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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Charles_Sanders_Peirce" title="Category:Charles Sanders Peirce">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Charles_Sanders_Peirce.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Charles_Sanders_Peirce.jpg/250px-Charles_Sanders_Peirce.jpg" decoding="async" width="250" height="335" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Charles_Sanders_Peirce.jpg/375px-Charles_Sanders_Peirce.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/58/Charles_Sanders_Peirce.jpg 2x" data-file-width="434" data-file-height="582" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Charles_Sanders_Peirce_bibliography" title="Charles Sanders Peirce bibliography">Bibliography</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Pragmatism" title="Pragmatism">Pragmatism</a> in epistemology</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Abductive_reasoning" title="Abductive reasoning">Abductive reasoning</a></li> <li><a href="/wiki/Fallibilism" title="Fallibilism">Fallibilism</a></li> <li><a href="/wiki/Pragmaticism" title="Pragmaticism">Pragmaticism</a> <ul><li><a href="/wiki/Pragmatic_maxim" title="Pragmatic maxim">as maxim</a></li> <li><a href="/wiki/Pragmatic_theory_of_truth" title="Pragmatic theory of truth">as theory of truth</a></li></ul></li> <li><a href="/wiki/Community_of_inquiry" title="Community of inquiry">Community of inquiry</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Logic</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Continuous_predicate" title="Continuous predicate">Continuous predicate</a></li> <li><a href="/wiki/Peirce%27s_law" title="Peirce's law">Peirce's law</a></li> <li><a href="/wiki/Entitative_graph" class="mw-redirect" title="Entitative graph">Entitative graph in Qualitative logic</a></li> <li><a class="mw-selflink selflink">Existential graph</a></li> <li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li> <li><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></li> <li><a href="/wiki/Logic_of_information" title="Logic of information">Logic of information</a></li> <li><a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li> <li><a href="/wiki/Logical_NOR" title="Logical NOR">Logical NOR</a></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order logic</a></li> <li><a href="/wiki/Trikonic" title="Trikonic">Trikonic</a></li> <li><a href="/wiki/Type-token_distinction" class="mw-redirect" title="Type-token distinction">Type-token distinction</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Semiotic_theory_of_Charles_Sanders_Peirce" title="Semiotic theory of Charles Sanders Peirce">Semiotic theory</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Indexicality" title="Indexicality">Indexicality</a></li> <li><a href="/wiki/Interpretant" title="Interpretant">Interpretant</a></li> <li><a href="/wiki/Semiosis" title="Semiosis">Semiosis</a></li> <li><a href="/wiki/Sign_relation" title="Sign relation">Sign relation</a></li> <li><a href="/wiki/Universal_rhetoric" title="Universal rhetoric">Universal rhetoric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Miscellaneous contributions</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Agapism" title="Agapism">Agapism</a></li> <li><a href="/wiki/Bell_triangle" title="Bell triangle">Bell triangle</a></li> <li><a href="/wiki/Categories_(Peirce)" title="Categories (Peirce)">Categories</a></li> <li><i><a href="/wiki/Phaneron" title="Phaneron">Phaneron</a></i></li> <li><a href="/wiki/Synechism" title="Synechism">Synechism</a></li> <li><a href="/wiki/Tychism" title="Tychism">Tychism</a></li> <li><a href="/wiki/Classification_of_the_sciences_(Peirce)" title="Classification of the sciences (Peirce)">Classification of sciences</a></li> <li><a href="/wiki/Listing_number" title="Listing number">Listing number</a></li> <li><a href="/wiki/Peirce_quincuncial_projection" title="Peirce quincuncial projection">Quincuncial projection</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Biographical</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Joseph_Morton_Ransdell" title="Joseph Morton Ransdell">Joseph Morton Ransdell</a></li> <li><a href="/wiki/Allan_Marquand" title="Allan Marquand">Allan Marquand</a></li> <li><a href="/wiki/Juliette_Peirce" title="Juliette Peirce">Juliette Peirce</a></li> <li><a href="/wiki/Charles_Santiago_Sanders_Peirce" title="Charles Santiago Sanders Peirce">Charles Santiago Sanders Peirce</a></li> <li><a href="/wiki/Roberta_Kevelson" title="Roberta Kevelson">Roberta Kevelson</a></li> <li><a href="/wiki/Christine_Ladd-Franklin" title="Christine Ladd-Franklin">Christine Ladd-Franklin</a></li> <li><a href="/wiki/Victoria,_Lady_Welby" title="Victoria, Lady Welby">Victoria, Lady Welby</a></li> <li><a href="/wiki/The_Metaphysical_Club" title="The Metaphysical Club">The Metaphysical Club</a> <ul><li><a href="/wiki/The_Metaphysical_Club:_A_Story_of_Ideas_in_America" title="The Metaphysical Club: A Story of Ideas in America">book</a></li></ul></li> <li><i><a href="/wiki/Peirce_Geodetic_Monument" title="Peirce Geodetic Monument">Peirce Geodetic Monument</a></i></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:C._S._Peirce_articles" title="Template:C. S. Peirce articles"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:C._S._Peirce_articles" title="Template talk:C. S. Peirce articles"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:C._S._Peirce_articles" title="Special:EditPage/Template:C. S. Peirce articles"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>An <b>existential graph</b> is a type of <a href="/wiki/Diagram" title="Diagram">diagrammatic</a> or visual notation for logical expressions, created by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, who wrote on graphical logic as early as 1882,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and continued to develop the method until his death in 1914. They include both a separate graphical notation for logical statements and a logical calculus, a formal system of rules of inference that can be used to derive theorems. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Peirce found the algebraic notation (i.e. symbolic notation) of logic, especially that of predicate logic,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which was still very new during his lifetime and which he himself played a major role in developing, to be philosophically unsatisfactory, because the symbols had their meaning by mere convention. In contrast, he strove for a style of writing in which the signs literally carry their meaning within them<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> – in the terminology of his theory of signs: a system of iconic signs that resemble or resemble the represented objects and relations.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Thus, the development of an iconic, graphic and – as he intended – intuitive and easy-to-learn logical system was a project that Peirce worked on throughout his life. After at least one aborted approach – the "Entitative Graphs" – the closed system of "Existential Graphs" finally emerged from 1896 onwards. Although considered by their creator to be a clearly superior and more intuitive system, as a mode of writing and as a calculus, they had no major influence on the history of logic. This has been attributed to the fact(s) that, for one, Peirce published little on this topic, and that the published texts were not written in a very understandable way;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and, for two, that the linear formula notation in the hands of experts is actually the less complex tool.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Hence, the existential graphs received little attention<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> or were seen as unwieldy.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> From 1963 onwards, works by Don D. Roberts and J. Jay Zeman, in which Peirce's graphic systems were systematically examined and presented, led to a better understanding; even so, they have today found practical use within only one modern application—the conceptual graphs introduced by John F. Sowa in 1976, which are used in computer science to represent knowledge. However, existential graphs are increasingly reappearing as a subject of research in connection with a growing interest in graphical logic,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> which is also expressed in attempts to replace the rules of inference given by Peirce with more intuitive ones.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The overall system of existential graphs is composed of three subsystems that build on each other, the alpha graphs, the beta graphs and the gamma graphs. The alpha graphs are a purely propositional logical system. Building on this, the beta graphs are a first order logical calculus. The gamma graphs, which have not yet been fully researched and were not completed by Peirce, are understood as a further development of the alpha and beta graphs. When interpreted appropriately, the gamma graphs cover higher-level predicate logic as well as modal logic. As late as 1903, Peirce began a new approach, the "Tinctured Existential Graphs," with which he wanted to replace the previous systems of alpha, beta and gamma graphs and combine their expressiveness and performance in a single new system. Like the gamma graphs, the "Tinctured Existential Graphs" remained unfinished. </p><p>As calculi, the alpha, beta and gamma graphs are sound (i.e., all expressions derived as graphs are semantically valid). The alpha and beta graphs are also complete (i.e., all propositional or predicate-logically semantically valid expressions can be derived as alpha or beta graphs). <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="The_graphs">The graphs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=2" title="Edit section: The graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Peirce proposed three systems of existential graphs: </p> <ul><li><i>alpha</i>, <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> and the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a>;</li> <li><i>beta</i>, isomorphic to <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> with identity, with all formulas closed;</li> <li><i>gamma</i>, (nearly) isomorphic to <a href="/wiki/Normal_modal_logic" title="Normal modal logic">normal modal logic</a>.</li></ul> <p><i>Alpha</i> nests in <i>beta</i> and <i>gamma</i>. <i>Beta</i> does not nest in <i>gamma</i>, quantified modal logic being more general than put forth by Peirce. </p> <div class="mw-heading mw-heading3"><h3 id="Alpha">Alpha</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=3" title="Edit section: Alpha"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PeirceAlphaGraphs.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/PeirceAlphaGraphs.svg/300px-PeirceAlphaGraphs.svg.png" decoding="async" width="300" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/PeirceAlphaGraphs.svg/450px-PeirceAlphaGraphs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/PeirceAlphaGraphs.svg/600px-PeirceAlphaGraphs.svg.png 2x" data-file-width="272" data-file-height="244" /></a><figcaption>Alpha graphs</figcaption></figure> <p>The <a href="/wiki/Syntax" title="Syntax">syntax</a> is: </p> <ul><li>The blank page;</li> <li>Single letters or phrases written anywhere on the page;</li> <li>Any graph may be enclosed by a <a href="/wiki/Simple_closed_curve" class="mw-redirect" title="Simple closed curve">simple closed curve</a> called a <i>cut</i> or <i>sep</i>. A cut can be empty. Cuts can nest and concatenate at will, but must never intersect.</li></ul> <p>Any well-formed part of a graph is a <b>subgraph</b>. </p><p>The <a href="/wiki/Semantics" title="Semantics">semantics</a> are: </p> <ul><li>The blank page denotes <b>Truth</b>;</li> <li>Letters, phrases, subgraphs, and entire graphs may be <b>True</b> or <b>False</b>;</li> <li>To enclose a subgraph with a cut is equivalent to logical <a href="/wiki/Negation" title="Negation">negation</a> or Boolean <a href="/wiki/Complement_(order_theory)" class="mw-redirect" title="Complement (order theory)">complementation</a>. Hence an empty cut denotes <b>False</b>;</li> <li>All subgraphs within a given cut are tacitly <a href="/wiki/Conjunction_(logic)" class="mw-redirect" title="Conjunction (logic)">conjoined</a>.</li></ul> <p>Hence the <i>alpha</i> graphs are a minimalist notation for <a href="/wiki/Sentential_logic" class="mw-redirect" title="Sentential logic">sentential logic</a>, grounded in the expressive adequacy of <b>And</b> and <b>Not</b>. The <i>alpha</i> graphs constitute a radical simplification of the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a> and the <a href="/wiki/Connective_(logic)" class="mw-redirect" title="Connective (logic)">truth functors</a>. </p><p>The <i>depth</i> of an object is the number of cuts that enclose it. </p><p><i>Rules of inference</i>: </p> <ul><li>Insertion - Any subgraph may be inserted into an odd numbered depth. The surrounding white page is depth 1. Depth 2 are the black letters and lines that encircle elements. Depth 3 is entering the next white area in an enclosed element.</li> <li>Erasure - Any subgraph in an even numbered depth may be erased.</li></ul> <p><i>Rules of equivalence</i>: </p> <ul><li>Double cut - A pair of cuts with nothing between them may be drawn around any subgraph. Likewise two nested cuts with nothing between them may be erased. This rule is equivalent to Boolean involution and <a href="/wiki/Double_negation" title="Double negation">double negation</a> elimination.</li> <li>Iteration/Deiteration – To understand this rule, it is best to view a graph as a <a href="/wiki/Tree_structure" title="Tree structure">tree structure</a> having <a href="/wiki/Node_(computer_science)" title="Node (computer science)">nodes</a> and <a href="/wiki/Tree_structure" title="Tree structure">ancestors</a>. Any subgraph <i>P</i> in node <i>n</i> may be copied into any node depending on <i>n</i>. Likewise, any subgraph <i>P</i> in node <i>n</i> may be erased if there exists a copy of <i>P</i> in some node ancestral to <i>n</i> (i.e., some node on which <i>n</i> depends). For an equivalent rule in an algebraic context, see <b>C2</b> in <i><a href="/wiki/Laws_of_Form" title="Laws of Form">Laws of Form</a></i>.</li></ul> <p>A proof manipulates a graph by a series of steps, with each step justified by one of the above rules. If a graph can be reduced by steps to the blank page or an empty cut, it is what is now called a <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautology</a> (or the complement thereof, a contradiction). Graphs that cannot be simplified beyond a certain point are analogues of the <a href="/wiki/Satisfiable" class="mw-redirect" title="Satisfiable">satisfiable</a> <a href="/wiki/Formula" title="Formula">formulas</a> of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Beta">Beta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=4" title="Edit section: Beta"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Beta-existential-graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Beta-existential-graph.png/220px-Beta-existential-graph.png" decoding="async" width="220" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Beta-existential-graph.png/330px-Beta-existential-graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Beta-existential-graph.png/440px-Beta-existential-graph.png 2x" data-file-width="1080" data-file-height="346" /></a><figcaption>Existential graph of the statement "There is something that is not a human"</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Existential_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Existential_graphs.png/220px-Existential_graphs.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Existential_graphs.png/330px-Existential_graphs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Existential_graphs.png/440px-Existential_graphs.png 2x" data-file-width="694" data-file-height="518" /></a><figcaption>beta existential graphs</figcaption></figure> <p>In the case of betagraphs, the atomic expressions are no longer propositional letters (P, Q, R,...) or statements ("It rains," "Peirce died in poverty"), but predicates in the sense of predicate logic (see there for more details), possibly abbreviated to predicate letters (F, G, H,...). A predicate in the sense of predicate logic is a sequence of words with clearly defined spaces that becomes a propositional sentence if you insert a proper noun into each space. For example, the word sequence "_ x is a human" is a predicate because it gives rise to the declarative sentence "Peirce is a human" if you enter the proper name "Peirce" in the blank space. Likewise, the word sequence "_<sub>1</sub> is richer than _<sub>2</sub>" is a predicate, because it results in the statement "Socrates is richer than Plato" if the proper names "Socrates" or "Plato" are inserted into the spaces. </p> <div class="mw-heading mw-heading3"><h3 id="Notation_of_betagraphs">Notation of betagraphs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=5" title="Edit section: Notation of betagraphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> The basic language device is the line of identity, a thickly drawn line of any form. The identity line docks onto the blank space of a predicate to show that the predicate applies to at least one individual. In order to express that the predicate "_ is a human being" applies to at least one individual – i.e. to say that there is (at least) one human being – one writes an identity line in the blank space of the predicate "_ is a human being:"</p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Existential_graph_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/51/Existential_graph_3.png" decoding="async" width="175" height="120" class="mw-file-element" data-file-width="175" data-file-height="120" /></a><figcaption>Existential graph of the statement "Some man eats a man"</figcaption></figure> <p>The beta graphs can be read as a system in which all formula are to be taken as closed, because all variables are implicitly quantified. If the "shallowest" part of a line of identity has even depth, the associated variable is tacitly <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existentially</a> (<a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universally</a>) quantified. </p><p>Zeman (1964) was the first to note that the <i>beta</i> graphs are <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> with <a href="/wiki/First-order_logic#Equality_and_its_axioms" title="First-order logic">equality</a> (also see Zeman 1967). However, the secondary literature, especially Roberts (1973) and Shin (2002), does not agree on how this is. Peirce's writings do not address this question, because first-order logic was first clearly articulated only after his death, in the 1928 first edition of <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Wilhelm Ackermann</a>'s <i><a href="/wiki/Principles_of_Mathematical_Logic" title="Principles of Mathematical Logic">Principles of Mathematical Logic</a></i>. </p> <div class="mw-heading mw-heading3"><h3 id="Gamma">Gamma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=6" title="Edit section: Gamma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Add to the syntax of <i>alpha</i> a second kind of <a href="/wiki/Simple_closed_curve" class="mw-redirect" title="Simple closed curve">simple closed curve</a>, written using a dashed rather than a solid line. Peirce proposed rules for this second style of cut, which can be read as the primitive <a href="/wiki/Unary_operation" title="Unary operation">unary operator</a> of <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>. </p><p>Zeman (1964) was the first to note that the <i>gamma</i> graphs are equivalent to the well-known <a href="/wiki/Modal_logic" title="Modal logic">modal logics S4</a> and <a href="/wiki/S5_(modal_logic)" title="S5 (modal logic)">S5</a>. Hence the <i>gamma</i> graphs can be read as a peculiar form of <a href="/wiki/Normal_modal_logic" title="Normal modal logic">normal modal logic</a>. This finding of Zeman's has received little attention to this day, but is nonetheless included here as a point of interest. </p> <div class="mw-heading mw-heading2"><h2 id="Peirce's_role"><span id="Peirce.27s_role"></span>Peirce's role</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=7" title="Edit section: Peirce's role"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The existential graphs are a curious offspring of <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a> the <a href="/wiki/Logic" title="Logic">logician</a>/mathematician with Peirce the founder of a major strand of <a href="/wiki/Semiotics" title="Semiotics">semiotics</a>. Peirce's graphical logic is but one of his many accomplishments in logic and mathematics. In a series of papers beginning in 1867, and culminating with his classic paper in the 1885 <i><a href="/wiki/American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a></i>, Peirce developed much of the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a>, <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional calculus</a>, <a href="/wiki/Quantification_(logic)" class="mw-redirect" title="Quantification (logic)">quantification</a> and the <a href="/wiki/First-order_logic" title="First-order logic">predicate calculus</a>, and some rudimentary <a href="/wiki/Set_theory" title="Set theory">set theory</a>. <a href="/wiki/Model_theory" title="Model theory">Model theorists</a> consider Peirce the first of their kind. He also extended <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a>'s <a href="/wiki/Relation_algebra" title="Relation algebra">relation algebra</a>. He stopped short of <a href="/wiki/Metalogic" title="Metalogic">metalogic</a> (which eluded even <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>). </p><p>But Peirce's evolving <a href="/wiki/Semiotic" class="mw-redirect" title="Semiotic">semiotic</a> theory led him to doubt the value of logic formulated using conventional linear notation, and to prefer that logic and mathematics be notated in two (or even three) dimensions. His work went beyond <a href="/wiki/Euler_diagram" title="Euler diagram">Euler's diagrams</a> and <a href="/wiki/John_Venn" title="John Venn">Venn</a>'s 1880 <a href="/wiki/Venn_diagram" title="Venn diagram">revision</a> thereof. <a href="/wiki/Frege" class="mw-redirect" title="Frege">Frege</a>'s 1879 work <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i> also employed a two-dimensional notation for logic, but one very different from Peirce's. </p><p>Peirce's first published paper on graphical logic (reprinted in Vol. 3 of his <i>Collected Papers</i>) proposed a system dual (in effect) to the <i>alpha</i> existential graphs, called the <a href="/wiki/Entitative_graph" class="mw-redirect" title="Entitative graph">entitative graphs</a>. He very soon abandoned this formalism in favor of the existential graphs. In 1911 <a href="/wiki/Victoria,_Lady_Welby" title="Victoria, Lady Welby">Victoria, Lady Welby</a> showed the existential graphs to <a href="/wiki/C._K._Ogden" class="mw-redirect" title="C. K. Ogden">C. K. Ogden</a> who felt they could usefully be combined with Welby's thoughts in a "less abstruse form."<sup id="cite_ref-Petrilli_12-0" class="reference"><a href="#cite_note-Petrilli-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Otherwise they attracted little attention during his life and were invariably denigrated or ignored after his death, until the PhD theses by Roberts (1964) and Zeman (1964). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Logical_NOR" title="Logical NOR">Nor operator</a></li> <li><a href="/wiki/Conceptual_graph" title="Conceptual graph">Conceptual graph</a></li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sander Peirce</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Peirce, C. S., "[On Junctures and Fractures in Logic]" (editors' title for MS 427 (the new numbering system), Fall–Winter 1882), and "Letter, Peirce to O. H. Mitchell" (L 294, 21 December 1882), <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#W" title="Charles Sanders Peirce bibliography">Writings of Charles S. Peirce</a></i>, v. 4, "Junctures" on pp. 391–393 (Google <a rel="nofollow" class="external text" href="https://archive.org/details/writingsofcharle0002peir">preview</a>) and the letter on pp. 394–399 (Google <a rel="nofollow" class="external text" href="https://archive.org/details/writingsofcharle0002peir">preview</a>). See <a href="/wiki/John_F._Sowa" title="John F. Sowa">Sowa, John F.</a> (1997), "Matching Logical Structure to Linguistic Structure", <i>Studies in the Logic of Charles Sanders Peirce</i>, Nathan Houser, Don D. Roberts, and James Van Evra, editors, Bloomington and Indianapolis: Indiana University Press, pp. 418–444, see 420, 425, 426, 428.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSmullyan1968" class="citation cs2">Smullyan, Raymond M. (1968), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-642-86718-7_13">"Prenex Tableaux"</a>, <i>First-Order Logic</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 117–121, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-86720-0" title="Special:BookSources/978-3-642-86720-0"><bdi>978-3-642-86720-0</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2024-07-10</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=First-Order+Logic&rft.atitle=Prenex+Tableaux&rft.pages=117-121&rft.date=1968&rft.isbn=978-3-642-86720-0&rft.aulast=Smullyan&rft.aufirst=Raymond+M.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-642-86718-7_13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExistential+graph" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">"Peirce wants a sign which will not merely be conventionally understood [...], but which will "wear its meaning on its sleeve," so to speak" (Zeman 1964, page 21, quoted from the online edition)</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">"[algebraic formulas] are not ‚iconic‘ – that is, they do not resemble the objects or relationships they represent. Peirce took this to be a defect.“ (Roberts 1973, Seite 17)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">"[Peirce's] graphical publications were few and not easy to understand, as he admitted himself." (Roberts 1973, page 12)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">"[T]he syntax of Peirce's graphs lacks, at least in general, the combinatorial elegance and simplicity of linear notations" (Hammer 1998, page 502)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Roberts points out that even in the standard work on the history of logic, Kneale/Kneale: <i>The Development of Logic.</i> Clarendon Press. Oxford 1962, ISBN 0-19-824773-7, the logical diagrams of Peirce are not mentioned.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">"One questions the efficacy of Peirce's diagrams [...]. Their basic machinery is too complex [...]." (Quine: Review of Collected Papers of Charles Sanders Peirce, Volume 4: The Simplest Mathematics, Isis 22, page 552, quoted in Roberts 1973, page 13)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">"Aside from their historic interest, Peirce's graphical formalisms are of current interest. Sowa's system of conceptual graphs [...] is based on Peirce's work. [Other work] also indicates increasing interest in the logic of graphical reasoning." (Hammer 1998, page 489)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">see, e.g., Sun-Joo Shin, "Reconstituting Beta Graphs into an Efficacious System," <i>Journal of Logic, Language and Information archive,</i> Volume 8, Issue 3, July 1999, 273–295.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">The evidence for this was provided by J. Jay Zeman in his dissertation in 1964 (see bibliography); for alpha graphs, see also the work of White, 1984</span> </li> <li id="cite_note-Petrilli-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Petrilli_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetrilli2017" class="citation book cs1">Petrilli, Susan (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gqs0DwAAQBAJ&q=Ogden+Welby&pg=PT244"><i>Victoria Welby and the Science of Signs: Significs, Semiotics, Philosophy of Language</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-29598-7" title="Special:BookSources/978-1-351-29598-7"><bdi>978-1-351-29598-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Victoria+Welby+and+the+Science+of+Signs%3A+Significs%2C+Semiotics%2C+Philosophy+of+Language&rft.pub=Routledge&rft.date=2017&rft.isbn=978-1-351-29598-7&rft.aulast=Petrilli&rft.aufirst=Susan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGqs0DwAAQBAJ%26q%3DOgden%2BWelby%26pg%3DPT244&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExistential+graph" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Primary_literature">Primary literature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=11" title="Edit section: Primary literature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>1931–1935 & 1958. <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#CP" title="Charles Sanders Peirce bibliography">The Collected Papers of Charles Sanders Peirce</a></i>. Volume 4, Book II: "Existential Graphs", consists of paragraphs 347–584. A discussion also begins in paragraph 617. <ul><li>Paragraphs 347–349 (II.1.1. "Logical Diagram")—Peirce's definition "Logical Diagram (or Graph)" in <a href="/wiki/James_Mark_Baldwin" title="James Mark Baldwin">Baldwin</a>'s <i>Dictionary of Philosophy and Psychology</i> (1902), <a rel="nofollow" class="external text" href="https://archive.org/details/beginningthirdr00randgoog/page/n58">v. 2, p. 28</a>. <i>Classics in the History of Psychology</i> <a rel="nofollow" class="external text" href="http://psychclassics.yorku.ca/Baldwin/Dictionary/defs/L4defs.htm#Logical%20Diagram">Eprint</a>.</li> <li>Paragraphs 350–371 (II.1.2. "Of Euler's Diagrams")—from "Graphs" (manuscript 479) c. 1903.</li> <li>Paragraphs 372–584 <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050901083355/http://www.existentialgraphs.com/#table2">Eprint</a>.</li> <li>Paragraphs 372–393 (II.2. "Symbolic Logic")—Peirce's part of "Symbolic Logic" in Baldwin's <i>Dictionary of Philosophy and Psychology</i> (1902) <a rel="nofollow" class="external text" href="https://archive.org/details/beginningthirdr00randgoog/page/n671">v. 2, pp. 645</a>–650, beginning (near second column's top) with "If symbolic logic be defined...". Paragraph 393 (Baldwin's DPP2 p. 650) is by Peirce and <a href="/wiki/Christine_Ladd-Franklin" title="Christine Ladd-Franklin">Christine Ladd-Franklin</a> ("C.S.P., C.L.F.").</li> <li>Paragraphs 394–417 (II.3. "Existential Graphs")—from Peirce's pamphlet <i>A Syllabus of Certain Topics of Logic</i>, pp. 15–23, Alfred Mudge & Son, Boston (1903).</li> <li>Paragraphs 418–509 (II.4. "On Existential Graphs, Euler's Diagrams, and Logical Algebra")—from "Logical Tracts, No. 2" (manuscript 492), c. 1903.</li> <li>Paragraphs 510–529 (II.5. "The Gamma Part of Existential Graphs")—from "Lowell Lectures of 1903," Lecture IV (manuscript 467).</li> <li>Paragraphs 530–572 (II.6.)—"Prolegomena To an Apology For Pragmaticism" (1906), <i><a href="/wiki/The_Monist" title="The Monist">The Monist</a></i>, v. XVI, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3KoLAAAAIAAJ&pg=RA2-PA492">n. 4, pp. 492</a>-546. Corrections (1907) in <i>The Monist</i> v. XVII, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RqsLAAAAIAAJ&pg=PA160">p. 160</a>.</li> <li>Paragraphs 573–584 (II.7. "An Improvement on the Gamma Graphs")—from "For the National Academy of Science, 1906 April Meeting in Washington" (manuscript 490).</li> <li>Paragraphs 617–623 (at least) (in Book III, Ch. 2, §2, paragraphs 594–642)—from "Some Amazing Mazes: Explanation of Curiosity the First", <i>The Monist</i>, v. XVIII, 1908, <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_CqsLAAAAIAAJ_2/page/n497">n. 3, pp. 416</a>-464, see starting <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_CqsLAAAAIAAJ_2/page/n521">p. 440</a>.</li></ul></li> <li>1992. "Lecture Three: The Logic of Relatives", <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#RLT" title="Charles Sanders Peirce bibliography">Reasoning and the Logic of Things</a></i>, pp. 146–164. Ketner, Kenneth Laine (editing and introduction), and <a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a> (commentary). <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>. Peirce's 1898 lectures in Cambridge, Massachusetts.</li> <li>1977, 2001. <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#SS" title="Charles Sanders Peirce bibliography">Semiotic and Significs</a>: The Correspondence between C.S. Peirce and <a href="/wiki/Victoria_Lady_Welby" class="mw-redirect" title="Victoria Lady Welby">Victoria Lady Welby</a></i>. Hardwick, C.S., ed. Lubbock TX: Texas Tech University Press. 2nd edition 2001.</li> <li><a rel="nofollow" class="external text" href="http://www.jfsowa.com/peirce/ms514.htm">A transcription of Peirce's MS 514</a> (1909), edited with commentary by <a href="/wiki/John_Sowa" class="mw-redirect" title="John Sowa">John Sowa</a>.</li></ul> <p>Currently, the chronological critical edition of Peirce's works, the <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#W" title="Charles Sanders Peirce bibliography">Writings</a></i>, extends only to 1892. Much of Peirce's work on <a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">logical graphs</a> consists of manuscripts written after that date and still unpublished. Hence our understanding of Peirce's graphical logic is likely to change as the remaining 23 volumes of the chronological edition appear. </p> <div class="mw-heading mw-heading3"><h3 id="Secondary_literature">Secondary literature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=12" title="Edit section: Secondary literature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hammer, Eric M. (1998), "Semantics for Existential Graphs," <i>Journal of Philosophical Logic 27</i>: 489–503.</li> <li>Ketner, Kenneth Laine <ul><li>(1981), "The Best Example of Semiosis and Its Use in Teaching Semiotics", <i>American Journal of Semiotics</i> v. I, n. 1–2, pp. 47–83. Article is an introduction to existential graphs.</li> <li>(1990), <i>Elements of Logic: An Introduction to Peirce's Existential Graphs</i>, Texas Tech University Press, Lubbock, TX, 99 pages, spiral-bound.</li></ul></li> <li>Queiroz, João & Stjernfelt, Frederik <ul><li>(2011), "Diagrammatical Reasoning and Peircean Logic Representation", <i>Semiotica</i> vol. 186 (1/4). (Special issue on Peirce's diagrammatic logic.) <a rel="nofollow" class="external autonumber" href="http://www.degruyter.com/view/j/semi.2011.2011.issue-186/issue-files/semi.2011.2011.issue-186.xml">[1]</a></li></ul></li> <li>Roberts, Don D. <ul><li>(1964), "Existential Graphs and Natural Deduction" in Moore, E. C., and Robin, R. S., eds., <i>Studies in the Philosophy of C. S. Peirce, 2nd series</i>. Amherst MA: <a href="/wiki/University_of_Massachusetts_Press" title="University of Massachusetts Press">University of Massachusetts Press</a>. The first publication to show any sympathy and understanding for Peirce's graphical logic.</li> <li>(1973). <i>The Existential Graphs of C.S. Peirce.</i> John Benjamins. An outgrowth of his 1963 thesis.</li></ul></li> <li><a href="/wiki/Sun-Joo_Shin" title="Sun-Joo Shin">Shin, Sun-Joo</a> (2002), <i>The Iconic Logic of Peirce's Graphs</i>. MIT Press.</li> <li><a href="/wiki/Fernando_Zalamea" title="Fernando Zalamea">Zalamea, Fernando</a>. <i>Peirce's Logic of Continuity.</i> Docent Press, Boston MA. 2012. ISBN 9 780983 700494. <ul><li>Part II: Peirce's Existential Graphs, pp. 76-162.</li></ul></li> <li>Zeman, J. J. <ul><li>(1964), <i><a rel="nofollow" class="external text" href="http://users.clas.ufl.edu/jzeman/">The Graphical Logic of C.S. Peirce.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180914015520/http://users.clas.ufl.edu/jzeman/">Archived</a> 2018-09-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i> Unpublished Ph.D. thesis submitted to the <a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a>.</li> <li>(1967), "A System of Implicit Quantification," <i>Journal of Symbolic Logic 32</i>: 480–504.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Existential_graph&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: <a rel="nofollow" class="external text" href="http://setis.library.usyd.edu.au/stanford/entries/peirce-logic/#SymIcoRep">Peirce's Logic</a> by <a href="/wiki/Sun-Joo_Shin" title="Sun-Joo Shin">Sun-Joo Shin</a> and Eric Hammer.</li> <li>Dau, Frithjof, <a rel="nofollow" class="external text" href="http://www.dr-dau.net/eg_readings.shtml">Peirce's Existential Graphs --- Readings and Links.</a> An annotated bibliography on the existential graphs.</li> <li>Gottschall, Christian, <a rel="nofollow" class="external text" href="http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-peirce.html">Proof Builder</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060212072303/http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-peirce.html">Archived</a> 2006-02-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> — Java applet for deriving Alpha graphs.</li> <li>Liu, Xin-Wen, "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20081022205810/http://philosophy.cass.cn/facu/liuxinwen/01.htm">The literature of C.S. Peirce’s Existential Graphs</a>" (via Wayback Machine), Institute of Philosophy, Chinese Academy of Social Sciences, Beijing, PRC.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSowa" class="citation web cs1"><a href="/wiki/John_Sowa" class="mw-redirect" title="John Sowa">Sowa, John F.</a> <a rel="nofollow" class="external text" href="http://www.jfsowa.com/pubs/laws.htm">"Laws, Facts, and Contexts: Foundations for Multimodal Reasoning"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-10-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Laws%2C+Facts%2C+and+Contexts%3A+Foundations+for+Multimodal+Reasoning&rft.aulast=Sowa&rft.aufirst=John+F.&rft_id=http%3A%2F%2Fwww.jfsowa.com%2Fpubs%2Flaws.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExistential+graph" class="Z3988"></span> (NB. Existential graphs and <a href="/wiki/Conceptual_graph" title="Conceptual graph">conceptual graphs</a>.)</li> <li>Van Heuveln, Bram, "<a rel="nofollow" class="external text" href="http://www.cogsci.rpi.edu/~heuveb/research/EG/index.html">Existential Graphs.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090829072353/http://www.cogsci.rpi.edu/~heuveb/research/EG/index.html">Archived</a> 2009-08-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>" Dept. of Cognitive Science, <a href="/wiki/Rensselaer_Polytechnic_Institute" title="Rensselaer Polytechnic Institute">Rensselaer Polytechnic Institute</a>. Alpha only.</li> <li>Zeman, Jay J., "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20050901083355/http://www.existentialgraphs.com/">Existential Graphs</a>". With <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050901083355/http://www.existentialgraphs.com/#table2">four online papers</a> by Peirce.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output 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