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many-sorted language

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A many-sorted language is a variation of the classical first-order language.  Whereas structures of a first-order language consist of a single universe, structures of a many-sorted language may contain many ``universes'', where each universe is ``named'' by a symbol called ``sort'', hence the name ``many-sorted''.  

To formalize the notion of a many-sorted language, we start with a non-empty set $S$ whose elements we call \emph{sorts}.  Let $S^+$ be the set of all (finite) non-empty words over $S$.  Elements of $S^+$ are called \emph{sort types} and are written as tuples.  So instead of writing $s_1s_2\cdots s_n \in S^+$, it is written $(s_1, s_2, \cdots, s_n)\in S^+$.

The next item to be defined is the underlying signature of a many-sorted language.  A signature $\Sigma$ consists of 
\begin{itemize}
\item a non-empty set $S$ of sorts, 
\item a set $F$ of function symbols, and 
\item a set $R$ of (non-logical) relation symbols, 
\end{itemize}
such that each element in $F\cup R$ corresponds to a sort type.  In other words, there is a function $t:F\cup R\to S^+$, and for every $r\in F\cup R$, $t(r)$ is its sort type.  An element $c\in F$ such that $t(c)\in S$ is called a constant symbol (of sort $t(c)$).

In addition to the signature $\Sigma$, we introduce additional symbols: 
\begin{itemize}
\item the set $V$ of variables, such that each sort $s\in S$ corresponds to a countably infinite subset $V_s\subseteq V$ of variables.  In other words, there is a function $v:V\to S$, such that for each $s\in S$, $v^{-1}(s)$ is countably infinite.  For each variable $x\in V$, its sort is defined to be $v(x)$.
\item logical predicates: $\vee, \neg, \forall$
\item the equality relation symbol: $=$, and
\item the left and right parentheses: $(,)$
\end{itemize}

Using $\Sigma$ and the additional symbols above, we can build terms inductively as follows:
\begin{itemize}
\item each variable $x\in V$ is a term of sort $v(x)$
\item if $f\in F$ is a function symbol of sort type $(s_1,\ldots,s_n)$, and for each $i=1,\ldots,n-1$, $t_i$ is a term of sort $s_i$, then $f(t_1,\ldots,t_{n-1})$ is a term of sort $s_n$.
\item all the terms are built this way.
\end{itemize}

Finally, from the terms, we inductively define formulas:
\begin{itemize}
\item if $t_1$ and $t_2$ are terms of the same sort, then $(t_1=t_2)$ is a formula
\item if $r\in R$ is a relation symbol of sort type $(s_1,\ldots,s_n)$, and for each $i=1,\ldots,n$, $t_i$ is a term of sort $s_i$, then $r(t_1,\ldots,t_n)$ is a formula
\item if $\alpha$ is a formula, then so is $(\neg \alpha)$
\item if $\alpha,\beta$ are formulas, then so is $(\alpha \vee \beta)$
\item if $\alpha$ is a formula and $x\in V$ is a variable, then so is $(\forall x (\alpha))$
\item all the formulas are formed this way.
\end{itemize}

The signature $\Sigma$, additional symbols, and terms and formulas subsequently defined constitute what is known as the \emph{many-sorted language} $L=L(\Sigma)$ on $\Sigma$.

As in first order language, the outer most parentheses may be eliminated without causing much harm, so that $(\neg \alpha)$ becomes $\neg \alpha$.  In addition, we may introduce other familiar logical symbols $\wedge, \to, \leftrightarrow$, and $\exists$ in terms of $\vee,\neg$, and $\forall$.  The specifics of how this is done can be found in the entry on first order language.

From this definition, we see at once that the classical first order language is a one-sorted language ($S=1$).  Sort types are identified with their lengths.  Thus, the sort type of a function or a relation symbol is its arity.

\textbf{Remark}.  It is not hard to show that a many-sorted language is not much different from a first-order language.  Provided that $V$ is countably infinite, a many-sorted language $L$ can be ``converted'' into a first-ordered language $L_1$.  Basically, all the function and relation symbols in $L$ are in $L'$, as well as the additional symbols such as variables and logical symbols.  For each sort $s\in S$, we introduce a new unary relation symbol $P_s$ in $L_1$.  For any formula that contains a subformula of the form $\forall x \phi(x)$, we replace each occurrence of such a subformula by a formula of the form $\forall x (P_s(x)\to \phi(x))$, where $x$ is a variable of sort $s$ and $\phi$ is some formula in which $x$ occurs as a free variable.  The result is that $L_1$ becomes a one-sorted language.
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