Definition:Conditional/Boolean Interpretation

Definition

Let $\mathbf A$ and $\mathbf B$ be propositional formulas.

Let $\implies$ denote the conditional operator.

The truth value of $\mathbf A \implies \mathbf B$ under a boolean interpretation $v$ is given by:

$\map v {\mathbf A \implies \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \F \text{ or } \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$

and the truth value of $\mathbf A \impliedby \mathbf B$ under a boolean interpretation $v$ is given by:

$\map v {\mathbf A \impliedby \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \T \text{ or } \map v {\mathbf B} = \F \\ \F & : \text{otherwise} \end{cases}$

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 6 \ (3)$
• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.3$: Boolean interpretations: Figure $2.7$
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.5$: Semantics of Propositional Logic