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\begin{document}

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\section*{proposition}

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

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\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{predicates}{Predicates}\dotfill \pageref*{predicates} \linebreak
\noindent\hyperlink{in_categorytheoretic_logic}{In category-theoretic logic}\dotfill \pageref*{in_categorytheoretic_logic} \linebreak
\noindent\hyperlink{InTypeTheory}{In type theory}\dotfill \pageref*{InTypeTheory} \linebreak
\noindent\hyperlink{propositional_and_predicate_logic}{Propositional and predicate logic}\dotfill \pageref*{propositional_and_predicate_logic} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In [[logic]], a \textbf{proposition} is intended to be interpreted [[semantics|semantically]] as having a [[truth value]]. In modern logic, it's cleanest to start by specifying a [[context]] and considering the propositions \emph{in} that context.

\hypertarget{predicates}{}\subsection*{{Predicates}}\label{predicates}

If (in a given context $\Gamma$) we have a [[type]] $A$, then we may extend $\Gamma$ to a context $\Delta \coloneqq \Gamma, x\colon A$ (assuming that the [[variable]] $x$ is not otherwise in use). We may then think of any proposition in $\Delta$ as a \textbf{predicate} $P$ in $\Gamma$ with the \textbf{free variable} $x$ of type $A$; this generalises to more complicated extensions of contexts (say by several variables).

If $P$ is a predicate with free variable $x$ of type $A$ and $t$ is a [[term]] of type $A$, then we get a proposition $P[t/x]$ by substituting $t$ for every instance of $x$ in $P$. Conversely, any proposition $Q$ may be interpreted as a predicate $Q[\hat{x}]$ in which the free variable $x$ simply doesn't appear. (We have $Q[\hat{x}][t/x] = Q$ for every term $t$.)

There is a more traditional approach of viewing a predicate as a [[function]] from terms to propositions, a \textbf{propositional function}. Then $P[t/x]$ is written $P(t)$, while $P$ itself from above is written $P(x)$ (since a variable is a term). In this approach, less care is usually taken with the context, so that $Q[\hat{x}]$ may be conflated with $Q$ (since $Q[\hat{x}](x) = Q$, or this would be so if $x$ were a term in $\Gamma$ instead of only in $\Delta$).

\hypertarget{in_categorytheoretic_logic}{}\subsection*{{In category-theoretic logic}}\label{in_categorytheoretic_logic}

In [[category-theoretic logic|categorial logic]]/[[categorical semantics]], we have a [[category]] $\mathcal{C}$ and a [[class of monomorphisms]] (often all [[monomorphisms]]) $\mathcal{M}$ in $\mathcal{C}$. Then a \textbf{context} is an [[object]] of $\mathcal{C}$ and a \textbf{proposition} in the context $\Gamma$ is an $\mathcal{M}$-[[subobject]] of $\Gamma$. We also have a class of [[display maps]] (often all [[morphisms]] in $\mathcal{C}$) such that $\mathcal{M}$ is closed under [[pullbacks]] both along display maps and along [[sections]] of display maps. These two ways of pulling back propositions in one context to propositions in another context correspond (respectively) to forming $Q[\hat{x}]$ and $P[t/x]$.

More specifically, if $\mathcal{C}$ is a [[finitely complete category]], then the objects of $\mathcal{C}$ may equivalently be viewed as contexts and as types in the [[internal language]] of $\mathcal{C}$; a morphism from $\Gamma$ to $A$ is a term of type $A$ in context $\Gamma$. The extension of $\Gamma$ by a variable $x$ of type $A$ is the [[product]] $\Gamma \times A$, and the display map to $\Gamma$ is simply the projection. Every term $t\colon \Gamma \to A$ defines a section of this display map, and we may literally construct $Q[\hat{x}]$ and $P[t/x]$ as pullbacks.

If $\mathcal{C}$ is even a [[topos]], then a proposition $Q$ in $\Gamma$ may be identified with a term whose type is the [[subobject classifier]] $\Omega$, and the predicate $Q[\hat{x}]$ is the [[composite]] $\Gamma \times A \to \Gamma \to \Omega$. Given a term $t\colon \Gamma \to A$ and a predicate $P\colon \Gamma \times A \to \Omega$, the proposition $P[t/x]$ is the composite $\Gamma \to \Gamma \times A \to \Omega$. Internalising a bit (by [[currying]]), we may view $Q$ as a [[global element]] $1 \to \Omega^\Gamma$ and $P$ as a [[morphism]] $A \to \Omega^\Gamma$, recovering the view that predicates are proposition-valued `functions' (morphisms).

In general, we may intuitively think of an object $A$ in the [[slice category]] $\mathcal{C}/\Gamma$ as the `set' (object) of possible values of terms $t$ of type $A$ in context $\Gamma$, and think of a predicate $P$ with a free variable of type $A$ (in the same context) as being the `subset' (subobject) on those $t$ for which the statement $P(t)$ is [[true]].

\hypertarget{InTypeTheory}{}\subsection*{{In type theory}}\label{InTypeTheory}

In [[type theory]] under the \emph{[[propositions as types]]} paradigm, every [[type]] represents the proposition that it is [[inhabited type|inhabited]]. Hence the types which have at most one term may be identified with propositions (``propositions as some types''). In [[homotopy type theory]] these are the [[(-1)-types]]. The [[reflective subuniverse|reflection]] that sends types to their underlying proposition qua [[(-1)-truncation]] is the [[n-truncation modality]] for $n = (-1)$, also called [[bracket type]]-formation.

\hypertarget{propositional_and_predicate_logic}{}\subsection*{{Propositional and predicate logic}}\label{propositional_and_predicate_logic}

In [[propositional logic]], we fix a single context (considered the [[empty context]]) and consider the logic of propositions in that context. In [[predicate logic]], we fix the empty context but work also in extensions of that context by free variables. Predicate logic uses [[quantifiers]] as a way to move between contexts, more specifically to move from a predicate $P$ in a given context $\Gamma$ (which is a proposition in some extension of $\Gamma$) to a proposition in $\Gamma$. The free variables in the predicate still appear in the written form of the proposition, but they are now \emph{bound} variables and are not free in the proposition's context; some logicians prefer to systematically replace bound variables with numbered placeholders (especially when defining [[Gödel number]]s and the like).

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

\begin{itemize}%
\item [[property]]


\item [[mere proposition]]


\item [[type of propositions]], [[bracket type]], [[h-proposition]]


\item [[propositional extensionality]]


\item [[theory]]


\item \textbf{proposition}/[[type]] ([[propositions as types]])


\item [[definition]]/[[proof]]/[[program]] ([[proofs as programs]])


\item [[theorem]], [[axiom]]


\item [[conjecture]], [[folklore]]



\end{itemize}
[[!redirects proposition]] [[!redirects propositions]]

[[!redirects predicate]] [[!redirects predicates]] [[!redirects propositional function]] [[!redirects propositional functions]]



\end{document}