Definition:Constructed Semantics/Instance 5

Definition
Let $\mathcal L_0$ be the language of propositional logic.

The constructed semantics $\mathscr C_5$ for $\mathcal L_0$ is used for the following results:


 * Hilbert Proof System Instance 2 Independence Results: Independence of $(A4)$

Structures
Define the structures of $\mathscr C_5$ as mappings $v$ by the Principle of Recursive Definition, as follows.

Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$.

Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2, 3 \}$ be given.

Next, regard the following as definitional abbreviations:

It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by:

Validity
Define validity in $\mathscr C_5$ by declaring:


 * $\models_{\mathscr C_5} \phi$ $v \left({\phi}\right) = 0$

Examples

 * Rule of Idempotence
 * Rule of Addition
 * Rule of Commutation