logical language LogicalLanguage 2002-08-28 18:52:02 2007-01-05 10:40:42 Definition term formula constant % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} In its most general form, a \emph{logical language} is a set of rules for constructing \emph{formulas} for some logic, which can then be assigned truth values based on the rules of that logic. A logical language $\mathcal{L}$ consists of: \begin{itemize} \item A set $F$ of function symbols (common examples include $+$ and $\cdot$) \item A set $R$ of relation symbols (common examples include $=$ and $<$) \item A set $C$ of logical connectives (usually $\neg$, $\wedge$, $\vee$, $\rightarrow$ and $\leftrightarrow$) \item A set $Q$ of quantifiers (usuallly $\forall$ and $\exists$) \item A set $V$ of variables \end{itemize} Every function symbol, relation symbol, and connective is associated with an arity (the set of $n$-ary function symbols is denoted $F_n$, and similarly for relation symbols and connectives). Each quantifier is a generalized quantifier associated with a quantifier type $\langle n_1,\ldots,n_n\rangle$. The underlying logic has a (possibly empty) set of types $T$. There is a function $\operatorname{Type}:F\cup V\rightarrow T$ which assignes a type to each function and variable. For each arity $n$ is a function $\operatorname{Inputs}_n:F_n\cup R_n\rightarrow T^n$ which gives the types of each of the arguments to a function symbol or relation. In addition, for each quantifier type $\langle n_1,\ldots,n_n\rangle$ there is a function $\operatorname{Inputs}_{\langle n_1,\ldots,n_n\rangle}$ defined on $Q_{\langle n_1,\ldots,n_n\rangle}$ (the set of quantifiers of that type) which gives an $n$-tuple of $n_i$-tuples of types of the arguments taken by formulas the quantifier applies to. The \emph{terms} of $\mathcal{L}$ of type $t\in T$ are built as follows: \begin{enumerate} \item If $v$ is a variable such that $\operatorname{Type}(v)=t$ then $v$ is a term of type $t$ \item If $f$ is an $n$-ary function symbol such that $\operatorname{Type}(f)=t$ and $t_1,\ldots,t_n$ are terms such that for each $i<n$ $\operatorname{Type}(t_i)=(\operatorname{Inputs}_n(f))_i$ then $ft_1,\ldots,t_n$ is a term of type $t$ \end{enumerate} The \emph{formulas} of $\mathcal{L}$ are built as follows: \begin{enumerate} \item If $r$ is an $n$-ary relation symbol and $t_1,\ldots,t_n$ are terms such that $\operatorname{Type}(t_i)=(\operatorname{Inputs}_n(r))_i$ then $rt_1,\ldots,t_n$ is a formula \item If $c$ is an $n$-ary connective and $f_1,\ldots,f_n$ are formulas then $cf_1,\ldots,f_n$ is a formula \item If $q$ is a quantifier of type $\langle n_1,\ldots,n_n\rangle$, $v_{1,1},\ldots,v_{1,n_1},v_{2,1},\ldots,v_{n,1},\ldots,v_{n,n_n}$ are a sequence of variables such that $\operatorname{Type}(v_{i,j})=((\operatorname{Inputs}_{\langle n_1,\ldots,n_n\rangle}(q))_j)_i$ and $f_1,\ldots,f_n$ are formulas then $qv_{1,1},\ldots,v_{1,n_1},v_{2,1},\ldots,v_{n,1},\ldots,v_{n,n_n}f_1,\ldots,f_n$ is a formula \end{enumerate} Generally the connectives, quantifiers, and variables are specified by the appropriate logic, while the function and relation symbols are specified for particular languages. Note that $0$-ary functions are usually called \emph{constants}. If there is only one type which is equated directly with truth values then this is essentially a propositional logic. If the standard quantifiers and connectives are used, there is only one type, and one of the relations is $=$ (with its usual semantics), this produces first order logic. If the standard quantifiers and connectives are used, there are two types, and the relations include $=$ and $\in$ with appropriate semantics, this is second order logic (a slightly different formulation replaces $\in$ with a $2$-ary function which represents function application; this views second order objects as functions rather than sets). Note that often connectives are written with infix notation with parentheses used to control order of operations.