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\begin{document}

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\section*{continuous logic}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_terms_of_enriched_category_theory}{In terms of enriched category theory}\dotfill \pageref*{in_terms_of_enriched_category_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Continuous logic is a logic whose truth values can take continuous values in $[0,1]$. The main variant used in model theory is motivated by the model theory of Banach spaces and similar structures. The language has connectives for each continuous function $c: [0,1]^n \to [0,1]$ and the quantifiers are interpreted as infimum and supremum. The models of this logic are bounded complete metric structures equipped with uniformly continuous maps and $[0,1]$-valued predicates.

As well as satisfying a form of [[completeness theorem]],

\begin{quote}%
\ldots{}continuous first-order logic satisfies suitably phrased forms of the compactness theorem, the L\"o{}wenheim-Skolem theorems, the diagram arguments, Craig's interpolation theorem, Beth's definability theorem, characterizations of quantifier elimination and model completeness, the existence of saturated and homogeneous models results, the omitting types theorem, fundamental results of stability theory, and nearly all other results of elementary model theory. (\hyperlink{YaacovPed10}{Yaacov \&{} Pedersen 10})


\end{quote}
\hypertarget{in_terms_of_enriched_category_theory}{}\subsection*{{In terms of enriched category theory}}\label{in_terms_of_enriched_category_theory}

Building on the proposal of [[Lawvere]] to understand a form of \href{metric+space#LawvereMetricSpace}{metric space} as a category [[enriched category theory|enriched]] in the [[monoidal category|monoidal]] [[poset]] $([0, \infty], \geq)$, there have been attempts to consider continuous logic as a similarly enriched logic. In (\hyperlink{AlbertHart}{Albert \&{} Hart}), the authors develop a parallel for [[conceptual completeness]] via the notion of a \emph{metric} [[pretopos]].

[[Simon Cho]] argues that the object of truth values of continuous logic, $[0,1]$, should be seen as a ``continuous subobject classifier'' in a similar sense to the [[subobject classifier]] of topos theory where subobjects are classified via [[pullbacks]] (\hyperlink{Cho19}{Cho 19}).

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

Continuous logic is introduced in

\begin{itemize}%
\item C. W. Henson, J. Iovino, \emph{Ultraproducts in analysis} in: Analysis and Logic, London Math. Soc. Lecture Note Series \textbf{262}, Cambridge University Press 2002.

\end{itemize}
motivated by

\begin{itemize}%
\item Chen-Chung Chang, Jerome H. Keisler, \emph{Continuous Model Theory}, Annals of Mathematics Studies \textbf{58}, 1966.

\end{itemize}
A recent version is in

\begin{itemize}%
\item [[Itaï Ben Yaacov]], \emph{Uncountable dense categoricity in cats}, J. Symb. Logic \textbf{70}, 829--860, 2005
\item [[Itaï Ben Yaacov]], \emph{Continuous first-order logic and logical stability}, \href{http://math.univ-lyon1.fr/~begnac/articles/cfo.pdf}{pdf}
\item [[Itaï Ben Yaacov]], Arthur Pedersen, \emph{A proof of completeness for continuous first order logic}, Journal of Symbolic Logic 75, 168–190, 2010, (\href{https://arxiv.org/abs/0903.4051}{arXiv:0903.4051}).
\item [[Itaï Ben Yaacov]], Alex Usvyatsov. \emph{Logic of metric spaces and Hausdorff CATs}
\item [[Itaï Ben Yaacov]], Alexander Berenstein, Ward C. Henson, Alexander Usvyatsov, \emph{Model Theory for metric structures}, in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press, 2008, \href{http://matematicas.uniandes.edu.co/~aberenst/mtfms.pdf}{pdf}
\item Alexander Berenstein, Andres Villaveces, \emph{Hilbert spaces with random predicates}, \href{http://matematicas.uniandes.edu.co/~aberenst/amalg10.pdf}{pdf}

\end{itemize}
A discussion of [[conceptual completeness]] in the setting of continuous logic is found in

\begin{itemize}%
\item Jean-Martin Albert, Bradd Hart. \emph{Metric logical categories and conceptual completeness for first order continuous logic}, (\href{https://arxiv.org/abs/1607.03068}{arXiv:1607.03068})

\end{itemize}
A treatment of metric space semantics for continuous logic as a variety of [[enriched categories]] is given in

\begin{itemize}%
\item [[Simon Cho]], \emph{Categorical semantics of metric spaces and continuous logic}, (\href{https://arxiv.org/abs/1901.09077}{arXiv:1901.09077})

\end{itemize}
See also his thesis

\begin{itemize}%
\item [[Simon Cho]], \emph{Continuity in enriched categories and metric model theory}, (\href{http://math.lsa.umich.edu/~simoncho/thesis.pdf}{thesis})

\end{itemize}
For [[syntax-semantics duality]] in the case of [[infinitary logic|infinitary]] continuous logic, see

\begin{itemize}%
\item Ruiyuan Chen, \emph{Representing Polish groupoids via metric structures}, (\href{https://arxiv.org/abs/1908.03268}{arXiv:1908.03268})

\end{itemize}
An older and rather \emph{different system} also called continuous logic of a Russian school is surveyed

\begin{itemize}%
\item Vitaly I. Levin, \emph{Basic concepts of continuous logic}, Studies in logic grammar and rethoric, 11 (24) 2007, \href{http://math.univ-lyon1.fr/~begnac/articles/cfo.pdf}{pdf}

\end{itemize}
There is a version of [[abstract elementary classes]] in the setting of continuous logic, [[metric abstract elementary class]]es.



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