True for every formula.

Question

For a formula $ \gamma = r \iff (p_1 \vee p_2) $ there is true that: $\rho \models \gamma \iff \rho(r) = \max(\rho(p_1), \rho(p_2))$

Is it a true for every formula r?

Answer 1

Your question is very terse and hard to understand. It is particularly confusing to use a symbol like $\iff$ both as an object language symbol (i.e., to form formulas) and as a metalanguage symbol (i.e., to abbreviate assertions about formulas). Let me write out (rather verbosely, by way of contrast) what I think you mean:

If $\gamma$ is the formula $r \Leftrightarrow (p_1 \lor p_2)$, is it true that under any assignment $\rho$, $\rho$ satisfies $\gamma$ iff the truth value $\rho(r)$ of $r$ under $\rho$ is equal to the larger of the truth values $\rho(p_1)$ and $\rho(p_2)$ (where we adopt the usual ordering of truth values so that truth is greater than falsehood)?

Assuming the usual definitions of everything involved, this is true and can be proved quite directly from the definition of the satisfaction relation $\models$. If you are having problems with this, you need to give some more details about the particular definitions you are working with and what you have tried.