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\begin{document}

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\section*{abstract model theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{link}{Link}\dotfill \pageref*{link} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Abstract model theory} is the study of the general properties of the [[model theory]] of extensions of (classical untyped) [[first-order logic]].

Originally motivated by \emph{[[Lindström's theorem]]} that characterizes first-order logic, the field has subsequently been extended to provide alternative characterizations and include different logics within its range.

The basic concept of abstract model theory is that of an \emph{abstract logic} which is a triple $\mathcal{L}=(S,\Phi ,\models)$ where $\models$ is a binary relation between the class of $\mathcal{L}$-`structures' $S$ and the class of $\mathcal{L}$-`sentences' $\Phi$ to be thought of as minimalistic version of the \emph{satisfaction} relation.

\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

As the logical relations studied by abstract model theory are of a functorial nature, some category theory entered the picture already in Barwise (1974). The theory of [[institution|institutions]], aka \emph{institution-independent model theory} (Diaconescu 2008), constitutes abstract categorical model theory proper. In a similar abstract categorical vein is the [[geometric theory\#FunctorialDefinition|functorial approach to geometric theories]] described in Johnstone (2002, sec. B4.2).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[model theory]]


\item [[Lindström's theorem]]


\item [[compactness theorem]]


\item [[Löwenheim-Skolem theorem]]


\item [[institution]]


\item [[Per Lindström]]



\end{itemize}
\hypertarget{link}{}\subsection*{{Link}}\label{link}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Abstract_model_theory}{Abstract model theory}}

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Jon Barwise, \emph{Axioms for abstract model theory} , Annals of Mathematical Logic 7 pp.221-265, 1974.


\item Barwise, Feferman (eds.), \emph{Model-theoretic Logics} , Springer Heidelberg 1985 (freely available online: \href{http://projecteuclid.org/euclid.pl/1235417263#toc}{toc}) .


\item Răzvan Diaconescu, \emph{Institution-independent Model Theory} , Birkh\"a{}user Basel 2008.


\item Marta Garc\'i{}a-Matos, [[Jouko Väänänen]], \emph{Abstract Model Theory as a Framework for Universal Logic} , Logica Universalis 2005 pp.1-33 (\href{http://www.math.helsinki.fi/logic/opetus/lt/unilog2005_proofs.pdf}{draft}) .


\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol. I} , Oxford UP 2002.


\item [[Jouko Väänänen]], \emph{The Craig Interpolation Theorem and abstract model theory} , Synthese 164:401 (2008) (freely available online)



\end{itemize}


\end{document}