NOR is not associative.
Proof 1
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\neg p \land r$ | Assumption | (None) | ||
2 | 1 | $\neg p$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
3 | 1 | $r$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
4 | 1 | $q \lor r$ | Rule of Addition: $\lor \II_2$ | 3 | ||
5 | 1 | $\neg \neg \paren {q \lor r}$ | Double Negation Introduction: $\neg \neg \II$ | 4 | ||
6 | 1 | $\neg \paren {q \downarrow r}$ | Sequent Introduction | 5 | Definition of Logical NOR | |
7 | 1 | $\neg p \land \neg \paren {q \downarrow r}$ | Rule of Conjunction: $\land \II$ | 2, 6 | ||
8 | 1 | $\neg \paren {p \lor \paren {q \downarrow r} }$ | Sequent Introduction | 7 | De Morgan's Laws: Conjunction of Negations | |
9 | 1 | $p \downarrow \paren {q \downarrow r}$ | Sequent Introduction | 8 | Definition of Logical NOR | |
10 | 1 | $\paren {p \downarrow q} \lor r$ | Rule of Addition: $\lor \II_2$ | 3 | ||
11 | 1 | $\neg \neg \paren {\paren {p \downarrow q} \lor r}$ | Double Negation Introduction: $\neg \neg \II$ | 10 | ||
12 | 1 | $\neg \paren {\paren {p \downarrow q} \downarrow r}$ | Sequent Introduction | 11 | Definition of Logical NOR | |
13 | 1 | $\neg \neg \paren {p \downarrow \paren {q \downarrow r} }$ | Double Negation Introduction: $\neg \neg \II$ | 9 | ||
14 | 1 | $\paren {\neg \neg \paren {p \downarrow \paren {q \downarrow r} } } \land \paren {\neg \paren {\paren {p \downarrow q} \downarrow r} }$ | Rule of Conjunction: $\land \II$ | 13, 12 | ||
15 | 1 | $\neg \paren {\neg \paren {p \downarrow \paren {q \downarrow r} } \lor \paren {\paren {p \downarrow q} \downarrow r} }$ | Sequent Introduction | 14 | De Morgan's Laws: Conjunction of Negations | |
16 | 1 | $\neg \paren {p \downarrow \paren {q \downarrow r} \implies \paren {p \downarrow q} \downarrow r}$ | Sequent Introduction | 15 | Rule of Material Implication |
Taking $p = \bot$ and $r = \top$, we have $\vdash \neg p \land r$, discharging the last assumption.
Hence the result.
$\blacksquare$
Proof by Truth Table
Apply the Method of Truth Tables:
- $\begin{array}{|ccccc||ccccc|} \hline
p & \downarrow & (q & \downarrow & r) & (p & \downarrow & q) & \downarrow & r \\ \hline \F & \F & \F & \T & \F & \F & \T & \F & \F & \F \\ \F & \T & \F & \F & \T & \F & \T & \F & \F & \T \\ \F & \T & \T & \F & \F & \F & \F & \T & \T & \F \\ \F & \T & \T & \F & \T & \F & \F & \T & \F & \T \\ \T & \F & \F & \T & \F & \T & \F & \F & \T & \F \\ \T & \F & \F & \F & \T & \T & \F & \F & \F & \T \\ \T & \F & \T & \F & \F & \T & \F & \T & \T & \F \\ \T & \F & \T & \F & \T & \T & \F & \T & \F & \T \\ \hline \end{array}$
As can be seen by inspection, the truth values under the main connectives do not match for all boolean interpretations.
$\blacksquare$
Also see
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.13$