Prove that $P \vDash X$ and $Q \vDash X$ implies $(P \vee Q) \vDash X$

Question

Let $P \vDash X$ and $Q \vDash X$. Prove that $(P \vee Q) \vDash X$.

This may seem to obvious but is there a formal way to prove this? couldn't I just state that $(P \vee Q) \vDash X$ is true since both sides of the connective are already true?

Answer 1

The formal way is quite easy.

We want to prove that :

if $P \vDash X$ and $Q \vDash X$, then $(P \lor Q) \vDash X$.

But $P \vDash X$ means that for every valuation $v$, if $v(P)=T$, then $v(X)=T$; and the same for $Q \vDash X$.

Consider now $P \lor Q$ and a valuation $v$ whatever such that $v(P \lor Q)=T$.

This means that $v(P)=T$ or $v(Q)=T$; in both cases : $v(X)=T$.

Thus, we have shown that for every valuation $v$ such that $v(P \lor Q)=T$, we have also $v(X)=T$, i.e. :

$(P \lor Q) \vDash X$.