Definition:Constructed Semantics/Instance 4/Rule of Idempotence

Theorem
The Rule of Idempotence:


 * $(p \lor p) \implies p$

is a tautology in Instance 4 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 \\ 0 & 2 & 2 & 2 & 0 & 2 \\ 2 & 3 & 3 & 3 & 0 & 3 \\ \hline \end{array}$