Definition:Beta-Formula/Table
< Definition:Beta-Formula(Redirected from Definition:Table of Beta-Formulas)
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Definition
From Classification of $\beta$-Formulas, we obtain the following table of $\beta$-formulas $\mathbf B$ and corresponding $\mathbf B_1$ and $\mathbf B_2$:
- $\begin{array}{ccc} \hline \mathbf B & \mathbf B_1 & \mathbf B_2\\ \hline \neg \paren {\mathbf B_1 \land \mathbf B_2} & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \mathbf B_1 \lor \mathbf B_2 & \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \implies \mathbf B_2 & \neg \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \mathbin \uparrow \mathbf B_2 & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \neg \paren {\mathbf B_1 \mathbin \downarrow \mathbf B_2} & \mathbf B_1 & \mathbf B_2 \\ \neg \paren {\mathbf B_1 \iff \mathbf B_2} & \neg \paren {\mathbf B_1 \implies \mathbf B_2} & \neg \paren {\mathbf B_2 \implies \mathbf B_1} \\ \mathbf B_1 \oplus \mathbf B_2 & \neg \paren {\mathbf B_1 \implies \mathbf B_2} & \neg \paren {\mathbf B_2 \implies \mathbf B_1} \\ \hline \end{array}$
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.6.2$: Figure $2.8$