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<title>Logical consequence</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Logical consequence</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Entailment" redirects here. For other uses, see <a href="Entail_(disambiguation)" class="mw-disambig" title="Entail (disambiguation)">Entail (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Therefore" redirects here. For the therefore symbol ∴, see <a href="Therefore_sign" title="Therefore sign">Therefore sign</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Logical implication" redirects here. For the binary connective, see <a href="Material_conditional" title="Material conditional">Material conditional</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"⊧" redirects here. For the symbol, see <a href="Double_turnstile" title="Double turnstile">Double turnstile</a>.</div>

<p><b>Logical consequence</b> (also <b>entailment</b> or <b>logical implication</b>) is a fundamental <a href="Concept" title="Concept">concept</a> in <a href="Logic" title="Logic">logic</a> which describes the relationship between <a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">statements</a> that hold true when one statement logically <i>follows from</i> one or more statements. A <a href="Validity_(logic)" title="Validity (logic)">valid</a> logical <a href="Argument" title="Argument">argument</a> is one in which the <a href="Consequent" title="Consequent">conclusion</a> is entailed by the <a href="Premise" title="Premise">premises</a>, because the conclusion is the consequence of the premises. The <a href="Philosophical_analysis" title="Philosophical analysis">philosophical analysis</a> of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?<sup id="cite_ref-sep_1-0" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> All of <a href="Philosophical_logic" title="Philosophical logic">philosophical logic</a> is meant to provide accounts of the nature of logical consequence and the nature of <a href="Logical_truth" title="Logical truth">logical truth</a>.<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>

</p><p>Logical consequence is <a href="Logical_truth" title="Logical truth">necessary</a> and <a href="Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formal</a>, by way of examples that explain with <a href="Formal_proof" title="Formal proof">formal proof</a> and <a href="Interpretation_(logic)" title="Interpretation (logic)">models of interpretation</a>.<sup id="cite_ref-sep_1-1" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A sentence is said to be a logical consequence of a set of sentences, for a given <a href="Formal_language" title="Formal language">language</a>, <a href="If_and_only_if" title="If and only if">if and only if</a>, using only logic (i.e., without regard to any <i>personal</i> interpretations of the sentences) the sentence must be true if every sentence in the set is true.<sup id="cite_ref-iep_3-0" class="reference"><a href="#cite_note-iep-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Logicians make precise accounts of logical consequence regarding a given <a href="Formal_language" title="Formal language">language</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math></span><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span>. The Polish logician <a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the <a href="Logical_form" title="Logical form">logical form</a> of the sentences: (2) The relation is <a href="A_priori_and_a_posteriori" title="A priori and a posteriori">a priori</a>, i.e., it can be determined with or without regard to <a href="Empirical_evidence" title="Empirical evidence">empirical evidence</a> (sense experience); and (3) The logical consequence relation has a <a href="Modal_logic" title="Modal logic">modal</a> component.<sup id="cite_ref-iep_3-1" class="reference"><a href="#cite_note-iep-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Formal_accounts">Formal accounts</h2></div>

<p>The most widely prevailing view on how best to account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on the structure or <a href="Logical_form" title="Logical form">logical form</a> of the statements without regard to the contents of that form.

</p><p>Syntactic accounts of logical consequence rely on <a href="Schema_(logic)" class="mw-redirect" title="Schema (logic)">schemes</a> using <a href="Inference_rule" class="mw-redirect" title="Inference rule">inference rules</a>. For instance, we can express the logical form of a valid argument as:

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<dd>Therefore, all <i>X</i> are <i>Z</i>.</dd></dl>

<p>This argument is formally valid, because every <a href="Substitution_(logic)" title="Substitution (logic)">instance</a> of arguments constructed using this scheme is valid.

</p><p>This is in contrast to an argument like "Fred is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called <a href="Material_conditional" title="Material conditional">material consequence</a> of "Fred is Mike's brother's son", not a formal consequence. A formal consequence must be true <i>in all cases</i>, however this is an incomplete definition of formal consequence, since even the argument "<i>P</i> is <i>Q</i><span class="nowrap" style="padding-left:0.1em;">'</span>s brother's son, therefore <i>P</i> is <i>Q</i><span class="nowrap" style="padding-left:0.1em;">'</span>s nephew" is valid in all cases, but is not a <i>formal</i> argument.<sup id="cite_ref-sep_1-2" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="A_priori_property_of_logical_consequence">A priori property of logical consequence</h2></div>

<p>If it is known that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> follows logically from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> will affect that knowledge. Our knowledge that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> is a logical consequence of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> cannot be influenced by <a href="A_priori_and_a_posteriori" title="A priori and a posteriori">empirical knowledge</a>.<sup id="cite_ref-sep_1-3" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.<sup id="cite_ref-sep_1-4" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.<sup id="cite_ref-sep_1-5" class="reference"><a href="#cite_note-sep-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Proofs_and_models">Proofs and models</h2></div>

<p>The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of <i>proofs</i> and via <i>models</i>. The study of the syntactic consequence (of a logic) is called (its) <a href="Proof_theory" title="Proof theory">proof theory</a> whereas the study of (its) semantic consequence is called (its) <a href="Model_theory" title="Model theory">model theory</a>.<sup id="cite_ref-ChiaraDoets1996_4-0" class="reference"><a href="#cite_note-ChiaraDoets1996-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Syntactic_consequence">Syntactic consequence</h3></div>

<p>A formula <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is a <b>syntactic consequence</b><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><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><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><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><sup id="cite_ref-Kleene52_9-0" class="reference"><a href="#cite_note-Kleene52-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> within some <a href="Formal_system" title="Formal system">formal system</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS}}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS}}}</annotation>

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</math></span><img src="./ab025245b89b5d8e91a0f027632b0c1764ab2a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.419ex; height:2.176ex;" alt="{\displaystyle {\mathcal {FS}}}" loading="lazy"></span> of a set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> of formulas if there is a <a href="Formal_proof" title="Formal proof">formal proof</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS}}}</annotation>

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</math></span><img src="./ab025245b89b5d8e91a0f027632b0c1764ab2a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.419ex; height:2.176ex;" alt="{\displaystyle {\mathcal {FS}}}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>. This is denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \vdash _{\mathcal {FS}}A}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma \vdash _{\mathcal {FS}}A}</annotation>

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</math></span><img src="./63b152eb8c8c41670800ead25a9d78cf0c6d4b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.556ex; height:2.509ex;" alt="{\displaystyle \Gamma \vdash _{\mathcal {FS}}A}" loading="lazy"></span>. The turnstile symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

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</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). <sup id="cite_ref-Kleene52_9-1" class="reference"><a href="#cite_note-Kleene52-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>Syntactic consequence does not depend on any <a href="Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> of the formal system.<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>

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<div class="mw-heading mw-heading3"><h3 id="Semantic_consequence">Semantic consequence</h3></div>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS}}}</annotation>

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</math></span><img src="./ab025245b89b5d8e91a0f027632b0c1764ab2a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.419ex; height:2.176ex;" alt="{\displaystyle {\mathcal {FS}}}" loading="lazy"></span> of a set of statements <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> if and only if there is no model <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation>

</semantics>

</math></span><img src="./0e9730a0ada0426927ff64141eb9f505eca132d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}" loading="lazy"></span> in which all members of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> are true and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma \models _{\mathcal {FS}}A}</annotation>

</semantics>

</math></span><img src="./416893c958d5190664850fa855483e737fb785bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.151ex; height:2.843ex;" alt="{\displaystyle \Gamma \models _{\mathcal {FS}}A}" loading="lazy"></span>. Or, in other words, the set of the interpretations that make all members of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> true is a subset of the set of the interpretations that make <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading2"><h2 id="Modal_accounts">Modal accounts</h2></div>

<p><a href="Modal_logic" title="Modal logic">Modal</a> accounts of logical consequence are variations on the following basic idea:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

</semantics>

</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> are true, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<p>Alternatively (and, most would say, equivalently):

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

</semantics>

</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> to be true and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<p>Such accounts are called "modal" because they appeal to the modal notions of <a href="Logical_truth" title="Logical truth">logical necessity</a> and <a href="Logical_possibility" title="Logical possibility">logical possibility</a>. 'It is necessary that' is often expressed as a <a href="Universal_quantification" title="Universal quantification">universal quantifier</a> over <a href="Possible_world" title="Possible world">possible worlds</a>, so that the accounts above translate as:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

</semantics>

</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> are true and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<p>Consider the modal account in terms of the argument given as an example above:

</p>

<dl><dd>All frogs are green.</dd>

<dd>Kermit is a frog.</dd>

<dd>Therefore, Kermit is green.</dd></dl>

<p>The conclusion is a logical consequence of the premises because we can not imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.

</p>

<div class="mw-heading mw-heading3"><h3 id="Modal-formal_accounts">Modal-formal accounts</h3></div>

<p>Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

</semantics>

</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>/<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Warrant-based_accounts">Warrant-based accounts</h3></div>

<p>The accounts considered above are all "truth-preservational", in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed "<a href="Theory_of_justification" class="mw-redirect" title="Theory of justification">warrant</a>-preservational" accounts, according to which the characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored by <a href="Intuitionist" class="mw-redirect" title="Intuitionist">intuitionists.</a>

</p>

<div class="mw-heading mw-heading3"><h3 id="Non-monotonic_logical_consequence">Non-monotonic logical consequence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Non-monotonic_logic" title="Non-monotonic logic">Non-monotonic logic</a> and <a href="Belief_revision#Non-monotonic_inference_relation" title="Belief revision">Belief revision § Non-monotonic inference relation</a></div>

<p>The accounts discussed above all yield <a href="Monotonicity_of_entailment" title="Monotonicity of entailment">monotonic</a> consequence relations, i.e. ones such that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>. It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of

</p>

<dl><dd>{Birds can typically fly, Tweety is a bird}</dd></dl>

<p>but not of

</p>

<dl><dd>{Birds can typically fly, Tweety is a bird, Tweety is a penguin}.</dd></dl>

/* start https://en.wikipedia.org/ */

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<ul><li><a href="Abstract_algebraic_logic" title="Abstract algebraic logic">Abstract algebraic logic</a></li>

<li><a href="Ampheck" class="mw-redirect" title="Ampheck">Ampheck</a></li>

<li><a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra (logic)</a></li>

<li><a href="Boolean_domain" title="Boolean domain">Boolean domain</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean function</a></li>

<li><a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a></li>

<li><a href="Causality" title="Causality">Causality</a></li>

<li><a href="Deductive_reasoning" title="Deductive reasoning">Deductive reasoning</a></li>

<li><a href="Logic_gate" title="Logic gate">Logic gate</a></li>

<li><a href="Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li>

<li><a href="Peirce's_law" title="Peirce's law">Peirce's law</a></li>

<li><a href="Probabilistic_logic" title="Probabilistic logic">Probabilistic logic</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Sole_sufficient_operator" class="mw-redirect" title="Sole sufficient operator">Sole sufficient operator</a></li>

<li><a href="Strawson_entailment" title="Strawson entailment">Strawson entailment</a></li>

<li><a href="Strict_conditional" title="Strict conditional">Strict conditional</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology (logic)</a></li>

<li><a href="Tautological_consequence" title="Tautological consequence">Tautological consequence</a></li>

<li><a href="Therefore_sign" title="Therefore sign">Therefore sign</a></li>

<li><a href="Turnstile_(symbol)" title="Turnstile (symbol)">Turnstile (symbol)</a></li>

<li><a href="Double_turnstile" title="Double turnstile">Double turnstile</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-sep-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-sep_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sep_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sep_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-sep_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-sep_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-sep_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text">Beall, JC and Restall, Greg, <i><a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/fall2009/entries/logical-consequence/">Logical Consequence</a></i> The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.).</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"><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine, Willard Van Orman</a>, <i>Philosophy of Logic</i>.</span>

</li>

<li id="cite_note-iep-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-iep_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-iep_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">McKeon, Matthew, <i><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/logcon/">Logical Consequence</a></i> Internet Encyclopedia of Philosophy.</span>

</li>

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</style><cite id="CITEREFKosta_Dosen1996" class="citation book cs1">Kosta Dosen (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TCthvF8xLIAC&pg=PA292">"Logical consequence: a turn in style"</a>. In <a href="Maria_Luisa_Dalla_Chiara" title="Maria Luisa Dalla Chiara">Maria Luisa Dalla Chiara</a>; Kees Doets; Daniele Mundici; Johan van Benthem (eds.). <i>Logic and Scientific Methods: Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995</i>. Springer. p. 292. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7923-4383-7</bdi>.</cite></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"><a href="Michael_Dummett" title="Michael Dummett">Dummett, Michael</a> (1993) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EYP7uCZIRQYC&q=syntactic+consequence%27%27Frege%3A&pg=PA82"><i>philosophy of language</i></a> Harvard University Press, p.82ff</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"><a href="Jonathan_Lear" title="Jonathan Lear">Lear, Jonathan</a> (1986) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lXI7AAAAIAAJ&q=syntactic+consequence%27%27Aristotle&pg=PA1"><i>and Logical Theory</i></a> Cambridge University Press, 136p.</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">Creath, Richard, and <a href="Michael_Friedman_(philosopher)" title="Michael Friedman (philosopher)">Friedman, Michael</a> (2007) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=87BcFLgJmxMC&q=syntactic+consequence%27%27The&pg=PA189"><i>Cambridge companion to Carnap</i></a> Cambridge University Press, 371p.</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"><a rel="nofollow" class="external text" href="http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence">FOLDOC: "syntactic consequence"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130403201417/http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence">Archived</a> 2013-04-03 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

</li>

<li id="cite_note-Kleene52-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kleene52_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kleene52_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">S. C. Kleene, <i><a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/523942">Introduction to Metamathematics</a></i> (1952), Van Nostrand Publishing. p.88.</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"><cite id="CITEREFHunter1996" class="citation book cs1"><a href="Geoffrey_Hunter_(logician)" title="Geoffrey Hunter (logician)">Hunter, Geoffrey</a> (1996) [1971]. <i>Metalogic: An Introduction to the Metatheory of Standard First-Order Logic</i>. University of California Press (published 1973). p. 101. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780520023567</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36312727">36312727</a>.</cite> (<a rel="nofollow" class="external text" href="https://archive.org/details/metalogicintrodu0000hunt">accessible to patrons with print disabilities</a>)</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"><a href="John_Etchemendy" title="John Etchemendy">Etchemendy, John</a>, <i>Logical consequence</i>, The Cambridge Dictionary of Philosophy</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Resources">Resources</h2></div>

<ul><li><cite id="CITEREFAndersonBelnap1975" class="citation cs2">Anderson, A.R.; Belnap, N.D. Jr. (1975), <i>Entailment</i>, vol. 1, Princeton, NJ: Princeton</cite>.</li>

<li><cite id="CITEREFAugusto2017" class="citation cs2">Augusto, Luis M. (2017), <i>Logical consequences. Theory and applications: An introduction.</i></cite> London: College Publications. Series: <a rel="nofollow" class="external text" href="http://www.collegepublications.co.uk/logic/mlf/?00029">Mathematical logic and foundations</a>.</li>

<li><cite id="CITEREFBarwiseEtchemendy2008" class="citation cs2"><a href="Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>; <a href="John_Etchemendy" title="John Etchemendy">Etchemendy, John</a> (2008), <i>Language, Proof and Logic</i>, Stanford: CSLI Publications</cite>.</li>

<li><cite id="CITEREFBrown2003" class="citation cs2">Brown, Frank Markham (2003), <i>Boolean Reasoning: The Logic of Boolean Equations</i></cite> 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.</li>

<li><cite id="CITEREFDavis1965" class="citation cs2">Davis, Martin, ed. (1965), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qW8x7sQ4JXgC&q=consequence"><i>The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions</i></a>, New York: Raven Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486432281</bdi></cite>. Papers include those by <a href="G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>, <a href="Alonzo_Church" title="Alonzo Church">Church</a>, <a href="J._Barkley_Rosser" title="J. Barkley Rosser">Rosser</a>, <a href="Kleene" class="mw-redirect" title="Kleene">Kleene</a>, and <a href="Emil_Leon_Post" title="Emil Leon Post">Post</a>.</li>

<li><cite id="CITEREFDummett1991" class="citation cs2">Dummett, Michael (1991), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lvsVFxK3BPcC&q=consequence"><i>The Logical Basis of Metaphysics</i></a>, Harvard University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780674537866</bdi></cite>.</li>

<li><cite id="CITEREFEdgington2001" class="citation cs2">Edgington, Dorothy (2001), <i>Conditionals</i>, Blackwell</cite> in Lou Goble (ed.), <i>The Blackwell Guide to Philosophical Logic</i>.</li>

<li><cite id="CITEREFEdgington2006" class="citation cs2">Edgington, Dorothy (2006), <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/conditionals">"Indicative Conditionals"</a>, <i>Conditionals</i>, Metaphysics Research Lab, Stanford University</cite> in Edward N. Zalta (ed.), <i>The Stanford Encyclopedia of Philosophy</i>.</li>

<li><cite id="CITEREFEtchemendy1990" class="citation cs2">Etchemendy, John (1990), <i>The Concept of Logical Consequence</i>, Harvard University Press</cite>.</li>

<li><cite id="CITEREFGoble2001" class="citation cs2">Goble, Lou, ed. (2001), <i>The Blackwell Guide to Philosophical Logic</i>, Blackwell</cite>.</li>

<li><cite id="CITEREFHanson1997" class="citation cs2">Hanson, William H (1997), "The concept of logical consequence", <i>The Philosophical Review</i>, <b>106</b> (3): <span class="nowrap">365–</span>409, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2998398">10.2307/2998398</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2998398">2998398</a></cite> 365–409.</li>

<li><cite id="CITEREFHendricks2005" class="citation cs2"><a href="Vincent_F._Hendricks" title="Vincent F. Hendricks">Hendricks, Vincent F.</a> (2005), <i>Thought 2 Talk: A Crash Course in Reflection and Expression</i>, New York: Automatic Press / VIP, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-87-991013-7-5</bdi></cite></li>

<li><cite id="CITEREFPlanchette2001" class="citation cs2">Planchette, P. A. (2001), <i>Logical Consequence</i></cite> in Goble, Lou, ed., <i>The Blackwell Guide to Philosophical Logic</i>. Blackwell.</li>

<li><cite id="CITEREFQuine1982" class="citation cs2"><a href="W.V._Quine" class="mw-redirect" title="W.V. Quine">Quine, W.V.</a> (1982), <i>Methods of Logic</i>, Cambridge, MA: Harvard University Press</cite> (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), (4th edition, 1982).</li>

<li><cite id="CITEREFShapiro2002" class="citation cs2"><a href="Stewart_Shapiro" title="Stewart Shapiro">Shapiro, Stewart</a> (2002), <i>Necessity, meaning, and rationality: the notion of logical consequence</i></cite> in D. Jacquette, ed., <i>A Companion to Philosophical Logic</i>. Blackwell.</li>

<li><cite id="CITEREFTarski1936" class="citation cs2"><a href="Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1936), <i>On the concept of logical consequence</i></cite> Reprinted in Tarski, A., 1983. <i>Logic, Semantics, Metamathematics</i>, 2nd ed. <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. Originally published in <a href="Polish_language" title="Polish language">Polish</a> and <a href="German_language" title="German language">German</a>.</li>

<li><cite id="CITEREFRyszard_Wójcicki1988" class="citation book cs1">Ryszard Wójcicki (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryoflogicalc0000wojc"><i>Theory of Logical Calculi: Basic Theory of Consequence Operations</i></a></span>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-277-2785-5</bdi>.</cite></li>

<li>A paper on 'implication' from math.niu.edu, <a rel="nofollow" class="external text" href="http://www.math.niu.edu/~richard/Math101/implies.pdf">Implication</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141021082239/http://www.math.niu.edu/~richard/Math101/implies.pdf">Archived</a> 2014-10-21 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li>A definition of 'implicant' <a rel="nofollow" class="external text" href="http://www.allwords.com/word-implicant.html">AllWords</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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</div>

<ul><li><cite id="CITEREFBeallRestall2013" class="citation encyclopaedia cs1"><a href="Jc_Beall" title="Jc Beall">Beall, Jc</a>; <a href="Greg_Restall" title="Greg Restall">Restall, Greg</a> (2013-11-19). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logical-consequence/">"Logical Consequence"</a>. In <a href="Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i> (Winter 2016 ed.).</cite></li>

<li><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://iep.utm.edu/logcon/">"Logical consequence"</a>. <i><a href="Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</cite></li>

<li><a rel="nofollow" class="external text" href="https://www.inphoproject.org/taxonomy/2409">Logical consequence</a> at the <a href="Indiana_Philosophy_Ontology_Project" class="mw-redirect" title="Indiana Philosophy Ontology Project">Indiana Philosophy Ontology Project</a></li>

<li><a rel="nofollow" class="external text" href="https://philpapers.org/browse/logical-consequence-and-entailment">Logical consequence</a> at <a href="PhilPapers" title="PhilPapers">PhilPapers</a></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Implication">"Implication"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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</style><div id="Logic605" style="font-size:114%;margin:0 4em"><a href="Logic" title="Logic">Logic</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_logic" title="History of logic">History</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logic_in_computer_science" title="Logic in computer science">Computer science</a></li>

<li><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical</a></li>

<li><a href="Informal_logic" title="Informal logic">Informal</a>

<ul><li><a href="Critical_thinking" title="Critical thinking">Critical thinking</a></li>

<li><a href="Reason" title="Reason">Reason</a></li></ul></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical</a></li>

<li><a href="Non-classical_logic" title="Non-classical logic">Non-classical</a></li>

<li><a href="Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Argumentation_theory" title="Argumentation theory">Argumentation</a></li>

<li><a href="Metalogic" title="Metalogic">Metalogic</a></li>

<li><a href="Metamathematics" title="Metamathematics">Metamathematics</a></li>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abductive_reasoning" title="Abductive reasoning">Abduction</a></li>

<li><a href="Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li>

<li><a href="Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li>

<li><a href="Consequent" title="Consequent">Consequent</a></li>

<li><a href="Contradiction" title="Contradiction">Contradiction</a>

<ul><li><a href="Paradox" title="Paradox">Paradox</a></li>

<li><a href="Antinomy" title="Antinomy">Antinomy</a></li></ul></li>

<li><a href="Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>

<li><a href="Deductive_closure" title="Deductive closure">Deductive closure</a></li>

<li><a href="Definition" title="Definition">Definition</a></li>

<li><a href="Description" title="Description">Description</a></li>

<li><a href="Dichotomy" title="Dichotomy">Dichotomy</a></li>

<li>

<ul><li><a href="Linguistic_entailment" title="Linguistic entailment">Linguistic</a></li></ul></li>

<li><a href="Inductive_reasoning" title="Inductive reasoning">Induction</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li>

<li><a href="Premise" title="Premise">Premise</a></li>

<li><a href="Probability" title="Probability">Probability</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statement</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Truth" title="Truth">Truth</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Index_of_logic_articles" class="mw-redirect" title="Index of logic articles">Topics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>

<li><a href="List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li>

<li><a href="List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_logicians" title="List of logicians">Logicians</a></li>

<li><a href="List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li>

<li><a href="List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li>

<li><a href="List_of_fallacies" title="List of fallacies">Fallacies</a></li>

<li><a href="List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>

<li><span class="noviewer" typeof="mw:File"><span title="Outline"></span></span> <a href="Outline_of_logic" title="Outline of logic">Outline</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3APhilosophy" title="Portal:Philosophy">Portal</a></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> WikiProject</li>

<li><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic344" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom" title="Axiom">Axiom</a>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>

<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>

<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>

<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>

<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>

<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>

<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>

<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>

<li><a href="Consistency" title="Consistency">Consistency</a>

<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>

<li><a href="Argument" title="Argument">Argument</a></li>

<li><a href="Soundness" title="Soundness">Soundness</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>

<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>

<li><a href="Truth_table" title="Truth table">Truth tables</a></li>

<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>

<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>

<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>

<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>

<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>

<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>

<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>

<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>

<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>

<li>Set operations:

<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>

<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Power_set" title="Power set">power set</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li>

<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>

<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>

<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>

<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>

<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>

<li><a href="Codomain" title="Codomain">codomain</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>

<li><a href="Injective_function" title="Injective function">In</a>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>

<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>

<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>

<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>

<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>

<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>

<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>

<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>

<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>

<li><a href="General_set_theory" title="General set theory">General</a></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>

<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>

<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>

<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br><a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>

<li><a href="Arity" title="Arity">Arity</a></li>

<li><a href="Automata_theory" title="Automata theory">Automata</a></li>

<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>

<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>

<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>

<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>

<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>

<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>

<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>

<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>

<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>

<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>

<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>

<li><a href="Formal_language" title="Formal language">Language</a></li>

<li><a href="Metalanguage" title="Metalanguage">Metalanguage</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connective</a>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>

<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>

<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>

<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>

<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>

<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>

<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>

<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>

<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>

<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>

<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>

<li><a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic<br>systems</a> <span style="font-size: 85%;">(<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:

<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>

<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>

<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>

<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>

<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>

<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>

<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>

<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>

<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>

<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>

<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>

<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:

<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:

<ul><li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i></a></li>

<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>

<ul><li><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>

<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>

<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Formal_system" title="Formal system">Systems</a>

<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>

<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>

<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>

<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>

<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>

<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>

<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>

<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>

<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>

<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>

<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>

<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>

<li><a href="Saturated_model" title="Saturated model">saturated</a></li>

<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>

<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>

<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>

<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>

<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>

<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>

<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>

<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>

<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>

<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>

<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>

<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>

<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>

<li><a href="T-schema" title="T-schema">T-schema</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>

<li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>

<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>

<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>

<li><a href="Computable_function" title="Computable function">Computable function</a></li>

<li><a href="Computable_set" title="Computable set">Computable set</a></li>

<li><a href="Decision_problem" title="Decision problem">Decision problem</a>

<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>

<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>

<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>

<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Concrete_category" title="Concrete category">Concrete</a>/<a href="Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li>

<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>

<li><a href="History_of_logic" title="History of logic">History of logic</a></li>

<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="Logicism" title="Logicism">Logicism</a></li>

<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Supertask" title="Supertask">Supertask</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_connectives157" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Common_logical_connectives157" style="font-size:114%;margin:0 4em">Common <a href="Logical_connective" title="Logical connective">logical connectives</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a>/<a href="Logical_truth" title="Logical truth">True</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \top }</annotation>

</semantics>

</math></span><img src="./cf12e436fef2365e76fcb1034a51179d8328bb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }" loading="lazy"></span></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> (<a href="NAND_gate" title="NAND gate">NAND gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\wedge }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\wedge }}}</annotation>

</semantics>

</math></span><img src="./2f076fb91b5d1276ab165ed8dfaa3cacab8d4cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\wedge }}}" loading="lazy"></span></li>

<li><a href="Converse_(logic)" title="Converse (logic)">Converse implication</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation>

</semantics>

</math></span><img src="./682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }" loading="lazy"></span></li>

<li><a href="Material_conditional" title="Material conditional">Implication</a> (<a href="IMPLY_gate" title="IMPLY gate">IMPLY gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation>

</semantics>

</math></span><img src="./469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span></li>

<li><a href="Logical_disjunction" title="Logical disjunction">Disjunction</a> (<a href="OR_gate" title="OR gate">OR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lor }</annotation>

</semantics>

</math></span><img src="./ab47f6b1f589aedcf14638df1d63049d233d851a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" loading="lazy"></span></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Negation" title="Negation">Negation</a> (<a href="Inverter_(logic_gate)" title="Inverter (logic gate)">NOT gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg }</annotation>

</semantics>

</math></span><img src="./fa78fd02085d39aa58c9e47a6d4033ce41e02fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }" loading="lazy"></span></li>

<li><a href="Exclusive_or" title="Exclusive or">Exclusive or</a> (<a href="XOR_gate" title="XOR gate">XOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation>

</semantics>

</math></span><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy"></span></li>

<li><a href="Logical_biconditional" title="Logical biconditional">Biconditional</a> (<a href="XNOR_gate" title="XNOR gate">XNOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \odot }</annotation>

</semantics>

</math></span><img src="./e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"></span></li>

<li><a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statement</a> (<a href="Digital_buffer" title="Digital buffer">Digital buffer</a>)</li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logical_NOR" title="Logical NOR">Joint denial</a> (<a href="NOR_gate" title="NOR gate">NOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\vee }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\vee }}}</annotation>

</semantics>

</math></span><img src="./a6f9bdf4cb18d1b79d370c396dc425e80f8340f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}" loading="lazy"></span></li>

<li><a href="Material_nonimplication" title="Material nonimplication">Nonimplication</a> (<a href="NIMPLY_gate" title="NIMPLY gate">NIMPLY gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nRightarrow }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nRightarrow }</annotation>

</semantics>

</math></span><img src="./3e05a42e88f019861cf404b6f982d8f729a4c4ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \nRightarrow }" loading="lazy"></span></li>

<li><a href="Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nLeftarrow }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nLeftarrow }</annotation>

</semantics>

</math></span><img src="./746948304d4d6a4903cc7bf82bf687b6f284eb92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \nLeftarrow }" loading="lazy"></span></li>

<li><a href="Logical_conjunction" title="Logical conjunction">Conjunction</a> (<a href="AND_gate" title="AND gate">AND gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

</semantics>

</math></span><img src="./d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"></span></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contradiction" title="Contradiction">Contradiction</a>/<a href="False_(logic)" title="False (logic)">False</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bot }</annotation>

</semantics>

</math></span><img src="./f282c7bc331cc3bfcf1c57f1452cc23c022f58de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }" loading="lazy"></span></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><span class="nowrap"><span class="noviewer" typeof="mw:File"><span></span></span> </span><a href="Portal%3APhilosophy" title="Portal:Philosophy">Philosophy portal</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_symbols352" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Common_logical_symbols352" style="font-size:114%;margin:0 4em">Common <a href="List_of_logic_symbols" title="List of logic symbols">logical symbols</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:100%;"><tbody><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Arrow_(symbol)" title="Arrow (symbol)">→</a> </div> <a href="Material_conditional" title="Material conditional">implies</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Horseshoe_(symbol)" title="Horseshoe (symbol)">⊃</a> </div> <a href="Material_conditional" title="Material conditional">implies</a>,<br><a href="Subset" title="Subset">superset</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Turned_A" title="Turned A">∀</a> </div> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0; line-height:1.15em"><a href="Universal_quantification" title="Universal quantification">universal<br>quantification</a></div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Existential_quantification" title="Existential quantification">∃</a> </div> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0; line-height:1.15em"><a href="Existential_quantification" title="Existential quantification">existential<br>quantification</a></div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Tee_(symbol)" title="Tee (symbol)">⊤</a> </div> <a href="True_(logic)" class="mw-redirect" title="True (logic)">true</a>,<br><a href="Tautology_(logic)" title="Tautology (logic)">tautology</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Up_tack" title="Up tack">⊥</a> </div> <a href="False_(logic)" title="False (logic)">false</a>,<br><a href="Contradiction" title="Contradiction">contradiction</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Turnstile_(symbol)" title="Turnstile (symbol)">⊢</a> </div> <a href="Turnstile_(symbol)" title="Turnstile (symbol)">entails,<br>proves</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Double_turnstile" title="Double turnstile">⊨</a> </div> <a href="Double_turnstile" title="Double turnstile">entails,<br>therefore</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Therefore_sign#Similar_signs" title="Therefore sign">∵</a> </div> <a href="Therefore_sign#Similar_signs" title="Therefore sign">because</a>

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<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Formal_semantics_(natural_language)78" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Formal_semantics_(natural_language)78" style="font-size:114%;margin:0 4em"><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Central concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Principle_of_compositionality" title="Principle of compositionality">Compositionality</a></li>

<li><a href="Denotation" title="Denotation">Denotation</a></li>

<li><a href="Linguistic_entailment" title="Linguistic entailment">Entailment</a></li>

<li><a href="Extension_(semantics)" title="Extension (semantics)">Extension</a></li>

<li><a href="Generalized_quantifier" title="Generalized quantifier">Generalized quantifier</a></li>

<li><a href="Intension" title="Intension">Intension</a></li>

<li><a href="Logical_form_(linguistics)" title="Logical form (linguistics)">Logical form</a></li>

<li><a href="Presupposition" title="Presupposition">Presupposition</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Scope_(formal_semantics)" title="Scope (formal semantics)">Scope</a></li>

<li><a href="Speech_act" title="Speech act">Speech act</a></li>

<li><a href="Syntax%E2%80%93semantics_interface" title="Syntax–semantics interface">Syntax–semantics interface</a></li>

<li><a href="Truth-conditional_semantics" title="Truth-conditional semantics">Truth conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anaphora_(linguistics)" title="Anaphora (linguistics)">Anaphora</a></li>

<li><a href="Ambiguity" title="Ambiguity">Ambiguity</a></li>

<li><a href="Binding_(linguistics)" title="Binding (linguistics)">Binding</a></li>

<li><a href="Conditional_sentence" title="Conditional sentence">Conditionals</a></li>

<li><a href="Definiteness" title="Definiteness">Definiteness</a></li>

<li><a href="Disjunction" class="mw-redirect" title="Disjunction">Disjunction</a></li>

<li><a href="Evidentiality" title="Evidentiality">Evidentiality</a></li>

<li><a href="Focus_(linguistics)" title="Focus (linguistics)">Focus</a></li>

<li><a href="Indexicality" title="Indexicality">Indexicality</a></li>

<li><a href="Lexical_semantics" title="Lexical semantics">Lexical semantics</a></li>

<li><a href="Modality_(semantics)" title="Modality (semantics)">Modality</a></li>

<li><a href="Negation" title="Negation">Negation</a></li>

<li><a href="Propositional_attitudes" class="mw-redirect" title="Propositional attitudes">Propositional attitudes</a></li>

<li><a href="Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantification</a></li>

<li><a href="Vagueness" title="Vagueness">Vagueness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antecedent-contained_deletion" title="Antecedent-contained deletion">Antecedent-contained deletion</a></li>

<li><a href="Cataphora" title="Cataphora">Cataphora</a></li>

<li><a href="Coercion_(linguistics)" title="Coercion (linguistics)">Coercion</a></li>

<li><a href="Conservativity" title="Conservativity">Conservativity</a></li>

<li><a href="Counterfactuals" class="mw-redirect" title="Counterfactuals">Counterfactuals</a></li>

<li><a href="Crossover_effects" title="Crossover effects">Crossover effects</a></li>

<li><a href="Cumulativity_(linguistics)" title="Cumulativity (linguistics)">Cumulativity</a></li>

<li><a href="De_dicto_and_de_re" title="De dicto and de re">De dicto and de re</a></li>

<li><a href="Deontic_modality" title="Deontic modality">Deontic modality</a></li>

<li><a href="Discourse_relation" title="Discourse relation">Discourse relations</a></li>

<li><a href="Donkey_anaphora" class="mw-redirect" title="Donkey anaphora">Donkey anaphora</a></li>

<li><a href="Epistemic_modality" title="Epistemic modality">Epistemic modality</a></li>

<li><a href="Exhaustivity" title="Exhaustivity">Exhaustivity</a></li>

<li><a href="Faultless_disagreement" title="Faultless disagreement">Faultless disagreement</a></li>

<li><a href="Free_choice_inference" title="Free choice inference">Free choice inferences</a></li>

<li><a href="Givenness" title="Givenness">Givenness</a></li>

<li><a href="Homogeneity_(linguistics)" class="mw-redirect" title="Homogeneity (linguistics)">Homogeneity (linguistics)</a></li>

<li><a href="Hurford_disjunction" title="Hurford disjunction">Hurford disjunction</a></li>

<li><a href="Inalienable_possession" title="Inalienable possession">Inalienable possession</a></li>

<li><a href="Intersective_modifier" title="Intersective modifier">Intersective modification</a></li>

<li><a href="Logophoricity" title="Logophoricity">Logophoricity</a></li>

<li><a href="Mirativity" title="Mirativity">Mirativity</a></li>

<li><a href="Modal_subordination" title="Modal subordination">Modal subordination</a></li>

<li><a href="Opaque_context" title="Opaque context">Opaque contexts</a></li>

<li><a href="Performative_utterance" title="Performative utterance">Performatives</a></li>

<li><a href="Polarity_item" title="Polarity item">Polarity items</a></li>

<li><a href="Privative_adjective" title="Privative adjective">Privative adjectives</a></li>

<li><a href="Quantificational_variability_effect" title="Quantificational variability effect">Quantificational variability effect</a></li>

<li><a href="Responsive_predicate" title="Responsive predicate">Responsive predicate</a></li>

<li><a href="Rising_declarative" title="Rising declarative">Rising declaratives</a></li>

<li><a href="Scalar_implicature" title="Scalar implicature">Scalar implicature</a></li>

<li><a href="Sloppy_identity" title="Sloppy identity">Sloppy identity</a></li>

<li><a href="Subsective_modifier" title="Subsective modifier">Subsective modification</a></li>

<li><a href="Subtrigging" title="Subtrigging">Subtrigging</a></li>

<li><a href="Telicity" title="Telicity">Telicity</a></li>

<li><a href="Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li>

<li><a href="Veridicality" title="Veridicality">Veridicality</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalism</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Formal systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternative_semantics" title="Alternative semantics">Alternative semantics</a></li>

<li><a href="Categorial_grammar" title="Categorial grammar">Categorial grammar</a></li>

<li><a href="Combinatory_categorial_grammar" title="Combinatory categorial grammar">Combinatory categorial grammar</a></li>

<li><a href="Discourse_representation_theory" title="Discourse representation theory">Discourse representation theory (DRT)</a></li>

<li><a href="Dynamic_semantics" title="Dynamic semantics">Dynamic semantics</a></li>

<li><a href="Generative_grammar" title="Generative grammar">Generative grammar</a></li>

<li><a href="Glue_semantics" title="Glue semantics">Glue semantics</a></li>

<li><a href="Inquisitive_semantics" title="Inquisitive semantics">Inquisitive semantics</a></li>

<li><a href="Intensional_logic" title="Intensional logic">Intensional logic</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Mereology" title="Mereology">Mereology</a></li>

<li><a href="Montague_grammar" title="Montague grammar">Montague grammar</a></li>

<li><a href="Segmented_discourse_representation_theory" class="mw-redirect" title="Segmented discourse representation theory">Segmented discourse representation theory (SDRT)</a></li>

<li><a href="Situation_semantics" title="Situation semantics">Situation semantics</a></li>

<li><a href="Supervaluationism" title="Supervaluationism">Supervaluationism</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autonomy_of_syntax" title="Autonomy of syntax">Autonomy of syntax</a></li>

<li><a href="Context_set" class="mw-redirect" title="Context set">Context set</a></li>

<li><a href="Continuation" title="Continuation">Continuation</a></li>

<li><a href="Conversational_scoreboard" title="Conversational scoreboard">Conversational scoreboard</a></li>

<li><a href="Downward_entailing" title="Downward entailing">Downward entailing</a></li>

<li><a href="Existential_closure" title="Existential closure">Existential closure</a></li>

<li><a href="Function_application" title="Function application">Function application</a></li>

<li><a href="Meaning_postulate" title="Meaning postulate">Meaning postulate</a></li>

<li><a href="Monad_(functional_programming)" title="Monad (functional programming)">Monads</a></li>

<li><a href="Plural_quantification" title="Plural quantification">Plural quantification</a></li>

<li><a href="Possible_world" title="Possible world">Possible world</a></li>

<li><a href="Quantifier_raising" class="mw-redirect" title="Quantifier raising">Quantifier raising</a></li>

<li><a href="Quantization_(linguistics)" title="Quantization (linguistics)">Quantization</a></li>

<li><a href="Question_under_discussion" title="Question under discussion">Question under discussion</a></li>

<li><a href="Semantic_parsing" title="Semantic parsing">Semantic parsing</a></li>

<li><a href="Squiggle_operator" title="Squiggle operator">Squiggle operator</a></li>

<li><a href="Strawson_entailment" title="Strawson entailment">Strawson entailment</a></li>

<li><a href="Strict_conditional" title="Strict conditional">Strict conditional</a></li>

<li><a href="Type_shifter" title="Type shifter">Type shifter</a></li>

<li><a href="Universal_grinder" title="Universal grinder">Universal grinder</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cognitive_semantics" title="Cognitive semantics">Cognitive semantics</a></li>

<li><a href="Computational_semantics" title="Computational semantics">Computational semantics</a></li>

<li><a href="Distributional_semantics" title="Distributional semantics">Distributional semantics</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Formal grammar</a></li>

<li><a href="Inferentialism" class="mw-redirect" title="Inferentialism">Inferentialism</a></li>

<li><a href="Logic_translation" title="Logic translation">Logic translation</a></li>

<li><a href="Linguistics_wars" title="Linguistics wars">Linguistics wars</a></li>

<li><a href="Philosophy_of_language" title="Philosophy of language">Philosophy of language</a></li>

<li><a href="Pragmatics" title="Pragmatics">Pragmatics</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li></ul>

</div></td></tr></tbody></table></div>

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