Definition:Constructed Semantics/Instance 4/Rule of Idempotence

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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:

$\begin{array}{|cccc|c|c|} \hline \neg & (p & \lor & p) & \lor & p \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 2 & 2 & 2 & 0 & 2 \\ 2 & 3 & 3 & 3 & 0 & 3 \\ \hline \end{array}$

$\blacksquare$