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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#construction'>Construction</a></li> <ul> <li><a href='#propositional_case'>Propositional case</a></li> <ul> <li><a href='#glivenkos_theorem'>Glivenko’s Theorem</a></li> </ul> <li><a href='#firstorder_case'>First-order case</a></li> <li><a href='#higherorder_case'>Higher-order case</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><strong>Double negation translation</strong> is a method for converting <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> <a class="existingWikiWord" href="/nlab/show/valid">valid</a> in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> into propositions valid in <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a>. It can be used to establish relative consistency results. For example, it may be possible to show that the proof of a <a class="existingWikiWord" href="/nlab/show/contradiction">contradiction</a> in a classical theory could be translated to the proof of a contradiction in a constructive theory. <a class="existingWikiWord" href="/nlab/show/Kurt+G%C3%B6del">Kurt Gödel</a> used this technique to show that <a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a> and <a class="existingWikiWord" href="/nlab/show/Heyting+arithmetic">Heyting arithmetic</a> are equiconsistent.</p> <p>Double negation translation was discovered independently by a number of mathematicians including <a class="existingWikiWord" href="/nlab/show/Kurt+G%C3%B6del">Kurt Gödel</a>, <a class="existingWikiWord" href="/nlab/show/Gerhard+Gentzen">Gerhard Gentzen</a>, and <a class="existingWikiWord" href="/nlab/show/Andrey+Kolmogorov">Andrey Kolmogorov</a>, and is also called the <strong>Gödel–Gentzen negative translation</strong>.</p> <h2 id="construction">Construction</h2> <p>The traditional descriptions are highly syntactic, but can be motivated by recalling some conceptual relationships between <a class="existingWikiWord" href="/nlab/show/Boolean+algebras">Boolean algebras</a> (which are algebras for classical propositional logic) and <a class="existingWikiWord" href="/nlab/show/Heyting+algebras">Heyting algebras</a> (which are algebras for intuitionistic propositional logic).</p> <h3 id="propositional_case">Propositional case</h3> <p>Recall from the article <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a> that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Bool</mi><mo>→</mo><mi>Heyt</mi></mrow><annotation encoding="application/x-tex">U \colon Bool \to Heyt</annotation></semantics></math></div> <p>is given objectwise by taking a Heyting algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> to the poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">L_{\neg\neg}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/regular+elements">regular elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">x \in L</annotation></semantics></math>, i.e., those that are fixed by <a class="existingWikiWord" href="/nlab/show/double+negation">double negation</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mo>¬</mo><mo>¬</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x = \neg \neg x</annotation></semantics></math>. It was shown that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">L_{\neg\neg}</annotation></semantics></math> is a Boolean algebra, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\neg \neg \colon L \to L_{\neg\neg}</annotation></semantics></math> is a Heyting algebra homomorphism which is universal among Heyting algebra maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>→</mo><mi>U</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">L \to U B</annotation></semantics></math> into Boolean algebras.</p> <p>In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Heyt(S)</annotation></semantics></math> is the free Heyting algebra on a set of <a class="existingWikiWord" href="/nlab/show/variables">variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, it follows by composing left adjoints</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mover><mo>→</mo><mi>free</mi></mover><mi>Heyt</mi><mover><mo>→</mo><mi>F</mi></mover><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Set \stackrel{free}{\to} Heyt \stackrel{F}{\to} Bool</annotation></semantics></math></div> <p>that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><msub><mo stretchy="false">)</mo> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Heyt(S)_{\neg\neg}</annotation></semantics></math> is the free Boolean algebra on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>Furthermore, it was shown in <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a> that for Heyting algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>,</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves finite <a class="existingWikiWord" href="/nlab/show/meets">meets</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves the <a class="existingWikiWord" href="/nlab/show/implication">implication</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math>.</p> </li> </ul> <p>Consequently, the inclusion of regular elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">L_{\neg\neg} \to L</annotation></semantics></math> also preserves meets and implications, <strong>strictly</strong>. This gives the following result.</p> <div class="num_thm"> <h6 id="glivenkos_theorem">Glivenko’s Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⇒</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \Rightarrow q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/tautology">tautology</a> in classical propositional logic, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\neg \neg p) \Rightarrow (\neg \neg q)</annotation></semantics></math> is a tautology in intuitionistic propositional logic, and conversely.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/term">term</a> expressions in variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">x_i \in S</annotation></semantics></math> over the signature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mo>⇒</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0, 1, \vee, \wedge, \Rightarrow)</annotation></semantics></math>. To say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⇒</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \Rightarrow q</annotation></semantics></math> is a classical tautology means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mo stretchy="false">(</mo><mi>p</mi><mo>⇒</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1 \leq (p \Rightarrow q)</annotation></semantics></math> holds when interpreted in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bool(S)</annotation></semantics></math>. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><msub><mo stretchy="false">)</mo> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Bool(S) = Heyt(S)_{\neg\neg}</annotation></semantics></math>, this is equivalent to saying that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mo>¬</mo><mo>¬</mo><mo stretchy="false">(</mo><mi>p</mi><mo>⇒</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1 \leq \neg\neg(p \Rightarrow q) = (\neg\neg p) \Rightarrow (\neg\neg q)</annotation></semantics></math></div> <p>when interpreted in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Heyt(S)</annotation></semantics></math>, which is to say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mo>¬</mo><mo>¬</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\neg \neg p) \Rightarrow (\neg\neg q)</annotation></semantics></math> is an intuitionistic tautology.</p> </div> <p>Continuing this thought: the join <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Heyt</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Bool(S) = Heyt_{\neg\neg}</annotation></semantics></math> is computed as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mo>∨</mo> <mi>Bool</mi></msub><mi>b</mi><mo>=</mo><mo>¬</mo><mo>¬</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mo>∨</mo> <mi>Heyt</mi></msub><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \vee_{Bool} b = \neg\neg(a \vee_{Heyt} b)</annotation></semantics></math></div> <p>since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><msub><mo stretchy="false">)</mo> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}</annotation></semantics></math> preserves joins (it is a left adjoint). Putting all this together, because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><msub><mo stretchy="false">)</mo> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}</annotation></semantics></math> preserves Heyting algebra structure, we arrive at the following syntactic translation.</p> <div class="un_def" id="defn"> <h6 id="definition">Definition</h6> <p>The double-negation translation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>↦</mo><msup><mi>p</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">p \mapsto p^{N}</annotation></semantics></math> on term expressions in the theory of Heyting algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is defined by induction as follows.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><mo>¬</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x^N = \neg\neg x</annotation></semantics></math> for variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>0</mn> <mi>N</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0^N = 0</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mi>N</mi></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1^N = 1</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>∧</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><msup><mi>p</mi> <mi>N</mi></msup><mo>∧</mo><msup><mi>q</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(p \wedge q)^N = p^N \wedge q^N</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves the meet operation)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>⇒</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><msup><mi>p</mi> <mi>N</mi></msup><mo>⇒</mo><msup><mi>q</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(p \Rightarrow q)^N = p^N \Rightarrow q^N</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\neg\neg \colon L \to L</annotation></semantics></math> preserves the implication operation)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>∨</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><mo>¬</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>N</mi></msup><mo>∨</mo><msup><mi>q</mi> <mi>N</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p \vee q)^N = \neg\neg(p^N \vee q^N)</annotation></semantics></math>.</p> </li> </ul> </div> <p>Thus, by Glivenko’s theorem, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a classical tautology if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">p^N</annotation></semantics></math> is an intuitionistic tautology. This result may be extended to theories as well: suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is an intuitionistic theory or Heyting algebra, given by a presentation as a coequalizer in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Heyt</mi></mrow><annotation encoding="application/x-tex">Heyt</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Heyt</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Heyt</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">Heyt(T) \stackrel{\to}{\to} Heyt(S) \to L.</annotation></semantics></math></div> <p>Then, since the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↦</mo><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">L \mapsto L_{\neg\neg}</annotation></semantics></math> is a left adjoint, it takes this coequalizer to a coequalizer</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Bool</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Bool(T) \stackrel{\to}{\to} Bool(S) \to L_{\neg\neg}</annotation></semantics></math></div> <p>so that an term expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a theorem in the classical theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">L_{\neg\neg}</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">p^N</annotation></semantics></math> is a theorem in the intuitionistic theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>.</p> <h3 id="firstorder_case">First-order case</h3> <p>Double negation translation for a formula, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, of a <a class="existingWikiWord" href="/nlab/show/first-order+language">first-order language</a> is defined inductively by the following clauses:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><mo>¬</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi^N = \neg \neg \phi</annotation></semantics></math>, for atomic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϕ</mi><mo>∧</mo><mi>ψ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><msup><mi>ϕ</mi> <mi>N</mi></msup><mo>∧</mo><msup><mi>ψ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(\phi \wedge \psi)^N = \phi^N \wedge \psi^N</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϕ</mi><mo>∨</mo><mi>ψ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><msup><mi>ϕ</mi> <mi>N</mi></msup><mo>∧</mo><mo>¬</mo><msup><mi>ψ</mi> <mi>N</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi \vee \psi)^N = \neg(\neg \phi^N \wedge \neg \psi^N)</annotation></semantics></math> (or equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>N</mi></msup><mo>∨</mo><msup><mi>ψ</mi> <mi>N</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\neg \neg (\phi^N \vee \psi^N)</annotation></semantics></math>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϕ</mi><mo>→</mo><mi>ψ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><msup><mi>ϕ</mi> <mi>N</mi></msup><mo>→</mo><msup><mi>ψ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(\phi \to \psi)^N = \phi^N \to \psi^N</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>¬</mo><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><msup><mi>ϕ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(\neg \phi)^N = \neg \phi^N</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∀</mo><mi>x</mi><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><mo>∀</mo><mi>x</mi><msup><mi>ϕ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(\forall x \phi)^N = \forall x \phi^N</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∃</mo><mi>x</mi><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup><mo>=</mo><mo>¬</mo><mo>∀</mo><mi>x</mi><mo>¬</mo><msup><mi>ϕ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">(\exists x \phi)^N = \neg \forall x \neg \phi^N</annotation></semantics></math> (or equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo><mo>∃</mo><mi>x</mi><msup><mi>ϕ</mi> <mi>N</mi></msup></mrow><annotation encoding="application/x-tex">\neg \neg \exists x \phi^N</annotation></semantics></math>).</p> </li> </ul> <h3 id="higherorder_case">Higher-order case</h3> <p>The basic idea here is that any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> gives rise to a <a class="existingWikiWord" href="/nlab/show/Boolean+topos">Boolean topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>¬</mo><mo>¬</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{\neg\neg}</annotation></semantics></math>.</p> <h2 id="references">References</h2> <p>Informal exposition of a tiny aspect of this double negation business is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <em><a href="http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/">Intuitionistic mathematics for physics</a></em>, 2008</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 10, 2017 at 08:09:12. 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