# Definition:Constructed Semantics/Instance 2/Rule of Addition

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## Theorem

The Rule of Addition:

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

is a tautology in Instance 2 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:

- $\begin{array}{|cc|c|ccc|} \hline \neg & q & \lor & (p & \lor & q) \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 2 & 2 & 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 2 & 2 & 0 & 1 & 2 & 2 \\ 1 & 0 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 2 & 2 & 1 \\ 2 & 2 & 0 & 2 & 0 & 2 \\ \hline \end{array}$

$\blacksquare$

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.6$: Independence