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\begin{document}

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\section*{true proposition}

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

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{in_classical_logic}{In classical logic}\dotfill \pageref*{in_classical_logic} \linebreak
\noindent\hyperlink{in_constructive_logic}{In constructive logic}\dotfill \pageref*{in_constructive_logic} \linebreak
\noindent\hyperlink{in_linear_logic}{In linear logic}\dotfill \pageref*{in_linear_logic} \linebreak
\noindent\hyperlink{in_a_topos}{In a topos}\dotfill \pageref*{in_a_topos} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{in_homotopy_type_theory}{In homotopy type theory}\dotfill \pageref*{in_homotopy_type_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_the_topos_}{In the topos $Set$}\dotfill \pageref*{in_the_topos_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In [[logic]], the \textbf{true} proposition, or \textbf{truth}, is the [[proposition]] which is always true.

The truth is commonly denoted $true$, $T$, $\top$, or $1$. These may be pronounced `true' even where it would be ungrammatical for an adjective to appear in ordinary English.

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

\hypertarget{in_classical_logic}{}\subsubsection*{{In classical logic}}\label{in_classical_logic}

In [[classical logic]], there are two [[truth values]]: true and [[false]]. Classical logic is perfectly symmetric between truth and falsehood; see [[de Morgan duality]].

\hypertarget{in_constructive_logic}{}\subsubsection*{{In constructive logic}}\label{in_constructive_logic}

In [[constructive logic]], $true$ is the [[top element]] in the [[poset]] of [[truth values]].

Constructive logic is still [[two-valued logic|two-valued]] in the sense that any truth value which is not true is [[false]].

\hypertarget{in_linear_logic}{}\subsubsection*{{In linear logic}}\label{in_linear_logic}

In [[linear logic]], there is both \emph{additive} truth, denoted $\top$, and \emph{multiplicative} truth, denoted $1$. As the notation suggests, it is $\top$ that is the [[top element]] of the lattice of linear truth values. (In particular, $1 \vdash \top$ but $\top \nvdash 1$.)

\hypertarget{in_a_topos}{}\subsubsection*{{In a topos}}\label{in_a_topos}

In terms of the [[internal logic]] of a [[topos]] (or other [[category]]), $true$ is the [[top element]] in the [[poset of subobjects]] of any given [[object]] (where each object corresponds to a [[context]] in the internal language).

However, not every topos is [[two-valued topos|two-valued]] (even if it is [[boolean topos|boolean]], so there may be other truth values besides $true$ and $false$.

\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

In [[type theory]] with [[propositions as types]], truth is represented by the [[unit type]].

\hypertarget{in_homotopy_type_theory}{}\subsubsection*{{In homotopy type theory}}\label{in_homotopy_type_theory}

In [[homotopy type theory]], truth is represented by any [[contractible type]].

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

\hypertarget{in_the_topos_}{}\subsubsection*{{In the topos $Set$}}\label{in_the_topos_}

In the archetypical [[topos]] [[Set]], the terminal object is the [[singleton]] [[set]] $\{*\}$ (the [[point]]) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$. Then truth is the singleton set $\{*\}$, seen as the [[improper subset]] of itself. (See \href{http://ncatlab.org/nlab/show/internal+logic#LogicOfSet}{Internal logic of Set} for more details).

The same is true in the archetypical [[(∞,1)-topos]] [[∞Grpd]]. From that perspective it makes good sense to think of

\begin{itemize}%
\item a set as a 0-[[truncated]] $\infty$-groupoid: a [[0-groupoid]];


\item a [[subsingleton]] set as a $(-1)$-[[truncated]] $\infty$-groupoid: a [[(?1)-groupoid]];


\item the singleton set as the $(-2)$-[[truncated]] $\infty$-groupoid: the unique (up to equivalence) [[(?2)-groupoid]].



\end{itemize}
In this sense, the object $true$ in [[Set]] or [[∞Grpd]] may canonically be thought of as being [[generalized the|the]] unique [[(?2)-groupoid]].

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

\begin{itemize}%
\item \textbf{true}, [[unit type]]


\item [[false]], [[empty type]]


\item [[contradiction]], [[inconsistency]], [[consistency]]


\item [[correspondence theory of truth]]



\end{itemize}
[[!include homotopy n-types - table]]

[[!redirects true]] [[!redirects True]] [[!redirects true proposition]] [[!redirects true propositions]] [[!redirects true statement]] [[!redirects true statements]] [[!redirects true sentence]] [[!redirects true sentences]]

[[!redirects truth]] [[!redirects Truth]] [[!redirects the truth]]



\end{document}