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\begin{document}

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\section*{decidable proposition}

\hypertarget{decidable_propositions}{}\section*{{Decidable propositions}}\label{decidable_propositions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ExternalDecidability}{Externally decidable propositions in logic}\dotfill \pageref*{ExternalDecidability} \linebreak
\noindent\hyperlink{InternalDecidability}{Internally decidable propositions in constructive mathematics}\dotfill \pageref*{InternalDecidability} \linebreak
\noindent\hyperlink{relation_between_these}{Relation between these}\dotfill \pageref*{relation_between_these} \linebreak
\noindent\hyperlink{categorial_interpretation}{Categorial interpretation}\dotfill \pageref*{categorial_interpretation} \linebreak


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

A [[proposition]] is decidable if we know whether it is [[true]] or [[false]]. This has (at least) two interpretations, which we will call `internal' and `external' (however, these adjectives are rarely used and must be guessed from the context).

\begin{itemize}%
\item \hyperlink{ExternalDecidability}{External decidability}: either $p$ or $\not p$ may be [[deduction|deduced]] in the theory. This is a statement in the [[metalanguage]].


\item \hyperlink{InternalDecidability}{Internal decidability}: $p \vee \not p$ may be deduced in the theory; in other words ``$p$ or not $p$'' holds in the [[object language]].



\end{itemize}
\hypertarget{ExternalDecidability}{}\subsection*{{Externally decidable propositions in logic}}\label{ExternalDecidability}

In [[logic]], a [[proposition]] $p$ in a given [[theory]] (or in a given [[context]] of a given theory) is \textbf{externally decidable} if there is in that theory (or in that context) a [[proof]] of $p$ or a refutation of $p$ (a proof of the [[negation]] $\neg{p}$). Of course, this only makes sense if the logic of the theory includes an operation of [[negation]].

Any statement that can be proved or refuted is decidable, and one might hope for a consistent [[foundation of mathematics]] in which every statement is decidable. However, G\"o{}del's [[incompleteness theorem]] dashes these hopes; any [[consistent theory]] strong enough to model [[arithmetic]] (actually a rather weak form of arithmetic) must contain undecidable statements.

For example, the [[continuum hypothesis]] is undecidable in [[ZFC]], assuming that $ZFC$ is consistent at all.

\hypertarget{InternalDecidability}{}\subsection*{{Internally decidable propositions in constructive mathematics}}\label{InternalDecidability}

In [[constructive mathematics]], a [[proposition]] $p$ is \textbf{internally decidable} if the [[law of excluded middle]] applies to $p$; that is, if $p \vee \neg{p}$ holds. Of course, in [[classical mathematics]], every statement is decidable in this sense. Even in constructive mathematics, some statements are decidable and no statement is undecidable; that is, $\neg{(p \vee \neg{p})}$ is always false, but this is not enough to guarantee that $p \vee \neg{p}$ is true.

For example, consider the [[Riemann hypothesis]] (or any of the many unsolved $\Pi_1$-[[Pi-1-proposition|propositions]] in [[number theory]]). This may be expressed as $\forall x, P(x)$ for $P$ some [[predicate]] on [[natural numbers]]. For each $x$, the statement $P(x)$ is decidable (that is, $\forall x, P(x) \vee \neg{P(x)}$ holds), and indeed one can in principle work out which with pencil and paper. However, the Riemann hypothesis itself has not yet been proved (constructively) to be decidable.

A [[set]] $A$ has \textbf{[[decidable equality]]} if every [[equality|equation]] between elements of $A$ (every proposition $x = y$ for $x, y$ in $A$) is decidable. A [[subset]] $B$ of a set $C$ is a \textbf{[[decidable subset]]} if every statement of membership in $B$ (every proposition $x \in B$ for $x$ in $A$) is decidable.

\hypertarget{relation_between_these}{}\subsection*{{Relation between these}}\label{relation_between_these}

In many foundations of constructive mathematics, the [[disjunction property]] holds (in the global context). That is, if $p \vee q$ can be proved, then either $p$ can be proved or $q$ can be proved. By G\"o{}del's incompleteness theorem, no consistent foundation of classical arithmetic can have this property, but some consistent foundations of intuitionistic arithmetic (such as [[Heyting arithmetic]]) do.

In any consistent logic with the disjunction property, a proposition is externally decidable if and only if it can be proved to be internally decidable. (Note that the claim that $p$ is externally decidable is a statement in the metalanguage, while the claim that $p$ is internally decidable is a statement in the object language.)

\hypertarget{categorial_interpretation}{}\subsection*{{Categorial interpretation}}\label{categorial_interpretation}

The [[internal logic]] of any [[Heyting category]] $C$ is a [[type theory]] in [[first-order logic|first-order]] [[intuitionistic logic]]; conversely, any such theory $T$ has a [[category of contexts]] which is a Heyting category.

Under this correspondence, the [[contexts]] of $T$ correspond to the [[objects]] of $C$, and the [[propositions]] in a given context correspond to the [[subobjects]] of that object. Then a proposition in a context $X$ is externally decidable if and only if it is, as a subobject of $X$, either $X$ itself (corresponding to being provable) or the [[initial object]] (corresponding to being refutable). And a proposition in the context $X$ is internally decidable if and only if, when thought of as a subobject $A$ of $X$, the [[union]] of $A$ and $\neg{A}$ is $X$.

The [[slice category]] $C/X$ is [[two-valued category|two-valued]] if and only if the context $X$ is consistent and every proposition in that context is externally decidable. $C$ is a [[Boolean category]] if and only if every proposition in every context is internally decidable.

This makes me think that we should define $p$ to be externally decidable iff $p$ is provable whenever it is not refutable, which in a constructive metalogic is weaker than the definition above. But then we lose the relationship with the disjunction property. I guess that we need intuitionistic, classical, and paraconsistent forms of external decidability, just as for two-valuedness.

[[!redirects decidable proposition]] [[!redirects decidable propositions]]

[[!redirects undecidable proposition]] [[!redirects undecidable propositions]]



\end{document}