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\section*{coherent logic}

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{coherent_logic}{}\section*{{Coherent logic}}\label{coherent_logic}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\textbf{Coherent logic} is a fragment of (finitary) [[first-order logic]] which allows only the connectives and quantifiers

\begin{itemize}%
\item $\wedge$ ([[and]]),


\item $\vee$ ([[or]]),


\item $\top$ ([[true]]),


\item $\bot$ ([[false]]),


\item $\exists$ ([[existential quantifier]]).



\end{itemize}
A \textbf{coherent formula} is a [[formula]] in coherent logic.

A \textbf{coherent sequent} is a [[sequent]] of the form $\varphi \vdash \psi$, where $\varphi$ and $\psi$ are coherent formulas, possibly with [[free variables]] $x_1,\dots,x_n$.

In full first-order logic, such a sequent is equivalent to the single formula

\begin{displaymath}
\forall x_1, \dots, \forall x_n (\varphi \Rightarrow \psi)
\end{displaymath}
(in the empty [[context]]). Of course, this latter formula is not coherent, but this shows that when we deal with coherent \emph{sequents} rather than merely formulas, it can also be thought of as allowing one instance of $\Rightarrow$ and a string of $\forall$s at the very outer level of a formula.

Coherent logic (including sequents, as above) is the [[internal logic]] of a [[coherent category]]. The [[classifying topos]] of a coherent theory is a [[coherent topos]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Any (finitary) [[algebraic theory]] is coherent.


\item A good example of a coherent theory that is not [[algebraic theory|algebraic]] (in any of the usual senses, although it comes from algebra) is the theory of a [[local ring]]; a similar example is the theory of a [[discrete field]].


\item The theory of a [[total order]] is coherent, though also not algebraic. The theory of a [[partial order]] is [[essentially algebraic theory|essentially algebraic]], but the totality axiom $\vdash_{x,y} (x\le y) \vee (y\le x)$ is coherent but not essentially algebraic.


\item The theory of a [[linear order]] is (seemingly) not coherent if we use the ``connectedness'' axiom $(x\nless y), (y\nless x) \vdash (x=y)$, which is not coherent since negation is not allowed in coherent formulas. We can express one outer negation, however, as in the irreflexivity axiom $(x\lt x)\vdash \bot$. Another solution is to use the ``trichotomy'' axiom $\top \vdash (x=y) \vee (x\lt y) \vee (y\lt x)$ instead, in order to get an axiomatisation of ``coherent'' linear orderings.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Coherent logic has many pleasing properties.

\begin{itemize}%
\item Every finitary [[first-order logic|first-order theory]] is equivalent, over [[classical logic]], to a coherent theory. This theory is called its \emph{Morleyization} and can be obtained by adding new relations representing each first-order formula and its negation, with axioms that guarantee (over classical logic) these relations are interpreted correctly (using the facts that $(P\Rightarrow Q) \dashv\vdash (\neg P \vee Q)$ and $(\forall x, P) \dashv\vdash (\neg \exists x, \neg P)$ in classical logic). See D1.5.13 in [[Sketches of an Elephant]], or Prop. 3.2.8 in Makkai-Par\'e{}.


\item By (one of the theorems called) [[Deligne completeness theorem|Deligne's theorem]], every [[coherent topos]] has [[point of a topos|enough points]]. In particular, this applies to the classifying toposes of coherent theories. It follows that models in [[Set]] are sufficient to detect provability in coherent logic. By Morleyization, we can obtain from this the classical [[completeness theorem for first-order logic]]. See for instance 6.2.2 in Makkai-Reyes.


\item Coherent logic also satisfies a \emph{definability theorem}: if a relation can be constructed in every [[Set]]-model of a coherent theory $T$, in a [[natural transformation|natural]] way, then that relation is named by some coherent formula in $T$. See chapter 7 of Makkai-Reyes or D3.5.1 in [[Sketches of an Elephant]].


\item It follows that if a morphism of coherent theories (i.e. an interpretation of one coherent theory in another) induces an equivalence of their categories of models in $Set$, then it is a [[Morita equivalence]] of theories (i.e. induces an equivalence of classifying toposes, hence an equivalence of categories of models in all [[Grothendieck toposes]]). This is called [[conceptual completeness]]; see 7.1.8 in Makkai-Reyes or D3.5.9 in [[Sketches of an Elephant]]. (Note, though, that two coherent theories can have equivalent categories of models in $Set$ without being Morita equivalent, if the former equivalence is not induced by a morphism of theories; see for instance D3.5.4 in the Elephant.)



\end{itemize}
However, here is a property which one might expect coherent theories to have, but which they do not.

\begin{itemize}%
\item The category of models of a coherent theory (in [[Set]]) is always an [[accessible category]] (this is true more generally for models of any logical theory). However, it need not be \emph{finitely} accessible (i.e. $\omega$-accessible). An example is given in Adamek-Rosicky 5.36: the theory of sets equipped with a binary relation $\prec$ such that for all $x$ there exists a $y$ with $x\prec y$. Then the model $(\mathbb{N},\lt)$ of this theory does not admit any morphism from an ($\omega$-)compact object. (However, many coherent theories do have a finitely accessible category of $Set$-models.)

\end{itemize}
\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

\begin{itemize}%
\item Sometimes coherent logic is called \emph{geometric logic}, but that term is now more commonly used for the analogous fragment of infinitary logic which allows disjunctions over arbitrary sets (though still only finitary conjunctions). See [[geometric logic]].


\item Conversely, geometric logic is sometimes called \emph{coherent} , e.g. in Reyes (\hyperlink{Reyes}{1977}), so that coherent logic in the nLab terminology corresponds to the \emph{finitary} fragment only.


\item Occasionally the [[existential quantifiers]] in coherent logic are further restricted to range only over \emph{finitely presented types}.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[finitely complete category]], [[cartesian functor]], [[cartesian logic]], [[cartesian theory]]


\item [[regular category]], [[regular functor]], [[regular logic]], [[regular theory]], [[regular coverage]], [[regular topos]]


\item [[coherent category]], [[coherent functor]], \textbf{coherent logic}, [[coherent theory]], [[coherent coverage]], [[coherent topos]]


\item [[geometric category]], [[geometric functor]], [[geometric logic]], [[geometric theory]]



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

A survey of results on geometric and coherent logic is in

\begin{itemize}%
\item [[Gonzalo Reyes]], \emph{Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi} , Cah. Top. Diff. G\'e{}o. Cat. \textbf{XVIII} no.2 (1977) pp.405-437. (\href{http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0}{numdam})

\end{itemize}
A standard textbook account of coherent logic (called `geometric logic' there) can be found in

\begin{itemize}%
\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994.

\end{itemize}
Properties of the generic model of a coherent theory are investigated in

\begin{itemize}%
\item Marc Bezem, [[Ulrik Buchholtz]], [[Thierry Coquand]], \emph{Syntactic Forcing Models for Coherent Logic} , arXiv:72.07743 (2017). (\href{https://arxiv.org/abs/1712.07743}{abstract})

\end{itemize}
Else consider the monographs

\begin{itemize}%
\item [[Michael Makkai]] and [[Gonzalo Reyes]], \emph{First Order Categorical Logic: Model-theoretical methods in the theory of topoi and related categories}, Springer-Verlag, 1977.


\item [[Michael Makkai]] and [[Robert Paré]], \emph{Accessible categories}


\item [[Peter Johnstone]], [[Sketches of an Elephant]] vol 2. Part D


\item [[Jiri Adamek]] and [[Jiri Rosicky]], \emph{Locally presentable and accessible categories}



\end{itemize}
[[!redirects coherent logic]] [[!redirects coherent logics]] [[!redirects coherent theory]] [[!redirects coherent theories]] [[!redirects coherent formula]] [[!redirects coherent formulas]] [[!redirects coherent sequent]] [[!redirects coherent sequents]] [[!redirects Morleyization]]



\end{document}