Consequence and Other Concepts

Now for a series of important definitions:

• A formula $\phi$ is a consequence of a set $\Phi$ of formulas, written $\Phi \models \phi$, iff every interpretation that models $\Phi$ also models $\phi$.
• A formula $\phi$ is said to be valid iff it's true on all interpretations. If this is true of a formula $\phi$, we write $\models \phi$.
• We say that a formula $\phi$ is satisfiable iff there is some $\cal I$ such that $\cal I$ $\models \phi$.
• Two formulas $\phi$ and $\psi$ are logically equivalent iff $\phi \models \psi$ and $\psi \models \phi$.