Variable

Free Variable

The set $FV(\phi )$ of free variables of a formula $\phi$ is defined inductively: If $\phi$ is an atomic formula then $FV(\phi )$ is the set of all variables that occur in $\phi$. $FV(\neg \phi )=FV(\phi )$.

• $FV(\phi \wedge \psi )=FV(\phi )\cup FV(\psi )$.
• $FV(\forall v\phi )=FV(\phi )\setminus \{v\}$. The symbol "$\setminus$" denotes set-theoretic subtraction. Regarding defined logical symbols, it follows that:
• $FV(\phi \vee \psi )=FV(\phi \to \psi )=FV(\phi \leftrightarrow \psi )=FV(\phi )\cup FV(\psi )$.
• $FV(\exists v\phi )=FV(\phi )\setminus \{v\}$.

The set $BV(\phi )$ of bound variables of $\phi$ is similarly defined by induction on complexity, and is the set of variables of $\phi$ occurring within the scope of a quantifier.

Sentence

A sentence is a formula with no free variables. So a sentence is always true or false in a structure, and this is independent of any assignment of values to variables.