Definition:Constructed Semantics/Instance 3

Definition

The constructed semantics $\mathscr C_3$ for $\LL_0$ is used for the following results:

Define the structures of $\mathscr C_3$ 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:

$(1)$	$:$	Conjunction	$\ds \mathbf A \land \mathbf B $	$\ds \stackrel {\text{def} } = $	$\ds \map \neg {\neg \mathbf A \lor \neg \mathbf B} $
$(2)$	$:$	Conditional	$\ds \mathbf A \implies \mathbf B $	$\ds \stackrel {\text{def} } = $	$\ds \neg \mathbf A \lor \mathbf B $
$(3)$	$:$	Biconditional	$\ds \mathbf A \iff \mathbf B $	$\ds \stackrel {\text{def} } = $	$\ds \paren {\mathbf A \implies \mathbf B} \land \paren {\mathbf B \implies \mathbf A} $

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

$\ds \map v {\neg \phi}$

$:=$

$\ds \begin{cases} 2 & : \text{if } \map v \phi = 0 \\ 1 & : \text{if } \map v \phi = 1 \\ 0 & : \text{if } \map v \phi = 2 \end{cases}$

$\ds \map v {\phi \lor \psi}$

$:=$

$\ds \begin{array}{c|ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 2 \\ \end{array}$

Define validity in $\mathscr C_3$ by declaring:

$\models_{\mathscr C_3} \phi$ if and only if $\map v \phi \in \set {0, 1}$

\((1)\)	$:$	Conjunction	\(\ds \mathbf A \land \mathbf B \)	\(\ds \stackrel {\text{def} } = \)	\(\ds \map \neg {\neg \mathbf A \lor \neg \mathbf B} \)
\((2)\)	$:$	Conditional	\(\ds \mathbf A \implies \mathbf B \)	\(\ds \stackrel {\text{def} } = \)	\(\ds \neg \mathbf A \lor \mathbf B \)
\((3)\)	$:$	Biconditional	\(\ds \mathbf A \iff \mathbf B \)	\(\ds \stackrel {\text{def} } = \)	\(\ds \paren {\mathbf A \implies \mathbf B} \land \paren {\mathbf B \implies \mathbf A} \)