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.