13  Truth-functional semantics for connectives

Below I include, for each of the logical connectives,

13.1 Negation (\(\neg\))

13.1.1 Truth-functional formula

\[ v_i(\neg \phi) = \begin{cases} 1 \text{ iff } v_i(\phi) = 0\\ 0 \text{ otherwise} \end{cases} \]

13.1.2 Truth-tabular representation

\[ \begin{array}{|c|c|}\hline \phi & \neg \phi \\\hline 1 & 0 \\ 0 & 1 \\\hline \end{array} \]

13.2 Disjunction

13.2.1 Truth-functional formula

\[ v_i(\phi \vee \psi) = \begin{cases} 0 \text{ iff } v_i(\phi) = 0 \text{ and } v_i(\psi)=0 \\ 1 \text{ otherwise} \end{cases} \]

13.2.2 Truth-tabular representation

\[ \begin{array}{|c|c|c|}\hline \phi & \psi & (\phi \vee \psi) \\\hline 1 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\\hline \end{array} \]

13.3 Conjunction

13.3.1 Truth-functional formula

\[ v_i(\phi \wedge \psi) = \begin{cases} 1 \text{ iff } v_i(\phi) = 1 \text{ and } v_i(\psi)=1 \\ 0 \text{ otherwise} \end{cases} \]

13.3.2 Truth-tabular representation

\[ \begin{array}{|c|c|c|}\hline \phi & \psi & (\phi \wedge \psi) \\\hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\\hline \end{array} \]

13.4 Conditional

13.4.1 Truth-functional formula

\[ v_i(\phi \to \psi) = \begin{cases} 0 \text{ iff } v_i(\phi) = 1 \text{ and } v_i(\psi)=0 \\ 1 \text{ otherwise} \end{cases} \]

13.4.2 Truth-tabular representation

\[ \begin{array}{|c|c|c|}\hline \phi & \psi & (\phi \to \psi) \\\hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\hline \end{array} \]

13.5 Biconditional

13.5.1 Truth-functional formula

\[ v_i(\phi \leftrightarrow \psi) = \begin{cases} 1 \text{ iff } v_i(\phi) = v_i(\psi) \\ 0 \text{ otherwise} \end{cases} \]

13.5.2 Truth-tabular representation

\[ \begin{array}{|c|c|c|}\hline \phi & \psi & (\phi \leftrightarrow \psi) \\\hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\hline \end{array} \]