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\begin{document}

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\section*{double negation translation}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{propositional_case}{Propositional case}\dotfill \pageref*{propositional_case} \linebreak
\noindent\hyperlink{firstorder_case}{First-order case}\dotfill \pageref*{firstorder_case} \linebreak
\noindent\hyperlink{higherorder_case}{Higher-order case}\dotfill \pageref*{higherorder_case} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Double negation translation} is a method for converting [[propositions]] [[valid]] in [[classical logic]] into propositions valid in [[intuitionistic logic]]. It can be used to establish relative consistency results. For example, it may be possible to show that the proof of a [[contradiction]] in a classical theory could be translated to the proof of a contradiction in a constructive theory. [[Kurt Gödel]] used this technique to show that [[Peano arithmetic]] and [[Heyting arithmetic]] are equiconsistent.

Double negation translation was discovered independently by a number of mathematicians including [[Kurt Gödel]], [[Gerhard Gentzen]], and [[Andrey Kolmogorov]], and is also called the \textbf{G\"o{}del--Gentzen negative translation}.

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

The traditional descriptions are highly syntactic, but can be motivated by recalling some conceptual relationships between [[Boolean algebras]] (which are algebras for classical propositional logic) and [[Heyting algebras]] (which are algebras for intuitionistic propositional logic).

\hypertarget{propositional_case}{}\subsubsection*{{Propositional case}}\label{propositional_case}

Recall from the article [[Heyting algebra]] that the [[left adjoint]] $F$ to the [[forgetful functor]]

\begin{displaymath}
U \colon Bool \to Heyt
\end{displaymath}
is given objectwise by taking a Heyting algebra $L$ to the poset $L_{\neg\neg}$ of [[regular elements]] $x \in L$, i.e., those that are fixed by [[double negation]]: $x = \neg \neg x$. It was shown that $L_{\neg\neg}$ is a Boolean algebra, and $\neg \neg \colon L \to L_{\neg\neg}$ is a Heyting algebra homomorphism which is universal among Heyting algebra maps $L \to U B$ into Boolean algebras.

In particular, if $Heyt(S)$ is the free Heyting algebra on a set of [[variables]] $S$, it follows by composing left adjoints

\begin{displaymath}
Set \stackrel{free}{\to} Heyt \stackrel{F}{\to} Bool
\end{displaymath}
that $Heyt(S)_{\neg\neg}$ is the free Boolean algebra on $S$.

Furthermore, it was shown in [[Heyting algebra]] that for Heyting algebras $L$,

\begin{itemize}%
\item $\neg\neg \colon L \to L$ preserves finite [[meets]];


\item $\neg\neg \colon L \to L$ preserves the [[implication]] operation $\Rightarrow$.



\end{itemize}
Consequently, the inclusion of regular elements $L_{\neg\neg} \to L$ also preserves meets and implications, \textbf{strictly}. This gives the following result.

\begin{thm}
\label{}\hypertarget{}{}
If $p \Rightarrow q$ is a [[tautology]] in classical propositional logic, then $(\neg \neg p) \Rightarrow (\neg \neg q)$ is a tautology in intuitionistic propositional logic, and conversely.

\end{thm}
\begin{proof}
Here $p$ and $q$ are [[term]] expressions in variables $x_i \in S$ over the signature $(0, 1, \vee, \wedge, \Rightarrow)$. To say $p \Rightarrow q$ is a classical tautology means that $1 \leq (p \Rightarrow q)$ holds when interpreted in $Bool(S)$. But since $Bool(S) = Heyt(S)_{\neg\neg}$, this is equivalent to saying that

\begin{displaymath}
1 \leq \neg\neg(p \Rightarrow q) = (\neg\neg p) \Rightarrow (\neg\neg q)
\end{displaymath}
when interpreted in $Heyt(S)$, which is to say that $(\neg \neg p) \Rightarrow (\neg\neg q)$ is an intuitionistic tautology.

\end{proof}
Continuing this thought: the join $\vee$ in $Bool(S) = Heyt_{\neg\neg}$ is computed as

\begin{displaymath}
a \vee_{Bool} b = \neg\neg(a \vee_{Heyt} b)
\end{displaymath}
since $\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}$ preserves joins (it is a left adjoint). Putting all this together, because $\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}$ preserves Heyting algebra structure, we arrive at the following syntactic translation.

\begin{udef}
The double-negation translation $p \mapsto p^{N}$ on term expressions in the theory of Heyting algebras $L$ is defined by induction as follows.

\begin{itemize}%
\item $x^N = \neg\neg x$ for variables $x$.


\item $0^N = 0$ (since $\neg\neg \colon L \to L$ preserves $0$)


\item $1^N = 1$ (since $\neg\neg \colon L \to L$ preserves $1$)


\item $(p \wedge q)^N = p^N \wedge q^N$ (since $\neg\neg \colon L \to L$ preserves the meet operation)


\item $(p \Rightarrow q)^N = p^N \Rightarrow q^N$ (since $\neg\neg \colon L \to L$ preserves the implication operation)


\item $(p \vee q)^N = \neg\neg(p^N \vee q^N)$.



\end{itemize}
\end{udef}
Thus, by Glivenko's theorem, $p$ is a classical tautology if and only if $p^N$ is an intuitionistic tautology. This result may be extended to theories as well: suppose $L$ is an intuitionistic theory or Heyting algebra, given by a presentation as a coequalizer in $Heyt$:

\begin{displaymath}
Heyt(T) \stackrel{\to}{\to} Heyt(S) \to L.
\end{displaymath}
Then, since the functor $L \mapsto L_{\neg\neg}$ is a left adjoint, it takes this coequalizer to a coequalizer

\begin{displaymath}
Bool(T) \stackrel{\to}{\to} Bool(S) \to L_{\neg\neg}
\end{displaymath}
so that an term expression $p$ is a theorem in the classical theory $L_{\neg\neg}$ if and only if $p^N$ is a theorem in the intuitionistic theory $L$.

\hypertarget{firstorder_case}{}\subsubsection*{{First-order case}}\label{firstorder_case}

Double negation translation for a formula, $\phi$, of a [[first-order language]] is defined inductively by the following clauses:

\begin{itemize}%
\item $\phi^N = \neg \neg \phi$, for atomic $\phi$.


\item $(\phi \wedge \psi)^N = \phi^N \wedge \psi^N$.


\item $(\phi \vee \psi)^N = \neg(\neg \phi^N \wedge \neg \psi^N)$ (or equivalently, $\neg \neg (\phi^N \vee \psi^N)$)


\item $(\phi \to \psi)^N = \phi^N \to \psi^N$.


\item $(\neg \phi)^N = \neg \phi^N$.


\item $(\forall x \phi)^N = \forall x \phi^N$.


\item $(\exists x \phi)^N = \neg \forall x \neg \phi^N$ (or equivalently, $\neg \neg \exists x \phi^N$).



\end{itemize}
\hypertarget{higherorder_case}{}\subsubsection*{{Higher-order case}}\label{higherorder_case}

The basic idea here is that any [[topos]] $E$ gives rise to a [[Boolean topos]] $E_{\neg\neg}$.

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

Informal exposition of a tiny aspect of this double negation business is in

\begin{itemize}%
\item [[Andrej Bauer]], \emph{\href{http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/}{Intuitionistic mathematics for physics}}, 2008

\end{itemize}
[[!redirects double negation translation]] [[!redirects double negation translations]] [[!redirects double-negation translation]] [[!redirects double-negation translations]] [[!redirects negative translation]] [[!redirects negative translations]]



\end{document}