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

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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