Questionnaire-2024/predicate_logic/0007_sol.tex

\begin{itemize}
    \item $\mathcal{V}$: Defines the set of variable symbols, e.g., $x,y,z$.
    \item $\mathcal{F}$: Defines the set of function symbols, e.g., $f,g,h$.
    \item $\mathcal{P}$: Defines the set of predicate symbols, e.g., $P,Q,R$. \\
\end{itemize}
Terms are defined as follows:
\begin{itemize}
    \item Any variable is a term.
    \item If $c \in \mathcal{F}$ is a nullary function, then $c$ is a term.
    \item If $t_1, t_2, \ldots t_n$ are terms and $f \in \mathcal{F}$ has arity $n > 0$, then $f(t_1, t_2, \ldots t_n)$ is a term.
    \item Nothing else is a term. \\
\end{itemize}
Formulas are defined as follows:
\begin{itemize}
    \item If $P \in \mathcal{P}$ is a predicate with arity $n > 0$ and $t_1, t_2, \ldots t_n$ are terms over $\mathcal{F}$, then $P(t_1, t_2, \ldots t_n)$ is a formula.
    \item If $\phi$ is a formula, then $\lnot \phi$ is a formula.
    \item If $\phi$ and $\psi$ are formulas, then $(\phi \land \psi)$, $(\phi \lor \psi)$, $(\phi \imp \psi)$ are formulas.
    \item If $\phi$ is a formula and $x$ is a variable, then $(\forall x \phi)$ and $(\exists x \phi)$ are formulas.
    \item Nothing else is a formula. 
\end{itemize}