Definition:Constructed Semantics/Instance 5/Rule of Idempotence
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Theorem
The Rule of Idempotence:
- $(p \lor p) \implies p$
is a tautology in Instance 5 of constructed semantics.
Proof
By the definitional abbreviation for the conditional:
- $\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Idempotence can be written as:
- $\neg \left({p \lor p}\right) \lor p$
This evaluates as follows:
- $\begin{array}{|cccc|c|c|} \hline
\neg & (p & \lor & p) & \lor & p \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 1 \\ 3 & 2 & 2 & 2 & 0 & 2 \\ 0 & 3 & 3 & 3 & 0 & 3 \\ \hline \end{array}$
$\blacksquare$