Definition:Constructed Semantics/Instance 4/Rule of Addition

Theorem
The Rule of Addition:


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

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 Addition can be written as:


 * $\neg q \lor \left({p \lor q}\right)$

This evaluates as follows:


 * $\begin{array}{|cc|c|ccc|} \hline

\neg & q & \lor & (p & \lor & q) \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 2 & 0 & 0 & 0 & 2 \\ 2 & 3 & 0 & 0 & 0 & 3 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 2 & 0 & 1 & 2 & 2 \\ 2 & 3 & 0 & 1 & 3 & 3 \\ 1 & 0 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 2 & 2 & 1 \\ 0 & 2 & 0 & 2 & 2 & 2 \\ 2 & 3 & 0 & 2 & 0 & 3 \\ 1 & 0 & 0 & 3 & 0 & 0 \\ 0 & 1 & 0 & 3 & 3 & 1 \\ 0 & 2 & 0 & 3 & 3 & 2 \\ 2 & 3 & 0 & 3 & 3 & 3 \\ \hline \end{array}$