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\section*{theory}

\begin{quote}%
This entry is about the notion in [[logic]]. For the notion of the same name in [[physics]] see at \emph{[[theory (physics)]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

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

[[!include type theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{SyntacticView}{Syntactic view}\dotfill \pageref*{SyntacticView} \linebreak
\noindent\hyperlink{SemanticView}{Semantic view}\dotfill \pageref*{SemanticView} \linebreak
\noindent\hyperlink{CategoricalView}{Categorical view}\dotfill \pageref*{CategoricalView} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classes_of_theories}{Classes of theories}\dotfill \pageref*{classes_of_theories} \linebreak
\noindent\hyperlink{hierarchy_of_theories_cartesian_regular_coherent_geometric}{Hierarchy of theories: cartesian, regular, coherent, geometric}\dotfill \pageref*{hierarchy_of_theories_cartesian_regular_coherent_geometric} \linebreak
\noindent\hyperlink{abelian_theories}{Abelian theories}\dotfill \pageref*{abelian_theories} \linebreak
\noindent\hyperlink{SpecificExamples}{Specific examples}\dotfill \pageref*{SpecificExamples} \linebreak
\noindent\hyperlink{models_for_a_theory}{Models for a theory}\dotfill \pageref*{models_for_a_theory} \linebreak
\noindent\hyperlink{settheoretic_models_for_a_firstorder_theory_in_syntactic_approach}{Set-theoretic models for a first-order theory in syntactic approach}\dotfill \pageref*{settheoretic_models_for_a_firstorder_theory_in_syntactic_approach} \linebreak
\noindent\hyperlink{categorical_point_of_view_and_models_in_topoi}{Categorical point of view and models in topoi}\dotfill \pageref*{categorical_point_of_view_and_models_in_topoi} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[mathematical logic]], a \emph{theory} is a formal [[language]] used to precisely [[axiom|axiomatize]] a certain class of [[models]].

In principle also all other notions of \emph{theory}, such as in the sense of [[physics]] should be special cases of this, but in practice of course there are many systems called ``theories'' which are not (yet) as fully formalized as in mathematical logic.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are several different viewpoint on theories:

\begin{itemize}%
\item The

\hyperlink{SyntacticView}{syntactic view}

is that the theory itself consists of a set of [[formulas]] in the first order language $Lang(\Sigma)$ of a [[signature (in logic)|signature]] $\Sigma$. Classically, these formulas are assumed to have no [[free variables]] (i.e. to be ``sentences''), but in weaker logics that lack [[universal quantification]] it is better to take them to be formulas-in-[[context]]. One also sometimes considers the theory to also include all logical consequences (aka theorems) of the axioms in $A$, relative to (some specified) fragment of [[first-order logic]] --- that is, to be ``saturated'' with respect to provability.


\item The

\hyperlink{SemanticView}{semantic view}

is that the theory is given by the class of its [[model]]s appropriate to that fragment of logic. G\"o{}del's [[completeness theorem]] is that a sentence in $\mathcal{T}$ is a theorem iff it is satisfied in every model.


\item The

\hyperlink{CategoricalView}{categorical view}

is that the logical formalism of a theory $\mathcal{T}$ can frequently be embodied in a [[syntactic category]] of terms $C_{\mathcal{T}}$, so that models of a theory $\mathcal{T}$ are identified with [[functor]]s

\begin{displaymath}
C_{\mathcal{T}} \to Set
\end{displaymath}
that preserve some (typically [[stuff, structure, property|property-like]]) [[stuff, structure, property|structures]] on $C_{\mathcal{T}}$, such as certain classes of [[colimit]]s or of [[limits]], pertinent to the fragment of logic at hand. Then a completeness theorem would be the statement that the canonical map

\begin{displaymath}
C_{\mathcal{T}} \to \prod_{models F in Set} Set
\end{displaymath}
is a [[full and faithful functor|full faithful embedding]] (one that preserves all relevant logical structure). For this reason, completeness theorems are also known as \emph{embedding theorems}.



\end{itemize}
Hm, is that the way it should be said?

In fact, the notion of model can be generalized away from $Set$ to more general [[categories]], namely those that have enough structure to ``internalize'' the fragment of logic at hand. From this very general point of view on model, the syntactic category $C_{\mathcal{T}}$ is the generic or universal model for $\mathcal{T}$, and if we simply call $C_{\mathcal{T}}$ the theory, then models and theories are placed on the same footing.

In this article we mostly consider the categorical view on ``theory''.

\hypertarget{SyntacticView}{}\subsubsection*{{Syntactic view}}\label{SyntacticView}

\begin{udefn}
In first-order [[logic]], a \textbf{theory} $\mathcal{T}$ is presented by

\begin{enumerate}%
\item a [[signature (in logic)|signature]] $\Sigma$, specifying the allowed [[type]]s of variables and the [[function]]s and [[relation]]s between these;


\item a set $A$ of [[sequent]]s of [[formula]]s in this signature, called the \textbf{axioms} of the theory: expressed in the first-order language $Lang(\Sigma)$ with equality generated by $\Sigma$



\end{enumerate}
\end{udefn}
For instance (\hyperlink{Johnstone}{Johnstone, def. D1.1.6}).

\hypertarget{SemanticView}{}\subsubsection*{{Semantic view}}\label{SemanticView}

(\ldots{})

\hypertarget{CategoricalView}{}\subsubsection*{{Categorical view}}\label{CategoricalView}

(\ldots{})

\begin{itemize}%
\item [[categorical semantics]]

\end{itemize}
(\ldots{})

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

\hypertarget{classes_of_theories}{}\subsubsection*{{Classes of theories}}\label{classes_of_theories}

There are many different kinds of ``theory'' depending on the strength of the ``logic'': a by-no-means complete list includes

\begin{itemize}%
\item [[equational logic]],


\item [[first-order theory]]


\item [[Horn theory]],


\item [[essentially algebraic theory]],


\item [[geometric logic]],


\item [[regular logic]],


\item [[exact logic]],


\item [[extensive logic]],


\item [[coherent logic]],


\item [[first-order logic]] (aka pretopos logic),



\end{itemize}
and corresponding theories for these logics.

\begin{itemize}%
\item [[essentially algebraic theory]]


\item [[generalized algebraic theory]]


\item [[algebraic theory]]

\begin{itemize}%
\item [[commutative algebraic theory]]


\item finitary algebraic theory: [[Lawvere theory]]

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

\end{itemize}


\end{itemize}


\end{itemize}
\hypertarget{hierarchy_of_theories_cartesian_regular_coherent_geometric}{}\paragraph*{{Hierarchy of theories: cartesian, regular, coherent, geometric}}\label{hierarchy_of_theories_cartesian_regular_coherent_geometric}

There is a hierarchy of theories that can be interpreted in the [[internal logic]] of a hierarchy of types of [[categories]]. Since [[predicates]] in the internal logic are represented by [[subobjects]], in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved. For instance:

\begin{itemize}%
\item \textbf{cartesian theories} Since $\wedge$ and $\top$ are represented by [[intersections]] and identities (top elements), these can be interpreted in any [[lex category]]. Theories involving only these are [[cartesian theory|cartesian theories]].


\item \textbf{regular theories} Since $\exists$ is represented by the [[image]] of a subobject, it can be interpreted in any [[regular category]]. Theories involving only $\wedge$, $\top$, and $\exists$ are [[regular theory|regular theories]].


\item \textbf{coherent theories} Since $\vee$ and $\bot$ are represented by [[union]] and [[bottom elements]], these can be interpreted in any [[coherent category]]. Theories which add these to regular logic are called [[coherent theory|coherent theories]].


\item \textbf{geometric theories} Finally, theories which also involve infinitary $\bigvee$, which is again represented by an infinitary union, can be interpreted in [[infinitary coherent categories]], aka \emph{[[geometric categories]]}. These are \textbf{[[geometric theories]]}.



\end{itemize}
Note that the axioms of one of these theories are actually of the form

\begin{displaymath}
\varphi \vdash_{\vec{x}} \psi
\end{displaymath}
where $\varphi$ and $\psi$ are formulas involving only the specified connectives and quantifiers, $\vdash$ means entailment, and $\vec{x}$ is a [[context]]. Such an axiom can also be written as

\begin{displaymath}
\forall \vec{x}. (\varphi \Rightarrow \psi)
\end{displaymath}
so that although $\Rightarrow$ and $\forall$ are not strictly part of any of the above logics, they can be applied ``once at top level.'' In an axiom of this form for geometric logic, the formulas $\varphi$ and $\psi$ which must be built out of $\top$, $\wedge$, $\bot$, $\bigvee$, and $\exists$ are sometimes called \emph{positive} formulas.

\hypertarget{abelian_theories}{}\subsubsection*{{Abelian theories}}\label{abelian_theories}

Interestingly, one form of logic which made an early appearance but is not ordinarily thought of as logic at all is the logic of [[abelian categories]], which is characterized by certain exactness properties. Here a small abelian category $A$ can be thought of as a syntactic site for some ``abelian theory''; models of the theory are exact additive functors with domain $A$. The classical models would in fact be exact additive functors $A \to Ab$, or exact additive functors to a category of modules. A ``Freyd-Heron-Lubkin-Mitchell'' embedding theorem is then a completeness theorem with respect to the classical models, and assures us that a statement in the language of abelian category theory is provable if and only if it is true when interpreted in any module category.

\hypertarget{SpecificExamples}{}\subsubsection*{{Specific examples}}\label{SpecificExamples}

The simplest nontrivial theory is the

\begin{itemize}%
\item [[theory of objects]]

\end{itemize}
A still pretty simple but very nontrivial theory is

\begin{itemize}%
\item [[elementary function arithmetic]]


\item [[second order arithmetic]]



\end{itemize}
The theories

\begin{itemize}%
\item $Th(Cat)$ of [[categories]]


\item $Th(Lex)$ of [[finitely complete categories]]


\item $Th(Topos)$ of [[elementary toposes]]


\item $Th(ETCS)$ of [[sets]] ([[ETCS]])



\end{itemize}
are discussed at \emph{[[fully formal ETCS]]}.

\hypertarget{models_for_a_theory}{}\subsection*{{Models for a theory}}\label{models_for_a_theory}

\hypertarget{settheoretic_models_for_a_firstorder_theory_in_syntactic_approach}{}\subsubsection*{{Set-theoretic models for a first-order theory in syntactic approach}}\label{settheoretic_models_for_a_firstorder_theory_in_syntactic_approach}

The basic concept is of a structure for a first-order language $L$: a set $M$ together with an interpretation of $L$ in $M$. A theory is specified by a language and a set of sentences in $L$. An $L$-structure $M$ is a \textbf{model} of $T$ if for every sentence $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true (``$\phi$ holds in $M$''). We say that $T$ is \textbf{consistent} or satisfiable (relative to the universe in which we do [[model theory]]) if there exist at least one model for $T$ (in our universe). Two theories, $T_1$, $T_2$ are said to be \textbf{equivalent} if they have the same models.

Given a class $K$ of structures for $L$, there is a theory $Th(K)$ consisting of all sentences in $L$ which hold in every structure from $K$. Two structures $M$ and $N$ are \textbf{elementary equivalent} (sometimes written by equality $M=N$, sometimes said ``elementarily equivalent'') if $Th(M)=Th(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $Th(K)$ is called a \textbf{set of axioms} of $K$. A theory is said to be \textbf{finitely axiomatizable} if there exist a finite set of axioms for $K$.

A theory is said to be \textbf{complete} if it is equivalent to $Th(M)$ for some structure $M$.

\hypertarget{categorical_point_of_view_and_models_in_topoi}{}\paragraph*{{Categorical point of view and models in topoi}}\label{categorical_point_of_view_and_models_in_topoi}

From the categorical point of view, for every theory $T$ there exists a [[category]] $C_T$ -- the \textbf{[[syntactic category]]} $C_T$ -- such that a model for $T$ is a [[functor]] $C_T \to \mathcal{T}$ into some [[topos]] $T$, satisfying certain conditions.

For instance the syntactic categories of [[Lawvere theory|Lawvere theories]] are precisely those categories that have finite cartesian [[product]]s and in which every object is [[isomorphic]] to a finite cartesian power $x^n$ of a distinguished object $x$. A model for a Lawvere theory is precisely a finite product preserving functor $C_T \to \mathcal{T}$.

We say a functor $\mathcal{T}_1 \to \mathcal{T}_2$ of toposes (for instance a [[logical morphism]] or a [[geometric morphism]]) \emph{preserves a theory} $T$ if for every model $C_T \to \mathcal{T}_1$ of $T$ in $\mathcal{T}_1$, the composite $C_T \to \mathcal{T}_1 \to \mathcal{T}_2$ is a model of $T$ in $\mathcal{T}_2$.

For instance, every [[geometric morphism]] preserves every [[Lawvere theory]] since, being a [[right adjoint]], it preserves [[limit]]s, hence finite products.

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

\begin{itemize}%
\item [[algebraic theory]], [[globular theory]]


\item [[type theory]]


\item [[theory of presheaf type]]


\item [[ordinal analysis]]



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

\begin{itemize}%
\item H. Jerome Keisler, \emph{Fundamentals of model theory}, A2 in JOhn Barwise, Handbook of mathematical logic (pp.47-103), North Holland 1977

\end{itemize}
A standard textbook reference for the [[categorical semantics]] is section D of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
A discussion of the relation between theories and their [[syntactic categories]] is at

\begin{itemize}%
\item n-category cafe 2010: \href{http://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html}{what is a theory}

\end{itemize}
Other references include

\begin{itemize}%
\item [[Mamuka Jibladze]], [[Teimuraz Pirashvili]], \emph{Cohomology of algebraic theories}, J. Algebra \textbf{137} (1991), no. 2, 253--296, 


\item wikipedia: \href{http://en.wikipedia.org/wiki/List_of_first-order_theories}{list of first-order theories}, \href{http://en.wikipedia.org/wiki/Morley%27s_categoricity_theorem}{Morley's categoricity theorem}, \href{http://en.wikipedia.org/wiki/Decidability_%28logic%29}{Decidability (logic)}, \href{http://en.wikipedia.org/wiki/Signature_%28logic%29}{signature (logic)}



\end{itemize}
In [[Coq]] theories are specified with the

\begin{itemize}%
\item \emph{[[Gallina specification language]]}.

\end{itemize}
[[!redirects theories]]

[[!redirects theory (logic)]]



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