Models for Propositional Logic

Theorem
This page gathers together some useful results that can be used in the derivation of proofs by propositional tableau.

Let $\mathcal M$ be a model for propositional logic, and let $\mathbf A$ and $\mathbf B$ be WFFs of propositional logic.

Then the following results hold.

The symbol $\models$ is used throughout to denote semantic consequence.

Double Negation

 * $\mathcal M \models \neg \neg \mathbf A$ iff $\mathcal M \models \mathbf A$

This is the rule of Double Negation.

And

 * $\mathcal M \models \left({\mathbf A \land \mathbf B}\right)$ iff both $\mathcal M \models \mathbf A$ and $\mathcal M \models \mathbf B$

This follows by definition of Conjunction.

Not And

 * $\mathcal M \models \neg \left({\mathbf A \land \mathbf B}\right)$ iff either $\mathcal M \models \neg \mathbf A$ or $\mathcal M \models \neg \mathbf B$

This follows from De Morgan's Laws: Disjunction of Negations.

Or

 * $\mathcal M \models \left({\mathbf A \lor \mathbf B}\right)$ iff either $\mathcal M \models \mathbf A$ or $\mathcal M \models \mathbf B$

This follows by definition of Disjunction.

Not Or

 * $\mathcal M \models \neg \left({\mathbf A \lor \mathbf B}\right)$ iff $\mathcal M \models \neg \mathbf A$ and $\mathcal M \models \neg \mathbf B$

This follows from De Morgan's Laws: Conjunction of Negations.

Implies

 * $\mathcal M \models \left({\mathbf A \implies \mathbf B}\right)$ iff either $\mathcal M \models \neg \mathbf A$ or $\mathcal M \models \mathbf B$

This follows from Disjunction and Implication.

Not Implies

 * $\mathcal M \models \neg \left({\mathbf A \implies \mathbf B}\right)$ iff $\mathcal M \models \mathbf A$ and $\mathcal M \models \neg \mathbf B$

This follows from Conjunction and Implication.

Iff

 * $\mathcal M \models \left({\mathbf A \iff \mathbf B}\right)$ iff either:
 * both $\mathcal M \models \mathbf A$ and $\mathcal M \models \mathbf B$

or:
 * both $\mathcal M \models \neg \mathbf A$ and $\mathcal M \models \neg \mathbf B$

This follows by definition of biconditional.

Exclusive Or

 * $\mathcal M \models \mathbf A \oplus \mathbf B$ iff either:
 * both $\mathcal M \models \mathbf A$ and $\mathcal M \models \neg \mathbf B$

or:
 * both $\mathcal M \models \neg \mathbf A$ and $\mathcal M \models \mathbf B$

This follows by definition of exclusive or.