Definition:Constructed Semantics/Instance 1/Rule of Addition
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Theorem
The Rule of Addition:
- $q \implies (q \lor p)$
is a tautology in Instance 1 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:
- $\begin{array}{|cc|c|ccc|} \hline
\neg & q & \lor & (p & \lor & q) \\ \hline 2 & 1 & 2 & 1 & 1 & 1 \\ 1 & 2 & 2 & 1 & 2 & 2 \\ 2 & 1 & 2 & 2 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 \\ \hline \end{array}$
$\blacksquare$