First Order Logic Consequence Relation

Question

I have been reading my text book for a bit now and am still puzzled on how to show something like in the following example problems:

For arbitrary formulas $\phi$, $\psi$, and $\chi$ show that:

1. $(\phi \lor \psi) \models \chi$ iff $\phi \models \chi$ and $\psi \models \chi$

2. $\emptyset \models (\phi \to \psi)$ iff $\phi \models \psi$

Thank you for helping me understand in advance.

Answer 1

The notion $\models$ is saying that every model which satisfies the formula on the left side satisfies the formulas on the right side.

1. Assume that $\phi\lor\psi\models \chi$. If $M\models \phi$, then $M\models \phi\vee \psi$ and thus $\phi\lor\psi\models \chi$ implies that $M\models \chi$, thus any arbitrary model which satisfies $\phi$ also satisfy $\chi$ i.e. $\phi\models \chi$. If we now assume $N\models\psi$ then we may in the same way show $N\models \chi$ and thus $\psi\models \chi$.

For the second direction assume that $\phi \models \chi$ and $\psi\models \chi$. If $M\models \phi\vee\psi$, then by the definition of $\vee$, we know that $M\models \phi$ or $M\models \psi$. If $M\models \phi$ then $\phi \models \chi$ implies that $M\models \chi$ on the other hand if $M\models\psi$ then $\psi \models \chi$ implies that $M\models \chi$. Thus we conclude that $M\models \chi$ and thus $\phi\lor\psi\models \chi$ hold.

2. Assume $\emptyset\models (\phi\rightarrow \psi)$. Thus each structure satisfies $\phi\rightarrow\psi$. Assume that $M\models \phi$. Since each structure satisfy $\phi\rightarrow\psi$ we know that $M\models \phi\rightarrow\psi$. Thus we may especially draw the conclusion that $M\models \psi$, hence $phi\models \psi$.

For the second direction assume $\phi\models \psi$ and let $M$ be any structure. If $M\not\models \phi$ then $M\models \phi\rightarrow \psi$ since the first part of the implication is false. On the other hand if $M\models \phi$ then $\phi\models \psi$ implies that $M\models \psi$, thus $M\models \phi\rightarrow \psi$ hold, as both parts of the implication are true. Hence any model satisfies $(\phi\rightarrow \psi)$, which make us able to conclude that $\emptyset\models (\phi\rightarrow \psi)$.