Definition:Constructed Semantics/Instance 2

Definition
Let $\LL_0$ be the language of propositional logic.

The constructed semantics $\mathscr C_2$ for $\LL_0$ is used for the following results:


 * Hilbert Proof System Instance 2 Independence Results: Independence of $(A1)$

Structures
Define the structures of $\mathscr C_2$ as mappings $v$ by the Principle of Recursive Definition, as follows.

Let $\PP_0$ be the vocabulary of $\LL_0$.

Let a mapping $v: \PP_0 \to \set {0, 1, 2}$ be given.

Next, regard the following as definitional abbreviations:

It only remains to define $\map v {\neg \phi}$ and $\map v {\phi \lor \psi}$ recursively, by:

Validity
Define validity in $\mathscr C_2$ by declaring:


 * $\models_{\mathscr C_2} \phi$ $\map v \phi = 0$

Examples

 * Rule of Addition
 * Rule of Commutation
 * Factor Principle