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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3815/#Item_14" title="Discuss this page in its dedicated thread on the nForum" 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>Ternary frames</title></head> <body> <h1 id="ternary_frames">Ternary frames</h1> <div class='maruku_toc'> <ul> <li><a href='#warning'>Warning</a></li> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#modeling_substructural_logic'>Modeling substructural logic</a></li> <li><a href='#additional_structure'>Additional structure</a></li> <ul> <li><a href='#truth_sets'>Truth sets</a></li> <li><a href='#compatibility_relations'>Compatibility relations</a></li> <li><a href='#falsity_sets'>Falsity sets</a></li> </ul> <li><a href='#dayconvolution'>From ternary frames to quantales</a></li> <ul> <li><a href='#generalizations'>Generalizations</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#phase_spaces'>Phase spaces</a></li> <li><a href='#pcas'>PCAs</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="warning">Warning</h2> <p><em>The term ‘frame’ is used in a different sense here than in <a class="existingWikiWord" href="/nlab/show/geometric+logic">geometric logic</a>; see <a class="existingWikiWord" href="/nlab/show/frame">frame</a>. The usage here is analogous to <a class="existingWikiWord" href="/nlab/show/Kripke+frames">Kripke frames</a> in <a class="existingWikiWord" href="/nlab/show/modal+logic">modal logic</a>.</em></p> <h2 id="idea">Idea</h2> <p>A <strong>ternary frame</strong> is a way of presenting a model for a <a class="existingWikiWord" href="/nlab/show/substructural+logic">substructural logic</a> (such as <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> and <a class="existingWikiWord" href="/nlab/show/relevant+logic">relevant logic</a>) in terms of a set of “worlds” or “states of information” and a ternary <a class="existingWikiWord" href="/nlab/show/relation">relation</a>.</p> <p>The construction generalizes from <a class="existingWikiWord" href="/nlab/show/posets">posets</a> to <a class="existingWikiWord" href="/nlab/show/categories">categories</a> using the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> of a <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal category</a> (<a href="#dayconvolution">see below</a>), or a “promagmal category” in the weakest version.</p> <h2 id="definition">Definition</h2> <p>A <strong>ternary frame</strong> is a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> together with a ternary relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>; we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> holds of three elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x,y,z\in A</annotation></semantics></math>.</p> <p>We may additionally ask that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> have a <a class="existingWikiWord" href="/nlab/show/partial+ordering">partial ordering</a>; in this case we demand the compatibility condition that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x'\le x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>′</mo><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">y'\le y</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≤</mo><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">z\le z'</annotation></semantics></math>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mo>′</mo><mi>y</mi><mo>′</mo><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R x' y' z'</annotation></semantics></math>.</p> <h2 id="modeling_substructural_logic">Modeling substructural logic</h2> <p>We can model <a class="existingWikiWord" href="/nlab/show/logic">logic</a> using a ternary frame with a “forcing” or “satisfaction” relation between points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and formulas. We begin by assigning to each <span class="newWikiWord">atomic formula<a href="/nlab/new/atomic+formula">?</a></span> a set of points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which satisfy it. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has a partial order, as above, then we ask each of these sets to be up-closed.</p> <p>The <a class="existingWikiWord" href="/nlab/show/logical+connectives">logical connectives</a> can then be defined inductively by clauses such as the following:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi><mi>&</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">x \Vdash P \& Q</annotation></semantics></math> (the negative <a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a>) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x\Vdash P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">x\Vdash Q</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi><mo>⊕</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">x \Vdash P \oplus Q</annotation></semantics></math> (the positive <a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a>) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash P</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">x\Vdash Q</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mstyle mathvariant="bold"><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">x \Vdash \mathbf{0}</annotation></semantics></math> (the positive <a class="existingWikiWord" href="/nlab/show/falsity">falsity</a>) never.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">x \Vdash \top</annotation></semantics></math> (the negative <a class="existingWikiWord" href="/nlab/show/truth">truth</a>) always.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi><mo>⊗</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">x \Vdash P \otimes Q</annotation></semantics></math> (the positive conjunction) if and only if there exist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y,z</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>y</mi><mi>z</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">R y z x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">y\Vdash P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>⊩</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">z\Vdash Q</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi><mo>⊸</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">x \Vdash P \multimap Q</annotation></semantics></math> (the one-sided linear implication) if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y,z</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">y\Vdash P</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>⊩</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">z\Vdash Q</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>Q</mi><mi>⟜</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash Q &#10204; P</annotation></semantics></math> (the dual linear implication) if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y,z</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">z\Vdash P</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊩</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">y\Vdash Q</annotation></semantics></math>.</li> </ul> <p>The logic obtained thereby will generally be <a class="existingWikiWord" href="/nlab/show/substructural+logic">substructural</a>: it need not satisfy the structural rules like <a class="existingWikiWord" href="/nlab/show/weakening+rule">weakening</a>, <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction</a>, <a class="existingWikiWord" href="/nlab/show/exchange+rule">exchange</a> or even associativity and unit for the tensor product. On this page, we have used the notation for substructural connectives from <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>.</p> <h2 id="additional_structure">Additional structure</h2> <p>We can impose properties or structure on the ternary frame to affect the logic. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>y</mi><mi>x</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R y x z</annotation></semantics></math>, then the logic we obtain will satisfy the exchange rule.</p> <p>We also need additional structure in order to model positive truth, negative falsity, negative disjunction, and negation.</p> <h3 id="truth_sets">Truth sets</h3> <p>A <strong>truth set</strong> in an ordered ternary frame is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">T\subseteq A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x\le y</annotation></semantics></math> if and only if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t\in T</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>t</mi><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">R t x y</annotation></semantics></math>, and if and only if there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">s\in T</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>s</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">R x s y</annotation></semantics></math>. (If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is commutative, as above, then the two conditions are equivalent.) Alternatively, given an unordered ternary frame <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, we could <em>define</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x\le y</annotation></semantics></math> in this way, and then require as a property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> that the resulting relation is a <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a truth set, then it makes sense to define</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">x \Vdash \mathbf{1}</annotation></semantics></math> (the positive truth) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">x\in T</annotation></semantics></math>.</li> </ul> <h3 id="compatibility_relations">Compatibility relations</h3> <p>One way to model negation (and thereby obtain negative falsity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math> and negative disjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\parr</annotation></semantics></math> by duality from positive truth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math> and positive conjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>) is with a <strong>compatibility relation</strong>, which is just a binary relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has a partial order, we demand additionally that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>C</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">x C y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x'\le x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≤</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">y\le y'</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mi>C</mi><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x' C y'</annotation></semantics></math>.</p> <p>Given such a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we define</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mo>¬</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash \neg P</annotation></semantics></math> if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>C</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">x C y</annotation></semantics></math> then not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">y\Vdash P</annotation></semantics></math>.</li> </ul> <h3 id="falsity_sets">Falsity sets</h3> <p>Negation can alternatively be modeled using a <em>false set</em>. Suppose given a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">F\subseteq A</annotation></semantics></math>, to be the interpretation of the negative falsity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mo>⊥</mo></mrow><annotation encoding="application/x-tex">x\Vdash \bot</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">x\in F</annotation></semantics></math>.</li> </ul> <p>We can then, if we wish, interpret negation and negative disjunction by defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mi>P</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>P</mi><mo>⊸</mo><mo>⊥</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\neg P \coloneqq (P\multimap \bot)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⅋</mo><mi>Q</mi><mo>≔</mo><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>⊗</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P\parr Q \coloneqq \neg (\neg P \otimes \neg Q)</annotation></semantics></math>.</p> <p>The latter is most sensible if negation is involutive, which it need not be in general — that is, if <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>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash \neg \neg P</annotation></semantics></math> we need not have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x\Vdash P</annotation></semantics></math>. One solution to this (if we want negation to be involutive) is to close up <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊩</mo></mrow><annotation encoding="application/x-tex">\Vdash</annotation></semantics></math> under double-negation. This entails replacing the clauses defining the interpretation of the positive connectives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{0}</annotation></semantics></math> with their double-negation closure. This is commonly done in the <a class="existingWikiWord" href="/nlab/show/phase+semantics">phase semantics</a> for linear logic (see below).</p> <h2 id="dayconvolution">From ternary frames to quantales</h2> <p>Since the models above associate to formulas <em>subsets</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, it seems natural to describe them in a purely algebraic way using structure on the <a class="existingWikiWord" href="/nlab/show/powerset">powerset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. In the case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a poset, instead of the powerset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> we must use the set of up-closed subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Since this subsumes the unordered case (use the discrete ordering), we henceforth assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to be a poset, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> having the assumed compatibility relation.</p> <p>In fact, this axiom (which we repeat here for the reader’s convenience):</p> <ul> <li>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x'\le x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>′</mo><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">y'\le y</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≤</mo><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">z\le z'</annotation></semantics></math>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mo>′</mo><mi>y</mi><mo>′</mo><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R x' y' z'</annotation></semantics></math>.</li> </ul> <p>says precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><msup><mi>A</mi> <mi>op</mi></msup><mi>⇸</mi><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A^{op} \times A^{op} &#8696; A^{op}</annotation></semantics></math>. (Here we are identifying posets with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math>-enriched categories, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/interval+category">interval category</a>.)</p> <p>Therefore, by <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> induces a binary tensor product on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math>-enriched <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math>, which is precisely the poset of up-closed sets in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. This tensor product is precisely the above interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>. By the usual Day convolution arguments, this tensor product functor has both left and right adjoints, which are precisely the interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math> and its dual.</p> <p>Of course, the interpretations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math> are just the categorical product and coproduct in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math>. Similarly, that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math> are the initial and terminal objects.</p> <p>A truth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> corresponds to a profunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mi>⇸</mi><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">1 &#8696; A^{op}</annotation></semantics></math> which is a unit for the pro-multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Therefore, in this case the tensor product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math> has a unit object, which is precisely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, the interpretation of the positive truth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math>.</p> <p>If we were to additionally add the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is <em>associative</em>, in the sense that</p> <ul> <li>there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>z</mi><mi>u</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">R z u v</annotation></semantics></math> if and only if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>w</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">R x w v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>y</mi><mi>u</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">R y u w</annotation></semantics></math></li> </ul> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A^{op}</annotation></semantics></math> would become a <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal</a> poset, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math> would be a complete and cocomplete closed monoidal poset, i.e. a <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a>.</p> <p>A false set <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> is of course just an arbitrary object of this quantale. Closing up under double-negation means restricting to the sub-poset of elements that are equal to their “double dual” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊸</mo><mi>F</mi><mo stretchy="false">)</mo><mo>⊸</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">x = (x \multimap F) \multimap F</annotation></semantics></math>. Since the self-adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊸</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\multimap F)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent</a>, this sub-poset is itself a quantale, and indeed a <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">*-autonomous</a> one.</p> <p>The quantale-theoretic content of a compatibility relation is somewhat trickier: as defined it is a profunctor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to itself, whereas the negation of a pro-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-autonomous poset would be a profunctor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A^{op}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. (This would induce a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">(\mathbf{2}^A)^{op} \to \mathbf{2}^{A}</annotation></semantics></math> due to the special property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo>≅</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}\cong \mathbf{2}^{op}</annotation></semantics></math>.) Moreover, the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mo>¬</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash \neg P</annotation></semantics></math> for a compatibility relation is also the (metatheoretic) <em>negation</em> of the map on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math> that would be induced profunctorially. If we put these together, we can see that negation ought to be the the composite of the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup><mo>→</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A \to \mathbf{2}^A</annotation></semantics></math> induced by the profunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup><mo>≅</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A \cong (\mathbf{2}^{A^{op}})^{op}</annotation></semantics></math> (using again that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo>≅</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}\cong \mathbf{2}^{op}</annotation></semantics></math>). At least if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is discrete, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≅</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\cong A^{op}</annotation></semantics></math>, then this has the correct domain and codomain, so we should be able to assert an axiom ensuring that it makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-autononmous.</p> <div class="query"> <p>To be completed…</p> </div> <h3 id="generalizations">Generalizations</h3> <p>The quantale-theoretic viewpoint suggests a generalization replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math> by any other quantale. That is, for any quantale <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>, we can define a notion of “<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>-valued ternary frame” that generates a new quantale by Day convolution. Everything goes through without significant change, except that compatibility relations seem to require <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> itself to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-autonomous.</p> <h2 id="examples">Examples</h2> <h3 id="phase_spaces">Phase spaces</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math>, then we can make it a ternary frame by defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> to mean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>y</mi><mo>=</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x \cdot y = z</annotation></semantics></math>. More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a poset equipped with a binary multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math>, we can make it an ordered ternary frame by defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> to mean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x \cdot y \le z</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math> has a unit object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mo stretchy="false">{</mo><mi>t</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">T = \{t\}</annotation></semantics></math> (in the unordered case) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">|</mo><mi>t</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">T = \{x | t \le x \}</annotation></semantics></math> (in the ordered case) is a truth set.</p> <p>Categorically, this corresponds to the usual way of regarding a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> as a <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal category</a>.</p> <p>In the special case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a commutative <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> equipped with a “false set” <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> as above (usually written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>) in this context, this semantics for linear logic is called <a class="existingWikiWord" href="/nlab/show/phase+semantics">phase semantics</a> (see there for more). It is usually expressed in terms of the quantale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{2}^A</annotation></semantics></math> obtained by Day convolution, but after passing to fixed points of the double-negation monad (in order to obtain an involutive negation). In this context, fixed points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding="application/x-tex">\neg\neg</annotation></semantics></math> are referred to as <em>facts</em>.</p> <p>It is also possible to interpret the exponential modalities <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>!</mo></mrow><annotation encoding="application/x-tex">!</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>?</mo></mrow><annotation encoding="application/x-tex">?</annotation></semantics></math> of linear logic using phase space semantics. For instance, we can define</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊩</mo><mo>!</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \Vdash !P</annotation></semantics></math> if and only if <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> belongs to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding="application/x-tex">\neg\neg</annotation></semantics></math>-closure of the set of all idempotents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊩</mo><mi>P</mi><mi>&</mi><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">y\Vdash P \& \mathbf{1}</annotation></semantics></math>.</li> </ul> <p>and obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>?</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">?P</annotation></semantics></math> by duality.</p> <p>Phase space semantics is <em>complete</em> for <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, in the sense that a formula is provable if and only if in any phase space semantics we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⊨</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">1\vDash P</annotation></semantics></math>, where <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> is the unit element of the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <h3 id="pcas">PCAs</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/partial+combinatory+algebra">partial combinatory algebra</a> (PCA), we can make it a ternary frame by defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">R x y z</annotation></semantics></math> to mean that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \cdot y</annotation></semantics></math> is defined and equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>. (Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an ordered PCA we can make it an ordered ternary frame.) The resulting interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math> almost coincides with the usual interpretation of implication in <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, and the combinators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">k,s</annotation></semantics></math> have the property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>⊩</mo><mi>P</mi><mo>⊸</mo><mo stretchy="false">(</mo><mi>Q</mi><mo>⊸</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k \Vdash P \multimap (Q \multimap P)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>⊩</mo><mo stretchy="false">(</mo><mi>P</mi><mo>⊸</mo><mi>Q</mi><mo>⊸</mo><mi>R</mi><mo stretchy="false">)</mo><mo>⊸</mo><mo stretchy="false">(</mo><mi>P</mi><mo>⊸</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>⊸</mo><mi>P</mi><mo>⊸</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">s \Vdash (P \multimap Q \multimap R) \multimap (P \multimap Q) \multimap P \multimap R</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">P,Q,R</annotation></semantics></math>, as would be expected from typed <a class="existingWikiWord" href="/nlab/show/combinatory+logic">combinatory logic</a>.</p> <h2 id="references">References</h2> <ul> <li> <p>R. Routley and R.K. Meyer, <em>The Semantics of Entailment I</em>, Truth, Syntax and Modality, ed. H. Leblanc, North-Holland Publishing Company (Amsterdam), pp. 199-243. (1973)</p> </li> <li> <p>R. Routley and R.K. Meyer, <em>The Semantics of Entailment, II-III</em>. Journal of Philosophical Logic, 1, 53-73 and 192-208. (1972)</p> </li> <li> <p>Natasha Kurtonina, <em>Frames and Labels - A modal analysis of categorial inference</em>. Ph.D. Thesis, Institute of Logic, Language and Information (ILLC), Amsterdam, 1994</p> </li> <li> <p>J. M. Dunn and R. K. Meyer, <em>Combinators and Structurally Free Logic</em>. Logic journal of the IGPL, 5(4):505-538, July 1997</p> </li> <li> <p>Greg Restall, <em>An introduction to substructural logic</em>. Routledge, 2000.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 15, 2024 at 10:56:27. 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