Definition:Constructed Semantics/Instance 3

Definition

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

The constructed semantics $\mathscr C_3$ for $\mathcal L_0$ is used for the following results:

Structures

Define the structures of $\mathscr C_3$ 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 $v: \mathcal P_0 \to \{ 0, 1, 2 \}$ be given.

Next, regard the following as definitional abbreviations:

 $(1)$ $:$ Conjunction $\displaystyle \mathbf A \land \mathbf B$ $\displaystyle =_{\text{def} }$ $\displaystyle \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right)$ $(2)$ $:$ Conditional $\displaystyle \mathbf A \implies \mathbf B$ $\displaystyle =_{\text{def} }$ $\displaystyle \neg \mathbf A \lor \mathbf B$ $(3)$ $:$ Biconditional $\displaystyle \mathbf A \iff \mathbf B$ $\displaystyle =_{\text{def} }$ $\displaystyle \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right)$

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

 $\displaystyle v \left({\neg \phi }\right)$ $:=$ $\displaystyle \begin{cases} 2 & : \text{if v \left({\phi}\right) = 0} \\ 1 & : \text{if v \left({\phi}\right) = 1} \\ 0 & : \text{if v \left({\phi}\right) = 2}\end{cases}$ $\displaystyle v \left({ \phi \lor \psi }\right)$ $:=$ $\displaystyle \begin{array}{c|ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 2 \\ \end{array}$

Validity

Define validity in $\mathscr C_3$ by declaring:

$\models_{\mathscr C_3} \phi$ if and only if $v \left({\phi}\right) \in \set{ 0, 1 }$