FirstOrderLogic 2002-08-28 22:11:13 2003-12-02 14:29:28 Definition % 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} A logic is \emph{first order} if it has exactly one type. Usually the term refers specifically to the logic with connectives $\neg$, $\vee$, $\wedge$, $\rightarrow$, and $\leftrightarrow$ and the quantifiers $\forall$ and $\exists$, all given the usual semantics: \begin{itemize} \item $\neg\phi$ is true iff $\phi$ is not true \item $\phi\vee\psi$ is true if either $\phi$ is true or $\psi$ is true \item $\forall x\phi(x)$ is true iff $\phi^t_x$ is true for every object $t$ (where $\phi^t_x$ is the result of replacing every unbound occurrence of $x$ in $\phi$ with $t$) \item $\phi\wedge\psi$ is the same as $\neg(\neg\phi\vee\neg\psi)$ \item $\phi\rightarrow\psi$ is the same as $(\neg\phi)\vee\psi$ \item $\phi\leftrightarrow\psi$ is the same as $(\phi\rightarrow\psi)\wedge(\psi\rightarrow\phi)$ \item $\exists x\phi(x)$ is the same as $\neg\forall x\neg\phi(x)$ \end{itemize} However languages with slightly different quantifiers and connectives are sometimes still called first order as long as there is only one type.