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 calculus, and let $$\mathbf A$$ and $$\mathbf B$$ be propositional WFFs.

Then the following results hold:

Double Negation

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

This is the rule of Double Negation.

And

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

This follows by definition of Conjunction.

Not And

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

This follows from De Morgan's Laws.

Or

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

This follows by definition of Disjunction.

Not Or

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

This follows from De Morgan's Laws.

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 Material Equivalence.

Not Iff

 * $$\mathcal M \models \neg \left({\mathbf A \iff \mathbf B}\right)$$ 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 Non-Equivalence.