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			<h1 id="firstHeading" class="firstHeading" lang="en">Logic</h1>
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				<div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div role="note" class="hatnote">This article is about the systematic study of the form of arguments. For other uses, see <a href="/wiki/Logic_(disambiguation)" class="mw-disambig" title="Logic (disambiguation)">Logic (disambiguation)</a>.</div>
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<p><b>Logic</b> (from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient&#160;Greek</a>: <span lang="grc" xml:lang="grc"><a href="//en.wiktionary.org/wiki/%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE" class="extiw" title="wikt:λογική">λογική</a>, <i>logike</i></span>)<sup id="cite_ref-argumentative_1-0" class="reference"><a href="#cite_note-argumentative-1">[1]</a></sup> originally meaning <i>the word</i>, or <i>what is spoken</i>, (but coming to mean <i>thought</i> or <i>reason</i>) is generally held to consist of the systematic study of the form of <a href="/wiki/Argument" title="Argument">arguments</a>. A valid argument is one where there is a specific relation of logical support between the assumptions of the argument and its conclusion. (In ordinary discourse, the conclusion of such an argument may be signified by words like 'therefore', 'hence', 'ergo' and so on.)</p>
<p>There is no universal agreement as to the exact scope and subject matter of logic (see 'Rival conceptions of logic', below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, and the study of fallacies and paradoxes. Historically, logic has been studied in <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> (since ancient times) and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> (since the mid-1800s), and recently logic has been studied in <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, <a href="/wiki/Psychology" title="Psychology">psychology</a>, and other fields.</p>
<p></p>
<div id="toc" class="toc">
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<h2>Contents</h2>
</div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#The_study_of_logic"><span class="tocnumber">1</span> <span class="toctext">The study of logic</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Logical_form"><span class="tocnumber">1.1</span> <span class="toctext">Logical form</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Semantics"><span class="tocnumber">1.2</span> <span class="toctext">Semantics</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Inference"><span class="tocnumber">1.3</span> <span class="toctext">Inference</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Logical_systems"><span class="tocnumber">1.4</span> <span class="toctext">Logical systems</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="#Logic_and_rationality"><span class="tocnumber">1.5</span> <span class="toctext">Logic and rationality</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Rival_conceptions_of_logic"><span class="tocnumber">1.6</span> <span class="toctext">Rival conceptions of logic</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-8"><a href="#History"><span class="tocnumber">2</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-9"><a href="#Types_of_logic"><span class="tocnumber">3</span> <span class="toctext"><span>Types of logic</span></span></a>
<ul>
<li class="toclevel-2 tocsection-10"><a href="#Syllogistic_logic"><span class="tocnumber">3.1</span> <span class="toctext">Syllogistic logic</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="#Propositional_logic"><span class="tocnumber">3.2</span> <span class="toctext">Propositional logic</span></a></li>
<li class="toclevel-2 tocsection-12"><a href="#Predicate_logic"><span class="tocnumber">3.3</span> <span class="toctext">Predicate logic</span></a></li>
<li class="toclevel-2 tocsection-13"><a href="#Modal_logic"><span class="tocnumber">3.4</span> <span class="toctext">Modal logic</span></a></li>
<li class="toclevel-2 tocsection-14"><a href="#Informal_reasoning_and_dialectic"><span class="tocnumber">3.5</span> <span class="toctext">Informal reasoning and dialectic</span></a></li>
<li class="toclevel-2 tocsection-15"><a href="#Mathematical_logic"><span class="tocnumber">3.6</span> <span class="toctext">Mathematical logic</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="#Philosophical_logic"><span class="tocnumber">3.7</span> <span class="toctext">Philosophical logic</span></a></li>
<li class="toclevel-2 tocsection-17"><a href="#Computational_logic"><span class="tocnumber">3.8</span> <span class="toctext">Computational logic</span></a></li>
<li class="toclevel-2 tocsection-18"><a href="#Non-classical_logics"><span class="tocnumber">3.9</span> <span class="toctext">Non-classical logics</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-19"><a href="#Controversies_in_logic"><span class="tocnumber">4</span> <span class="toctext">Controversies in logic</span></a>
<ul>
<li class="toclevel-2 tocsection-20"><a href="#.22Is_logic_empirical.3F.22"><span class="tocnumber">4.1</span> <span class="toctext">"Is logic empirical?"</span></a></li>
<li class="toclevel-2 tocsection-21"><a href="#Implication:_strict_or_material.3F"><span class="tocnumber">4.2</span> <span class="toctext">Implication: strict or material?</span></a></li>
<li class="toclevel-2 tocsection-22"><a href="#Tolerating_the_impossible"><span class="tocnumber">4.3</span> <span class="toctext">Tolerating the impossible</span></a></li>
<li class="toclevel-2 tocsection-23"><a href="#Rejection_of_logical_truth"><span class="tocnumber">4.4</span> <span class="toctext">Rejection of logical truth</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-24"><a href="#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-25"><a href="#Notes_and_references"><span class="tocnumber">6</span> <span class="toctext">Notes and references</span></a></li>
<li class="toclevel-1 tocsection-26"><a href="#Bibliography"><span class="tocnumber">7</span> <span class="toctext">Bibliography</span></a></li>
<li class="toclevel-1 tocsection-27"><a href="#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<p></p>
<h2><span class="mw-headline" id="The_study_of_logic">The study of logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=1" title="Edit section: The study of logic">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
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<td style="padding:0 10px;">Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy: Do not block the way of inquiry.</td>
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<td class="rquotecite" colspan="3" style="padding-top:10px; font-size:smaller; line-height:1em; text-align:right;"><cite>—&#160;<a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, "First Rule of Logic"</cite></td>
</tr>
</table>
<p>The concept of <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">logical form</a> is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional <a href="/wiki/Syllogism" title="Syllogism">Aristotelian syllogistic logic</a> and modern symbolic logic are examples of formal logic.</p>
<ul>
<li><b><a href="/wiki/Informal_logic" title="Informal logic">Informal logic</a></b> is the study of <a href="/wiki/Natural_language" title="Natural language">natural language</a> <a href="/wiki/Logical_argument" class="mw-redirect" title="Logical argument">arguments</a>. The study of <a href="/wiki/Fallacies" class="mw-redirect" title="Fallacies">fallacies</a> is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on many conceptions of logic, informal logic is not logic at all. See 'Rival conceptions of logic', below.</li>
<li><b><a href="/wiki/Mathematical_formalism" class="mw-redirect" title="Mathematical formalism">Formal logic</a></b> is the study of <a href="/wiki/Inference" title="Inference">inference</a> with purely formal content. An inference possesses a <i>purely formal content</i> if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle.<sup id="cite_ref-The_Basic_Works_2-0" class="reference"><a href="#cite_note-The_Basic_Works-2">[2]</a></sup> In many definitions of logic, <a href="/wiki/Logical_consequence" title="Logical consequence">logical inference</a> and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.</li>
<li><b><a href="/wiki/Mathematical_logic#Symbolic_logic" title="Mathematical logic">Symbolic logic</a></b> is the study of symbolic abstractions that capture the formal features of logical inference.<sup id="cite_ref-Principia_3-0" class="reference"><a href="#cite_note-Principia-3">[3]</a></sup><sup id="cite_ref-Hamilton_4-0" class="reference"><a href="#cite_note-Hamilton-4">[4]</a></sup> Symbolic logic is often divided into two main branches: <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> and <a href="/wiki/Predicate_logic" title="Predicate logic">predicate logic</a>.</li>
<li><b><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></b> is an extension of symbolic logic into other areas, in particular to the study of <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and <a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>.</li>
</ul>
<p>However, agreement on what logic is has remained elusive, and although the field of <a href="/wiki/Universal_logic" title="Universal logic">universal logic</a> has studied the common structure of logics, in 2007 Mossakowski et al. commented that "it is embarrassing that there is no widely acceptable formal definition of 'a logic'".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup></p>
<h3><span class="mw-headline" id="Logical_form">Logical form</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=2" title="Edit section: Logical form">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Logical_form" title="Logical form">Logical form</a></div>
<p>Logic is generally considered <b>formal</b> when it analyzes and represents the <i>form</i> of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic.</p>
<p>This is called showing the <i>logical form</i> of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features irrelevant to logic (such as gender and declension, if the argument is in Latin), replacing conjunctions irrelevant to logic (such as "but") with logical conjunctions like "and" and replacing ambiguous, or alternative logical expressions ("any", "every", etc.) with expressions of a standard type (such as "all", or the universal quantifier ∀).</p>
<p>Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression "all As are Bs" shows the logical form common to the sentences "all men are mortals", "all cats are carnivores", "all Greeks are philosophers", and so on.</p>
<p>The importance of form was recognised from ancient times. Aristotle uses variable letters to represent valid inferences in <i><a href="/wiki/Prior_Analytics" title="Prior Analytics">Prior Analytics</a></i>, leading <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan ?ukasiewicz">Jan ?ukasiewicz</a> to say that the introduction of variables was "one of Aristotle's greatest inventions".<sup id="cite_ref-Aristotle.27s_syllogistic_from_the_standpoint_of_modern_formal_logic_6-0" class="reference"><a href="#cite_note-Aristotle.27s_syllogistic_from_the_standpoint_of_modern_formal_logic-6">[6]</a></sup> According to the followers of Aristotle (such as <a href="/wiki/Ammonius_Saccas" title="Ammonius Saccas">Ammonius</a>), only the logical principles stated in schematic terms belong to logic, not those given in concrete terms. The concrete terms "man", "mortal", etc., are analogous to the substitution values of the schematic placeholders <i>A</i>, <i>B</i>, <i>C</i>, which were called the "matter" (Greek <i>hyle</i>) of the inference.</p>
<p>The fundamental difference between modern formal logic and traditional, or Aristotelian logic, lies in their differing analysis of the logical form of the sentences they treat.</p>
<ul>
<li>In the traditional view, the form of the sentence consists of (1) a subject (e.g., "man") plus a sign of quantity ("all" or "some" or "no"); (2) the <a href="/wiki/Copula_(linguistics)" title="Copula (linguistics)">copula</a>, which is of the form "is" or "is not"; (3) a predicate (e.g., "mortal"). Thus: 'all men are mortal'. The logical constants such as "all", "no" and so on, plus sentential connectives such as "and" and "or" were called <a href="/wiki/Syncategorematic" class="mw-redirect" title="Syncategorematic">syncategorematic</a> terms (from the Greek <i>kategorei</i> – to predicate, and <i>syn</i> – together with). This is a fixed scheme, where each judgment has a specific quantity and copula, determining the logical form of the sentence.</li>
<li>The modern view is more complex, since a single judgement of Aristotle's system involves two or more logical connectives. For example, the sentence "All men are mortal" involves, in term logic, two non-logical terms "is a man" (here <i>M</i>) and "is mortal" (here <i>D</i>): the sentence is given by the judgement A(M,D). In <a href="/wiki/Predicate_logic" title="Predicate logic">predicate logic</a>, the sentence involves the same two non-logical concepts, here analyzed as <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>m</mi>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle m(x)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7426128eeeea7766bd61d64cf1d13c3b18dd9138" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.221ex; height:2.843ex;" alt="m(x)" /></span> and <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>d</mi>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle d(x)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74c31c749da4bca087f03265cda1e09b0f26cb7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.397ex; height:2.843ex;" alt="d(x)" /></span>, and the sentence is given by <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi>
        <mi>x</mi>
        <mo>.</mo>
        <mo stretchy="false">(</mo>
        <mi>m</mi>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">&#x2192;<!-- → --></mo>
        <mi>d</mi>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \forall x.(m(x)\rightarrow d(x))}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f044a9ce3354eeda2251632d154e2371a077f6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.76ex; height:2.843ex;" alt="\forall x.(m(x)\rightarrow d(x))" /></span>, involving the logical connectives for <a href="/wiki/Universal_quantification" title="Universal quantification">universal quantification</a> and <a href="/wiki/Material_conditional" title="Material conditional">implication</a>.</li>
<li>The more complex modern view comes with more power. On the modern view, the fundamental form of a simple sentence is given by a recursive schema, like natural language and involving <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a>, which are joined by juxtaposition to other sentences, which in turn may have logical structure. Medieval logicians recognized the <a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">problem of multiple generality</a>, where Aristotelian logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.</li>
</ul>
<h3><span class="mw-headline" id="Semantics">Semantics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=3" title="Edit section: Semantics">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></div>
<p>The validity of an argument depends upon the meaning or <i>semantics</i> of the sentences that make it up.</p>
<p>Aristotle's <a href="/wiki/Organon" title="Organon">Organon</a>, especially <a href="/wiki/On_Interpretation" class="mw-redirect" title="On Interpretation">On Interpretation</a>, gives a cursory outline of semantics which the <a href="/wiki/Scholastic_logic" class="mw-redirect" title="Scholastic logic">scholastic logicians</a>, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called <a href="/wiki/Supposition_Theory" class="mw-redirect" title="Supposition Theory">Supposition Theory</a>. This showed how the truth of simple sentences, expressed schematically, depend on how the terms 'supposit' or <i>stand for</i> certain extra-linguistic items. For example, in book II of his <a href="/wiki/Sum_of_Logic" title="Sum of Logic">Summa Logicae</a>, <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a> presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not. Thus 'every A is B' is true if and only if there is something for which 'A' stands for, and there is nothing for which 'A' stands for, which 'B' does not also stand for.</p>
<p>Early modern logic defined semantics purely as a relation between ideas. <a href="/wiki/Antoine_Arnauld" title="Antoine Arnauld">Antoine Arnauld</a> in the <a href="/wiki/Port_Royal_Logic" class="mw-redirect" title="Port Royal Logic">Port Royal Logic</a>, says that 'after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called <i>affirming</i> or <i>denying</i>, and in general <i>judging'</i>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">[7]</a></sup> Thus truth and falsity are no more than the agreement or disagreement of ideas. This suggests obvious difficulties, leading Locke to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">[8]</a></sup> This view (<a href="/wiki/Psychologism" title="Psychologism">psychologism</a>) was taken to the extreme in the nineteenth century, and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century.</p>
<p>Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantification</a>, needed to solve the <a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">problem of multiple generality</a>, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics. The main modern approach is <i>model-theoretic semantics</i>, based on <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>'s <a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic theory of truth</a>. The approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of <a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">interpretation functions</a> from them to some predefined <a href="/w/index.php?title=Mathematical_domain&amp;action=edit&amp;redlink=1" class="new" title="Mathematical domain (page does not exist)">mathematical domains</a>: an interpretation of <a href="/wiki/First-order_predicate_logic" class="mw-redirect" title="First-order predicate logic">first-order predicate logic</a> is given by a mapping from terms to a universe of <a href="/wiki/Individual" title="Individual">individuals</a>, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics is one of the fundamental concepts of <a href="/wiki/Model_theory" title="Model theory">model theory</a>. Modern semantics also admits rival approaches, such as the <a href="/wiki/Proof-theoretic_semantics" title="Proof-theoretic semantics">proof-theoretic semantics</a> that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> on <a href="/wiki/Structural_proof_theory" title="Structural proof theory">structural proof theory</a> and is heavily influenced by <a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a>'s later philosophy, especially his aphorism "meaning is use".</p>
<h3><span class="mw-headline" id="Inference">Inference</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=4" title="Edit section: Inference">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p><i>Inference</i> is not to be confused with <i>implication</i>. An implication is a sentence of the form 'If p then q', and can be true or false. <a href="/wiki/Stoic_logic" class="mw-redirect" title="Stoic logic">The Stoic logician</a> <a href="/wiki/Philo_the_Dialectician" title="Philo the Dialectician">Philo of Megara</a> was the first to define the truth conditions of such an implication: false only when the antecedent p is true and the consequent q is false, in all other cases true. An inference, on the other hand, consists of two separately asserted propositions of the form 'p therefore q'. An inference is not true or false, but valid or invalid. However, there is a connection between implication and inference, as follows: if the implication 'if p then q' is <i>true</i>, the inference 'p therefore q' is <i>valid</i>. This was given an apparently paradoxical formulation by Philo, who said that the implication 'if it is day, it is night' is true only at night, so the inference 'it is day, therefore it is night' is valid in the night, but not in the day.</p>
<p>The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a> and <a href="/wiki/Walter_Burley" title="Walter Burley">Walter Burley</a>. It is uniquely medieval, though it has its origins in Aristotle's <a href="/wiki/Topics_(Aristotle)" title="Topics (Aristotle)">Topics</a> and <a href="/wiki/Boethius" title="Boethius">Boethius</a>' <i>De Syllogismis hypotheticis</i>. This is why many terms in logic are Latin. For example, the rule that licenses the move from the implication 'if p then q' plus the assertion of its antecedent p, to the assertion of the consequent q is known as <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> (or 'mode of positing'). Its Latin formulation is 'Posito antecedente ponitur consequens'. The Latin formulations of many other rules such as 'ex falso quodlibet' (anything follows from a falsehood), 'reductio ad absurdum' (disproof by showing the consequence is absurd) also date from this period.</p>
<p>However, the theory of consequences, or of the so-called '<a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a>' was never fully integrated into the theory of the 'categorical syllogism'. This was partly because of the resistance to reducing the categorical judgment 'Every S is P' to the so-called hypothetical judgment 'if anything is S, it is P'. The first was thought to imply 'some S is P', the second was not, and as late as 1911 in the <a href="/wiki/Encyclopedia_Britannica" class="mw-redirect" title="Encyclopedia Britannica">Encyclopedia Britannica</a> article on Logic, we find the Oxford logician T.H. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition. Cf. <a href="/wiki/Problem_of_existential_import" class="mw-redirect" title="Problem of existential import">problem of existential import</a></p>
<h3><span class="mw-headline" id="Logical_systems">Logical systems</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=5" title="Edit section: Logical systems">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Formal_system" title="Formal system">Formal system</a></div>
<p>A formal system is an organisation of terms used for the analysis of deduction. It consists of an alphabet, a language over the alphabet to construct sentences, and a rule for deriving sentences. Among the important properties that <a href="/wiki/Logical_system" class="mw-redirect" title="Logical system">logical systems</a> can have are:</p>
<ul>
<li><b><a href="/wiki/Consistency_proof" class="mw-redirect" title="Consistency proof">Consistency</a></b>, which means that no theorem of the system contradicts another.<sup id="cite_ref-Bergmann.2C_Merrie_2009_9-0" class="reference"><a href="#cite_note-Bergmann.2C_Merrie_2009-9">[9]</a></sup></li>
<li><b><a href="/wiki/Validity" title="Validity">Validity</a></b>, which means that the system's rules of proof never allow a false inference from true premises.</li>
<li><b><a href="/wiki/Completeness_(logic)" title="Completeness (logic)">Completeness</a></b>, which means that if a formula is true, it can be proven, i.e. is a <i>theorem</i> of the system.</li>
<li><b><a href="/wiki/Soundness" title="Soundness">Soundness</a></b>, meaning that if any formula is a theorem of the system, it is true. This is the converse of completeness. (Note that in a distinct philosophical use of the term, an argument is sound when it is both valid and its premises are true).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10">[10]</a></sup></li>
</ul>
<p>Some logical systems do not have all four properties. As an example, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>'s <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a> show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;<sup id="cite_ref-Hamilton_4-1" class="reference"><a href="#cite_note-Hamilton-4">[4]</a></sup> however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.<sup id="cite_ref-Introduction_to_Mathematical_Logic_11-0" class="reference"><a href="#cite_note-Introduction_to_Mathematical_Logic-11">[11]</a></sup></p>
<h3><span class="mw-headline" id="Logic_and_rationality">Logic and rationality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=6" title="Edit section: Logic and rationality">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Logic_and_rationality" title="Logic and rationality">Logic and rationality</a></div>
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<p>As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential important to <a href="/wiki/Rationality" title="Rationality">rationality</a>.</p>
<p><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deductive reasoning</a> concerns the <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> of given premises. On a narrow conception of logic (see below) logic concerns just deductive reasoning, although such a narrow conception controversially excludes most of what is called informal logic from the discipline. Other forms of reasoning are sometimes also taken to be part of logic, such as <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>. Similarly, it is important to distinguish deductive validity and inductive validity (called "strength"). An inference is deductively valid <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> there is no possible situation in which all the premises are true but the conclusion false. An inference is inductively strong if and only if its premises give some degree of probability to its conclusion.</p>
<p>The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of <a href="/wiki/Semantics" title="Semantics">semantics</a>. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use logical association <a href="/wiki/Rule_induction" title="Rule induction">rule induction</a>, while others may use <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical models</a> of probability such as <a href="/wiki/Decision_tree" title="Decision tree">decision trees</a>. For the most part this discussion of logic deals only with deductive logic.</p>
<p><a href="/wiki/Abductive_reasoning" title="Abductive reasoning">Abduction</a><sup id="cite_ref-abduction_12-0" class="reference"><a href="#cite_note-abduction-12">[12]</a></sup> is a form of logical inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> (1839–1914) first introduced the term as "guessing".<sup id="cite_ref-guess_13-0" class="reference"><a href="#cite_note-guess-13">[13]</a></sup> Peirce said that to <i>abduce</i> a hypothetical explanation <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>a</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle a}</annotation>
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<h3><span class="mw-headline" id="Rival_conceptions_of_logic">Rival conceptions of logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=7" title="Edit section: Rival conceptions of logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Rival_conceptions_of_logic" title="Rival conceptions of logic">Rival conceptions of logic</a></div>
<p>Logic arose (see below) from a concern with correctness of <a href="/wiki/Argumentation" class="mw-redirect" title="Argumentation">argumentation</a>. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a> that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of <a href="/wiki/Rationality" title="Rationality">rationality</a>. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations".<sup id="cite_ref-stanford-logic-onthology_15-0" class="reference"><a href="#cite_note-stanford-logic-onthology-15">[15]</a></sup></p>
<p>By contrast, <a href="/wiki/Immanuel_Kant" title="Immanuel Kant">Immanuel Kant</a> argued that logic should be conceived as the science of judgement, an idea taken up in <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>'s logical and philosophical work. But Frege's work is ambiguous in the sense that it is both concerned with the "laws of thought" as well as with the "laws of truth", i.e. it both treats logic in the context of a theory of the mind, and treats logic as the study of abstract formal structures.</p>
<p>Logic has been defined as "the study of arguments correct in virtue of their form". This has not been the definition taken in this article, but the idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to <a href="/wiki/Logicism" title="Logicism">logicism</a> in mathematics and the advent of the influence of mathematical logic on philosophy. A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths, with logic equivalently being the study of logical truth, and excludes many of the original objects of study of logic that are treated as informal logic. <a href="/wiki/Robert_Brandom" title="Robert Brandom">Robert Brandom</a> has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of <a href="/wiki/Material_inference" title="Material inference">material inference</a> (in the teminology of <a href="/wiki/Wilfred_Sellars" class="mw-redirect" title="Wilfred Sellars">Wilfred Sellars</a>), with logic making explicit the commitments that were originally implicit in informal inference.<sup id="cite_ref-brandom-2000_16-0" class="reference"><a href="#cite_note-brandom-2000-16">[16]</a></sup></p>
<h2><span class="mw-headline" id="History">History</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=8" title="Edit section: History">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div role="note" class="hatnote">Main article: <a href="/wiki/History_of_logic" title="History of logic">History of logic</a></div>
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<div class="thumbinner" style="width:152px;"><a href="/wiki/File:Aristotle_Altemps_Inv8575.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/150px-Aristotle_Altemps_Inv8575.jpg" width="150" height="201" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/225px-Aristotle_Altemps_Inv8575.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/300px-Aristotle_Altemps_Inv8575.jpg 2x" data-file-width="1700" data-file-height="2275" /></a>
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<a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, 384–322&#160;BCE.</div>
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<p>In Europe, logic was first developed by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>.<sup id="cite_ref-achievement_17-0" class="reference"><a href="#cite_note-achievement-17">[17]</a></sup> <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a> became widely accepted in science and mathematics and remained in wide use in the West until the early 19th&#160;century.<sup id="cite_ref-mtu_18-0" class="reference"><a href="#cite_note-mtu-18">[18]</a></sup> Aristotle's system of logic was responsible for the introduction of <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a>,<sup id="cite_ref-google_19-0" class="reference"><a href="#cite_note-google-19">[19]</a></sup> <a href="/wiki/Temporal_logic" title="Temporal logic">temporal</a> <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>,<sup id="cite_ref-google1_20-0" class="reference"><a href="#cite_note-google1-20">[20]</a></sup><sup id="cite_ref-google2_21-0" class="reference"><a href="#cite_note-google2-21">[21]</a></sup> and <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive logic</a>,<sup id="cite_ref-google3_22-0" class="reference"><a href="#cite_note-google3-22">[22]</a></sup> as well as influential terms such as <a href="/wiki/Terminology" title="Terminology">terms</a>, <a href="/wiki/Predicables" class="mw-redirect" title="Predicables">predicables</a>, <a href="/wiki/Syllogisms" class="mw-redirect" title="Syllogisms">syllogisms</a> and <a href="/wiki/Proposition" title="Proposition">propositions</a>. In <a href="/wiki/Europe" title="Europe">Europe</a> during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with <a href="/wiki/Christian" title="Christian">Christian</a> faith. During the <a href="/wiki/High_Middle_Ages" title="High Middle Ages">High Middle Ages</a>, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of <a href="/wiki/Scholasticism" title="Scholasticism">scholasticism</a>. In 1323, <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a>'s influential <i><a href="/wiki/Sum_of_Logic" title="Sum of Logic">Summa Logicae</a></i> was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in <a href="/wiki/Ludvig_Holberg" title="Ludvig Holberg">Holberg</a>'s satirical play <i><a href="/wiki/Erasmus_Montanus" title="Erasmus Montanus">Erasmus Montanus</a></i>.</p>
<p>The <a href="/wiki/Logic_in_China" title="Logic in China">Chinese logical</a> philosopher <a href="/wiki/Gongsun_Long" title="Gongsun Long">Gongsun Long</a> (<span class="nowrap">c. 325–250 BCE</span>) proposed the paradox "One and one cannot become two, since neither becomes two."<sup id="cite_ref-propositions_23-0" class="reference"><a href="#cite_note-propositions-23">[23]</a></sup> In China, the tradition of scholarly investigation into logic, however, was repressed by the <a href="/wiki/Qin_dynasty" title="Qin dynasty">Qin dynasty</a> following the legalist philosophy of <a href="/wiki/Han_Feizi" title="Han Feizi">Han Feizi</a>.</p>
<p>In India, innovations in the scholastic school, called <a href="/wiki/Nyaya" title="Nyaya">Nyaya</a>, continued from ancient times into the early 18th&#160;century with the <a href="/wiki/Navya-Nyaya" class="mw-redirect" title="Navya-Nyaya">Navya-Nyaya</a> school. By the 16th&#160;century, it developed theories resembling modern logic, such as <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>'s "distinction between sense and reference of proper names" and his "definition of number", as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modern <a href="/wiki/Set_theory" title="Set theory">set theory</a>.<sup id="cite_ref-Chakrabarti_24-0" class="reference"><a href="#cite_note-Chakrabarti-24">[24]</a></sup> Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as <a href="/wiki/Charles_Babbage" title="Charles Babbage">Charles Babbage</a>, <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a>, and <a href="/wiki/George_Boole" title="George Boole">George Boole</a>.<sup id="cite_ref-Indian_logic:_a_reader_25-0" class="reference"><a href="#cite_note-Indian_logic:_a_reader-25">[25]</a></sup> In the 20th&#160;century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.</p>
<p>The <a href="/wiki/Syllogism" title="Syllogism">syllogistic</a> logic developed by Aristotle predominated in the West until the mid-19th&#160;century, when interest in the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a> stimulated the development of symbolic logic (now called <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>). In 1854, George Boole published <i><a href="/wiki/The_Laws_of_Thought" title="The Laws of Thought">An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities</a></i>, introducing symbolic logic and the principles of what is now known as <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a>. In 1879, Gottlob Frege published <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i>, which inaugurated modern logic with the invention of <a href="/wiki/Quantification_(logic)" class="mw-redirect" title="Quantification (logic)">quantifier</a> notation. From 1910 to 1913, <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> published <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i><sup id="cite_ref-Principia_3-1" class="reference"><a href="#cite_note-Principia-3">[3]</a></sup> on the foundations of mathematics, attempting to derive mathematical truths from <a href="/wiki/Axiom" title="Axiom">axioms</a> and <a href="/wiki/Inference_rule" class="mw-redirect" title="Inference rule">inference rules</a> in symbolic logic. In 1931, <a href="/wiki/G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a> raised serious problems with the foundationalist program and logic ceased to focus on such issues.</p>
<p>The development of logic since Frege, Russell, and <a href="/wiki/Wittgenstein" class="mw-redirect" title="Wittgenstein">Wittgenstein</a> had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">Analytic philosophy</a>), and <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a>. Logic, especially sentential logic, is implemented in computer <a href="/wiki/Digital_electronics" title="Digital electronics">logic circuits</a> and is fundamental to <a href="/wiki/Computer_science" title="Computer science">computer science</a>. Logic is commonly taught by university philosophy departments, often as a compulsory discipline.</p>
<h2><span class="mw-headline" id="Types_of_logic"><span id="Types_of_logic">Types of logic</span></span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=9" title="Edit section: Types of logic">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Syllogistic_logic">Syllogistic logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=10" title="Edit section: Syllogistic logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
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A depiction from the 15th century of the <a href="/wiki/Square_of_opposition" title="Square of opposition">square of opposition</a>, which expresses the fundamental dualities of syllogistic.</div>
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<div role="note" class="hatnote">Main article: <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a></div>
<p>The <i><a href="/wiki/Organon" title="Organon">Organon</a></i> was <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s body of work on logic, with the <i><a href="/wiki/Prior_Analytics" title="Prior Analytics">Prior Analytics</a></i> constituting the first explicit work in formal logic, introducing the syllogistic.<sup id="cite_ref-Aristotle_26-0" class="reference"><a href="#cite_note-Aristotle-26">[26]</a></sup> The parts of syllogistic logic, also known by the name <a href="/wiki/Term_logic" title="Term logic">term logic</a>, are the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of <a href="/wiki/Syllogism" title="Syllogism">syllogisms</a> that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises.</p>
<p>Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. However, it was not alone: the <a href="/wiki/Stoics" class="mw-redirect" title="Stoics">Stoics</a> proposed a system of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> that was studied by medieval logicians. Also, the <a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">problem of multiple generality</a> was recognized in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.</p>
<p>Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of propositional logic and the <a href="/wiki/Predicate_calculus" class="mw-redirect" title="Predicate calculus">predicate calculus</a>. Others use Aristotle in <a href="/wiki/Argumentation_theory" title="Argumentation theory">argumentation theory</a> to help develop and critically question argumentation schemes that are used in <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> and <a href="/wiki/Legal" class="mw-redirect" title="Legal">legal</a> arguments.</p>
<h3><span class="mw-headline" id="Propositional_logic">Propositional logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=11" title="Edit section: Propositional logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></div>
<p>A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining <a href="/wiki/Atomic_propositions" class="mw-redirect" title="Atomic propositions">atomic propositions</a> using <a href="/wiki/Logical_connectives" class="mw-redirect" title="Logical connectives">logical connectives</a>, and in which a system of formal proof rules establishes certain formulae as "theorems". An example of a theorem of propositional logic is <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
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<h3><span class="mw-headline" id="Predicate_logic">Predicate logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=12" title="Edit section: Predicate logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
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<a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>'s <i><a href="/wiki/Begriffschrift" class="mw-redirect" title="Begriffschrift">Begriffschrift</a></i> introduced the notion of quantifier in a graphical notation, which here represents the judgement that <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
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        <mi>x</mi>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \forall x.F(x)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba23799cf40a6e9b0a427c429dc89fbb8a74bfee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.609ex; height:2.843ex;" alt="{\displaystyle \forall x.F(x)}" /></span> is true.</div>
</div>
</div>
<div role="note" class="hatnote">Main article: <a href="/wiki/Predicate_logic" title="Predicate logic">Predicate logic</a></div>
<p>Predicate logic is the generic term for symbolic formal systems such as <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>, <a href="/wiki/Many-sorted_logic" title="Many-sorted logic">many-sorted logic</a>, and <a href="/wiki/Infinitary_logic" title="Infinitary logic">infinitary logic</a>. It provides an account of <a href="/wiki/Quantifiers_(logic)" class="mw-redirect" title="Quantifiers (logic)">quantifiers</a> general enough to express a wide set of arguments occurring in natural language. For example, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s famous <a href="/wiki/Barber_paradox" title="Barber paradox">barber paradox</a>, "there is a man who shaves all and only men who do not shave themselves" can be formalised by the sentence <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">(</mo>
        <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>man</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
        <mo>&#x2227;<!-- ∧ --></mo>
        <mo stretchy="false">(</mo>
        <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi>
        <mi>y</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>man</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>y</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">&#x2192;<!-- → --></mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>shaves</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo>,</mo>
        <mi>y</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">&#x2194;<!-- ↔ --></mo>
        <mi mathvariant="normal">&#x00AC;<!-- ¬ --></mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>shaves</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>y</mi>
        <mo>,</mo>
        <mi>y</mi>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">)</mo>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle (\exists x)({\text{man}}(x)\wedge (\forall y)({\text{man}}(y)\rightarrow ({\text{shaves}}(x,y)\leftrightarrow \neg {\text{shaves}}(y,y))))}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac306aa72d70cbe45a54a684cc799ea40bacd5c7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:64.484ex; height:2.843ex;" alt="(\exists x)({\text{man}}(x)\wedge (\forall y)({\text{man}}(y)\rightarrow ({\text{shaves}}(x,y)\leftrightarrow \neg {\text{shaves}}(y,y))))" /></span>, using the non-logical predicate <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>man</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\text{man}}(x)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716ede5163dc136f13005a1d151d53c88b5ff7ba" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.593ex; height:2.843ex;" alt="{\displaystyle {\text{man}}(x)}" /></span> to indicate that <i>x</i> is a man, and the non-logical relation <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>shaves</mtext>
        </mrow>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo>,</mo>
        <mi>y</mi>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\text{shaves}}(x,y)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8ff072bf81f845faa91db93a334ed1d48b903a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.991ex; height:2.843ex;" alt="{\displaystyle {\text{shaves}}(x,y)}" /></span> to indicate that <i>x</i> is shaved by <i>y</i>; all other symbols of the formulae are logical, expressing the universal and existential <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifiers</a>, <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a>, <a href="/wiki/Material_conditional" title="Material conditional">implication</a>, <a href="/wiki/Negation" title="Negation">negation</a> and <a href="/wiki/Biconditional" class="mw-redirect" title="Biconditional">biconditional</a>.</p>
<p>Whilst Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take, predicate logic allows sentences to be analysed into subject and argument in several additional ways—allowing predicate logic to solve the <a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">problem of multiple generality</a> that had perplexed medieval logicians.</p>
<p>The development of predicate logic is usually attributed to <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, who is also credited as one of the founders of <a href="/wiki/Analytical_philosophy" class="mw-redirect" title="Analytical philosophy">analytical philosophy</a>, but the formulation of predicate logic most often used today is the first-order logic presented in <a href="/wiki/Principles_of_Mathematical_Logic" title="Principles of Mathematical Logic">Principles of Mathematical Logic</a> by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Wilhelm Ackermann</a> in 1928. The analytical generality of predicate logic allowed the formalization of mathematics, drove the investigation of <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and allowed the development of <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>'s approach to <a href="/wiki/Model_theory" title="Model theory">model theory</a>. It provides the foundation of modern <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>.</p>
<p>Frege's original system of predicate logic was second-order, rather than first-order. <a href="/wiki/Second-order_logic" title="Second-order logic">Second-order logic</a> is most prominently defended (against the criticism of <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a> and others) by <a href="/wiki/George_Boolos" title="George Boolos">George Boolos</a> and <a href="/wiki/Stewart_Shapiro" title="Stewart Shapiro">Stewart Shapiro</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2016)">citation needed</span></a></i>]</sup></p>
<h3><span class="mw-headline" id="Modal_logic">Modal logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=13" title="Edit section: Modal logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a></div>
<p>In languages, <a href="/wiki/Linguistic_modality" title="Linguistic modality">modality</a> deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. For example, "<i>We go to the games</i>" can be modified to give "<i>We should go to the games</i>", and "<i>We can go to the games</i>" and perhaps "<i>We will go to the games</i>". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.</p>
<p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous <a href="/wiki/Problem_of_future_contingents" title="Problem of future contingents">sea-battle argument</a> in <i><a href="/wiki/De_Interpretatione" title="De Interpretatione">De Interpretatione</a></i> § 9, that are now seen as anticipations of modal logic and its connection with <a href="/wiki/Potentiality" class="mw-redirect" title="Potentiality">potentiality</a> and time, the earliest formal system of modal logic was developed by <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a>, whom ultimately developed a theory of "<a href="/wiki/Temporal_logic" title="Temporal logic">temporally</a> <a href="/wiki/Modal_logic" title="Modal logic">modalized</a>" syllogistic.<sup id="cite_ref-Britannica_27-0" class="reference"><a href="#cite_note-Britannica-27">[27]</a></sup></p>
<p>While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of <a href="/wiki/Clarence_Irving_Lewis" title="Clarence Irving Lewis">Clarence Irving Lewis</a> in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include <a href="/wiki/Deontic_logic" title="Deontic logic">deontic logic</a> and <a href="/wiki/Epistemic_logic" class="mw-redirect" title="Epistemic logic">epistemic logic</a>. The seminal work of <a href="/wiki/Arthur_Prior" title="Arthur Prior">Arthur Prior</a> applied the same formal language to treat <a href="/wiki/Temporal_logic" title="Temporal logic">temporal logic</a> and paved the way for the marriage of the two subjects. <a href="/wiki/Saul_Kripke" title="Saul Kripke">Saul Kripke</a> discovered (contemporaneously with rivals) his theory of <a href="/wiki/Kripke_semantics" title="Kripke semantics">frame semantics</a>, which revolutionized the formal technology available to modal logicians and gave a new <a href="/wiki/Graph_theory" title="Graph theory">graph-theoretic</a> way of looking at modality that has driven many applications in <a href="/wiki/Computational_linguistics" title="Computational linguistics">computational linguistics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, such as <a href="/wiki/Dynamic_logic_(modal_logic)" title="Dynamic logic (modal logic)">dynamic logic</a>.</p>
<h3><span class="mw-headline" id="Informal_reasoning_and_dialectic">Informal reasoning and dialectic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=14" title="Edit section: Informal reasoning and dialectic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main articles: <a href="/wiki/Informal_logic" title="Informal logic">Informal logic</a> and <a href="/wiki/Logic_and_dialectic" title="Logic and dialectic">Logic and dialectic</a></div>
<p>The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle's <a href="/wiki/Organon" title="Organon">Organon</a> treat inference as it occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of <a href="/wiki/Rhetoric" title="Rhetoric">rhetoric</a>.</p>
<p>This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically <a href="/wiki/Dialectic" title="Dialectic">dialectical</a> logic forms the heart of a course in <a href="/wiki/Critical_thinking" title="Critical thinking">critical thinking</a>, a compulsory course at many universities. Dialectic has been linked to logic since ancient times, but it has not been until recent decades that European and American logicians have attempted to provide mathematical foundations for logic and dialectic by formalising dialectical logic. <a href="/wiki/Dialectical_logic" title="Dialectical logic">Dialectical logic</a> is also the name given to the special treatment of dialectic in <a href="/wiki/Hegel" class="mw-redirect" title="Hegel">Hegelian</a> and <a href="/wiki/Marx" class="mw-redirect" title="Marx">Marxist</a> thought. There have been pre-formal treatises on argument and dialectic, from authors such as <a href="/wiki/Stephen_Toulmin" title="Stephen Toulmin">Stephen Toulmin</a> (<i>The Uses of Argument</i>), <a href="/wiki/Nicholas_Rescher" title="Nicholas Rescher">Nicholas Rescher</a> (<i>Dialectics</i>),<sup id="cite_ref-28" class="reference"><a href="#cite_note-28">[28]</a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29">[29]</a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30">[30]</a></sup> and van Eemeren and Grootendorst (<a href="/wiki/Pragma-dialectics" title="Pragma-dialectics">Pragma-dialectics</a>). Theories of <a href="/wiki/Defeasible_reasoning" title="Defeasible reasoning">defeasible reasoning</a> can provide a foundation for the formalisation of dialectical logic and dialectic itself can be formalised as moves in a game, where an advocate for the truth of a proposition and an opponent argue. Such games can provide a formal <a href="/wiki/Game_semantics" title="Game semantics">game semantics</a> for many logics.</p>
<p><a href="/wiki/Argumentation_theory" title="Argumentation theory">Argumentation theory</a> is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> and <a href="/wiki/Law" title="Law">law</a>.</p>
<h3><span class="mw-headline" id="Mathematical_logic">Mathematical logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=15" title="Edit section: Mathematical logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div>
<p>Mathematical logic comprises two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.<sup id="cite_ref-Introduction_to_Elementary_Mathematical_Logic_31-0" class="reference"><a href="#cite_note-Introduction_to_Elementary_Mathematical_Logic-31">[31]</a></sup></p>
<p>The earliest use of mathematics and <a href="/wiki/Geometry" title="Geometry">geometry</a> in relation to logic and philosophy goes back to the ancient Greeks such as <a href="/wiki/Euclid" title="Euclid">Euclid</a>, <a href="/wiki/Plato" title="Plato">Plato</a>, and <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>.<sup id="cite_ref-The_Cambridge_Companion_to_Aristotle_32-0" class="reference"><a href="#cite_note-The_Cambridge_Companion_to_Aristotle-32">[32]</a></sup> Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.<sup id="cite_ref-Prior_Analytics_33-0" class="reference"><a href="#cite_note-Prior_Analytics-33">[33]</a></sup></p>
<p>One of the boldest attempts to apply logic to mathematics was the <a href="/wiki/Logicism" title="Logicism">logicism</a> pioneered by philosopher-logicians such as <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>. Mathematical theories were supposed to be logical <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a>, and the programme was to show this by means to a reduction of mathematics to logic.<sup id="cite_ref-Principia_3-2" class="reference"><a href="#cite_note-Principia-3">[3]</a></sup> The various attempts to carry this out met with failure, from the crippling of Frege's project in his <i>Grundgesetze</i> by <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>, to the defeat of <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a> by <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorems</a>.</p>
<p>Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>.<sup id="cite_ref-Introduction_to_Mathematical_Logic5_34-0" class="reference"><a href="#cite_note-Introduction_to_Mathematical_Logic5-34">[34]</a></sup> Despite the negative nature of the incompleteness theorems, <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>, a result in <a href="/wiki/Model_theory" title="Model theory">model theory</a> and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's <a href="/wiki/Proof_calculus" title="Proof calculus">proof calculus</a> is enough to <i>describe</i> the whole of mathematics, though not <i>equivalent</i> to it.</p>
<p>If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35">[35]</a></sup> <a href="/wiki/Set_theory" title="Set theory">Set theory</a> originated in the study of the infinite by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, and it has been the source of many of the most challenging and important issues in mathematical logic, from <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a>, through the status of the <a href="/wiki/Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a> and the question of the independence of the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>, to the modern debate on <a href="/wiki/Large_cardinal" title="Large cardinal">large cardinal</a> axioms.</p>
<p><a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">Recursion theory</a> captures the idea of computation in logical and <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> terms; its most classical achievements are the undecidability of the <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> by <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>, and his presentation of the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>.<sup id="cite_ref-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity_36-0" class="reference"><a href="#cite_note-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity-36">[36]</a></sup> Today recursion theory is mostly concerned with the more refined problem of <a href="/wiki/Complexity_class" title="Complexity class">complexity classes</a>—when is a problem efficiently solvable?—and the classification of <a href="/wiki/Turing_degree" title="Turing degree">degrees of unsolvability</a>.<sup id="cite_ref-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity6_37-0" class="reference"><a href="#cite_note-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity6-37">[37]</a></sup></p>
<h3><span class="mw-headline" id="Philosophical_logic">Philosophical logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=16" title="Edit section: Philosophical logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical logic</a></div>
<p><a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical logic</a> deals with formal descriptions of ordinary, non-specialist <a href="/wiki/Natural_language" title="Natural language">("natural") language</a>. Most philosophers assume that the bulk of everyday reasoning can be captured in logic if a method or methods to translate ordinary language into that logic can be found. Philosophical logic is essentially a continuation of the traditional discipline called "logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g. <a href="/wiki/Free_logic" title="Free logic">free logics</a>, <a href="/wiki/Tense_logic" class="mw-redirect" title="Tense logic">tense logics</a>) as well as various extensions of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> (e.g. <a href="/wiki/Modal_logic" title="Modal logic">modal logics</a>) and non-standard semantics for such logics (e.g. <a href="/wiki/Saul_Kripke" title="Saul Kripke">Kripke</a>'s <a href="/wiki/Supervaluationism" title="Supervaluationism">supervaluationism</a> in the semantics of logic).</p>
<p>Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38">[38]</a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39">[39]</a></sup></p>
<h3><span class="mw-headline" id="Computational_logic">Computational logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=17" title="Edit section: Computational logic">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic in computer science</a></div>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="/wiki/File:Register_transfer_level_-_example_toggler.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Register_transfer_level_-_example_toggler.svg/300px-Register_transfer_level_-_example_toggler.svg.png" width="300" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Register_transfer_level_-_example_toggler.svg/450px-Register_transfer_level_-_example_toggler.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Register_transfer_level_-_example_toggler.svg/600px-Register_transfer_level_-_example_toggler.svg.png 2x" data-file-width="300" data-file-height="200" /></a>
<div class="thumbcaption">
<div class="magnify"><a href="/wiki/File:Register_transfer_level_-_example_toggler.svg" class="internal" title="Enlarge"></a></div>
A simple toggling circuit is expressed using a logic gate and a synchronous register.</div>
</div>
</div>
<p>Logic cut to the heart of computer science as it emerged as a discipline: <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s work on the <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i> followed from <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>'s work on the <a href="/wiki/Incompleteness_theorems" class="mw-redirect" title="Incompleteness theorems">incompleteness theorems</a>. The notion of the general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.</p>
<p>In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with <a href="/wiki/Mathematical_notation" title="Mathematical notation">mathematical notation</a>, it would be possible to create a machine that reasons, or artificial intelligence. This was more difficult than expected because of the complexity of human reasoning. In <a href="/wiki/Logic_programming" title="Logic programming">logic programming</a>, a program consists of a set of axioms and rules. Logic programming systems such as <a href="/wiki/Prolog" title="Prolog">Prolog</a> compute the consequences of the axioms and rules in order to answer a query.</p>
<p>Today, logic is extensively applied in the fields of <a href="/wiki/Artificial_Intelligence" class="mw-redirect" title="Artificial Intelligence">Artificial Intelligence</a> and <a href="/wiki/Computer_Science" class="mw-redirect" title="Computer Science">Computer Science</a>, and these fields provide a rich source of problems in formal and informal logic. <a href="/wiki/Argumentation_theory" title="Argumentation theory">Argumentation theory</a> is one good example of how logic is being applied to artificial intelligence. The <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a> in particular regards:</p>
<ul>
<li>Section F.3 on <a href="/w/index.php?title=Logics_and_meanings_of_programs&amp;action=edit&amp;redlink=1" class="new" title="Logics and meanings of programs (page does not exist)">Logics and meanings of programs</a> and F.4 on <a href="/w/index.php?title=Mathematical_logic_and_formal_languages&amp;action=edit&amp;redlink=1" class="new" title="Mathematical logic and formal languages (page does not exist)">Mathematical logic and formal languages</a> as part of the theory of computer science: this work covers <a href="/wiki/Formal_semantics_of_programming_languages" class="mw-redirect" title="Formal semantics of programming languages">formal semantics of programming languages</a>, as well as work of <a href="/wiki/Formal_methods" title="Formal methods">formal methods</a> such as <a href="/wiki/Hoare_logic" title="Hoare logic">Hoare logic</a>;</li>
<li><a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> as fundamental to computer hardware: particularly, the system's section B.2 on <a href="/wiki/Arithmetic_and_logic_structures" class="mw-redirect" title="Arithmetic and logic structures">Arithmetic and logic structures</a>, relating to operatives AND, NOT, and OR;</li>
<li>Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a> and <a href="/wiki/Default_logic" title="Default logic">default logic</a> in <a href="/wiki/Knowledge_representation_formalisms_and_methods" class="mw-redirect" title="Knowledge representation formalisms and methods">Knowledge representation formalisms and methods</a>, <a href="/wiki/Horn_clause" title="Horn clause">Horn clauses</a> in logic programming, and <a href="/wiki/Description_logic" title="Description logic">description logic</a>.</li>
</ul>
<p>Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>, the machines can find and check proofs, as well as work with proofs too lengthy to write out by hand.</p>
<h3><span class="mw-headline" id="Non-classical_logics">Non-classical logics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=18" title="Edit section: Non-classical logics">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical logic</a></div>
<p>The logics discussed above are all "<a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">bivalent</a>" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions. <a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical logics</a> are those systems that reject various rules of <a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a>.</p>
<p>Hegel developed his own <a href="/wiki/Hegelian_dialectic" class="mw-redirect" title="Hegelian dialectic">dialectic logic</a> that extended <a href="/wiki/Immanuel_Kant" title="Immanuel Kant">Kant</a>'s transcendental logic but also brought it back to ground by assuring us that "neither in heaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself".<sup id="cite_ref-philosophical_40-0" class="reference"><a href="#cite_note-philosophical-40">[40]</a></sup></p>
<p>In 1910, <a href="/wiki/Nicolai_A._Vasiliev" title="Nicolai A. Vasiliev">Nicolai A. Vasiliev</a> extended the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction.<sup id="cite_ref-Brenner2008_41-0" class="reference"><a href="#cite_note-Brenner2008-41">[41]</a></sup> In the early 20th&#160;century <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan ?ukasiewicz">Jan ?ukasiewicz</a> investigated the extension of the traditional true/false values to include a third value, "possible", so inventing <a href="/wiki/Ternary_logic" class="mw-redirect" title="Ternary logic">ternary logic</a>, the first <a href="/wiki/Multi-valued_logic" class="mw-redirect" title="Multi-valued logic">multi-valued logic</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42">[42]</a></sup></p>
<p>Logics such as <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> have since been devised with an infinite number of "degrees of truth", represented by a <a href="/wiki/Real_number" title="Real number">real number</a> between 0 and 1.<sup id="cite_ref-stanford_43-0" class="reference"><a href="#cite_note-stanford-43">[43]</a></sup></p>
<p><a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">Intuitionistic logic</a> was proposed by <a href="/wiki/L.E.J._Brouwer" class="mw-redirect" title="L.E.J. Brouwer">L.E.J. Brouwer</a> as the correct logic for reasoning about mathematics, based upon his rejection of the <a href="/wiki/Law_of_the_excluded_middle" class="mw-redirect" title="Law of the excluded middle">law of the excluded middle</a> as part of his <a href="/wiki/Intuitionism" title="Intuitionism">intuitionism</a>. Brouwer rejected formalization in mathematics, but his student <a href="/wiki/Arend_Heyting" title="Arend Heyting">Arend Heyting</a> studied intuitionistic logic formally, as did <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a>. Intuitionistic logic is of great interest to computer scientists, as it is a <a href="/wiki/Constructive_logic" class="mw-redirect" title="Constructive logic">constructive logic</a> and can be applied for extracting verified programs from proofs.</p>
<p><a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a> is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalized with the principle of the excluded middle, and its <a href="/wiki/Relational_semantics" class="mw-redirect" title="Relational semantics">relational semantics</a> is bivalent, so this inclusion is disputable.</p>
<h2><span class="mw-headline" id="Controversies_in_logic">Controversies in logic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=19" title="Edit section: Controversies in logic">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id=".22Is_logic_empirical.3F.22">"Is logic empirical?"</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=20" title="Edit section: &quot;Is logic empirical?&quot;">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Is_logic_empirical%3F" title="Is logic empirical?">Is logic empirical?</a></div>
<p>What is the <a href="/wiki/Epistemology" title="Epistemology">epistemological</a> status of the <a href="/wiki/Classical_logic" title="Classical logic">laws of logic</a>? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled "Is logic empirical?"<sup id="cite_ref-Is_Logic_Empirical.3F_44-0" class="reference"><a href="#cite_note-Is_Logic_Empirical.3F-44">[44]</a></sup> <a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a>, building on a suggestion of <a href="/wiki/W._V._Quine" class="mw-redirect" title="W. V. Quine">W. V. Quine</a>, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> or of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be <a href="/wiki/Philosophical_realism" title="Philosophical realism">realists</a> about the physical phenomena described by quantum theory, then we should abandon the <a href="/wiki/Principle_of_distributivity" title="Principle of distributivity">principle of distributivity</a>, substituting for classical logic the <a href="/wiki/Quantum_logic" title="Quantum logic">quantum logic</a> proposed by <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a> and <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>.<sup id="cite_ref-The_Logic_of_Quantum_Mechanics_45-0" class="reference"><a href="#cite_note-The_Logic_of_Quantum_Mechanics-45">[45]</a></sup></p>
<p>Another paper of the same name by <a href="/wiki/Michael_Dummett" title="Michael Dummett">Michael Dummett</a> argues that Putnam's desire for realism mandates the law of distributivity.<sup id="cite_ref-Truth_and_Other_Enigmas_46-0" class="reference"><a href="#cite_note-Truth_and_Other_Enigmas-46">[46]</a></sup> Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, "Is logic empirical?" can be seen to lead naturally into the fundamental controversy in <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a> on <a href="/wiki/Realism_versus_anti-realism" class="mw-redirect" title="Realism versus anti-realism">realism versus anti-realism</a>.</p>
<h3><span class="mw-headline" id="Implication:_strict_or_material.3F">Implication: strict or material?</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=21" title="Edit section: Implication: strict or material?">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Paradox_of_entailment" class="mw-redirect" title="Paradox of entailment">Paradox of entailment</a></div>
<p>The notion of implication formalized in classical logic does not comfortably translate into natural language by means of "if&#160;... then&#160;...", due to a number of problems called the <a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a>.</p>
<p>The first class of paradoxes involves counterfactuals, such as <i>If the moon is made of green cheese, then 2+2=5</i>, which are puzzling because natural language does not support the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a>. Eliminating this class of paradoxes was the reason for <a href="/wiki/C._I._Lewis" class="mw-redirect" title="C. I. Lewis">C. I. Lewis</a>'s formulation of <a href="/wiki/Strict_implication" class="mw-redirect" title="Strict implication">strict implication</a>, which eventually led to more radically revisionist logics such as <a href="/wiki/Relevance_logic" title="Relevance logic">relevance logic</a>.</p>
<p>The second class of paradoxes involves redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus "if that man gets elected, granny will die" is materially true since granny is mortal, regardless of the man's election prospects. Such sentences violate the <a href="/wiki/Gricean_maxim" class="mw-redirect" title="Gricean maxim">Gricean maxim</a> of relevance, and can be modelled by logics that reject the principle of <a href="/wiki/Monotonicity_of_entailment" title="Monotonicity of entailment">monotonicity of entailment</a>, such as relevance logic.</p>
<h3><span class="mw-headline" id="Tolerating_the_impossible">Tolerating the impossible</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=22" title="Edit section: Tolerating the impossible">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div role="note" class="hatnote">Main article: <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">Paraconsistent logic</a></div>
<p><a href="/wiki/Hegel" class="mw-redirect" title="Hegel">Hegel</a> was deeply critical of any simplified notion of the <a href="/wiki/Law_of_Non-Contradiction" class="mw-redirect" title="Law of Non-Contradiction">Law of Non-Contradiction</a>. It was based on <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>'s idea that this law of logic also requires a sufficient ground to specify from what point of view (or time) one says that something cannot contradict itself. A building, for example, both moves and does not move; the ground for the first is our solar system and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable.</p>
<p>Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate <a href="/wiki/Inconsistency" class="mw-redirect" title="Inconsistency">inconsistency</a>. <a href="/wiki/Relevance_logic" title="Relevance logic">Relevance logic</a> and <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">paraconsistent logic</a> are the most important approaches here, though the concerns are different: a key consequence of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> and some of its rivals, such as <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, is that they respect the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a>, which means that the logic collapses if it is capable of deriving a contradiction. <a href="/wiki/Graham_Priest" title="Graham Priest">Graham Priest</a>, the main proponent of <a href="/wiki/Dialetheism" title="Dialetheism">dialetheism</a>, has argued for paraconsistency on the grounds that there are in fact, true contradictions.<sup id="cite_ref-stanford7_47-0" class="reference"><a href="#cite_note-stanford7-47">[47]</a></sup></p>
<h3><span class="mw-headline" id="Rejection_of_logical_truth">Rejection of logical truth</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=23" title="Edit section: Rejection of logical truth">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a>. Observe that this is opposite to the usual views in <a href="/wiki/Philosophical_skepticism" title="Philosophical skepticism">philosophical skepticism</a>, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of <a href="/wiki/Sextus_Empiricus" title="Sextus Empiricus">Sextus Empiricus</a>.</p>
<p><a href="/wiki/Friedrich_Nietzsche" title="Friedrich Nietzsche">Friedrich Nietzsche</a> provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "...&#160;mobile army of metaphors, metonyms, and anthropomorphisms—in short&#160;... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins."<sup id="cite_ref-nietzsche_48-0" class="reference"><a href="#cite_note-nietzsche-48">[48]</a></sup> His rejection of truth did not lead him to reject the idea of either inference or logic completely, but rather suggested that "logic [came] into existence in man's head [out] of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished".<sup id="cite_ref-nietzsche8_49-0" class="reference"><a href="#cite_note-nietzsche8-49">[49]</a></sup> Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic, too, also rests on assumptions that do not correspond to anything in the real world".<sup id="cite_ref-nietzsche9_50-0" class="reference"><a href="#cite_note-nietzsche9-50">[50]</a></sup></p>
<p>This position held by Nietzsche however, has come under extreme scrutiny for several reasons. He fails to demonstrate the validity of his claims and merely asserts them rhetorically. Although, since he is criticising the established criteria of validity, this does not undermine his position for one could argue that the demonstration of validity provided in the name of logic was just as rhetorically based. Some philosophers, such as <a href="/wiki/J%C3%BCrgen_Habermas" title="Jürgen Habermas">Jürgen Habermas</a>, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.<sup id="cite_ref-nietzsche10_51-0" class="reference"><a href="#cite_note-nietzsche10-51">[51]</a></sup> Again, it is unclear if this is a decisive critique for the criteria of coherency and consistent theory are exactly what is under question. <a href="/wiki/Georg_Luk%C3%A1cs" class="mw-redirect" title="Georg Lukács">Georg Lukács</a>, in his book <i>The Destruction of Reason</i>, asserts that, "Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible."<sup id="cite_ref-marxists_52-0" class="reference"><a href="#cite_note-marxists-52">[52]</a></sup> Still, in this respect his "theory" would be a much better depiction of a confused and chaotic reality than any consistent and compatible theory. <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book <i>A History of Western Philosophy</i>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53">[53]</a></sup></p>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=24" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<table class="metadata mbox-small" style="border:1px solid #aaa;background-color:#f9f9f9">
<tr>
<td class="mbox-image"><img alt="Book icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/30px-Office-book.svg.png" width="30" height="30" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/45px-Office-book.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/60px-Office-book.svg.png 2x" data-file-width="48" data-file-height="48" /></td>
<td class="mbox-text plainlist">
<ul style="font-weight: bold">
<li><a href="/wiki/Book:Logic" title="Book:Logic">Book: Logic</a></li>
</ul>
</td>
</tr>
</table>
<div class="div-col columns column-count column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;">
<ul>
<li><a href="/wiki/Digital_electronics" title="Digital electronics">Digital electronics</a> (also known as <i><a href="/wiki/Digital_logic" class="mw-redirect" title="Digital logic">digital logic</a></i> or <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a>)</li>
<li><a href="/wiki/Fallacies" class="mw-redirect" title="Fallacies">Fallacies</a></li>
<li><a href="/wiki/List_of_logicians" title="List of logicians">List of logicians</a></li>
<li><a href="/wiki/List_of_logic_journals" title="List of logic journals">List of logic journals</a></li>
<li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li>
<li><a href="/wiki/Logic_puzzle" title="Logic puzzle">Logic puzzle</a></li>
<li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a>
<ul>
<li><a href="/wiki/List_of_mathematics_articles" class="mw-redirect" title="List of mathematics articles">List of mathematics articles</a></li>
<li><a href="/wiki/Outline_of_mathematics" title="Outline of mathematics">Outline of mathematics</a></li>
</ul>
</li>
<li><a href="/wiki/Metalogic" title="Metalogic">Metalogic</a></li>
<li><a href="/wiki/Outline_of_logic" title="Outline of logic">Outline of logic</a></li>
<li><a href="/wiki/Philosophy" title="Philosophy">Philosophy</a>
<ul>
<li><a href="/wiki/List_of_philosophy_topics" class="mw-redirect" title="List of philosophy topics">List of philosophy topics</a></li>
<li><a href="/wiki/Outline_of_philosophy" title="Outline of philosophy">Outline of philosophy</a></li>
</ul>
</li>
<li><a href="/wiki/Reason" title="Reason">Reason</a></li>
<li><i><a href="/wiki/Straight_and_Crooked_Thinking" title="Straight and Crooked Thinking">Straight and Crooked Thinking</a></i> (book)</li>
<li><a href="/wiki/Truth" title="Truth">Truth</a></li>
<li><a href="/wiki/Vector_logic" title="Vector logic">Vector logic</a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Notes_and_references">Notes and references</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=25" title="Edit section: Notes and references">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-argumentative-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-argumentative_1-0">^</a></b></span> <span class="reference-text">"possessed of reason, intellectual, dialectical, argumentative", also related to <a href="//en.wiktionary.org/wiki/%CE%BB%CF%8C%CE%B3%CE%BF%CF%82" class="extiw" title="wiktionary:λόγος">λόγος</a> (<i>logos</i>), "word, thought, idea, argument, account, reason, or principle" (Liddell &amp; Scott 1999; Online Etymology Dictionary 2001).</span></li>
<li id="cite_note-The_Basic_Works-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-The_Basic_Works_2-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (2001). "<a href="/wiki/Posterior_Analytics" title="Posterior Analytics">Posterior Analytics</a>". In Mckeon, Richard. <i>The Basic Works</i>. Modern Library. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-375-75799-6" title="Special:BookSources/0-375-75799-6">0-375-75799-6</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Posterior+Analytics&amp;rft.au=Aristotle&amp;rft.btitle=The+Basic+Works&amp;rft.date=2001&amp;rft.genre=bookitem&amp;rft.isbn=0-375-75799-6&amp;rft.pub=Modern+Library&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Principia-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Principia_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Principia_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Principia_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead, Alfred North</a>; <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell, Bertrand</a> (1967). <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica to *56</a></i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-62606-4" title="Special:BookSources/0-521-62606-4">0-521-62606-4</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Alfred+North&amp;rft.aulast=Whitehead&amp;rft.au=Russell%2C+Bertrand&amp;rft.btitle=Principia+Mathematica+to+%2A56&amp;rft.date=1967&amp;rft.genre=book&amp;rft.isbn=0-521-62606-4&amp;rft.pub=Cambridge+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Hamilton-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hamilton_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hamilton_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">For a more modern treatment, see <cite class="citation book">Hamilton, A. G. (1980). <i>Logic for Mathematicians</i>. Cambridge University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-29291-3" title="Special:BookSources/0-521-29291-3">0-521-29291-3</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=A.+G.&amp;rft.aulast=Hamilton&amp;rft.btitle=Logic+for+Mathematicians&amp;rft.date=1980&amp;rft.genre=book&amp;rft.isbn=0-521-29291-3&amp;rft.pub=Cambridge+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></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">T. Mossakowski, <a href="/wiki/Joseph_Goguen" title="Joseph Goguen">J. A. Goguen</a>, R. Diaconescu, A. Tarlecki, "What is a Logic?", <a href="/wiki/Logica_Universalis" title="Logica Universalis">Logica Universalis</a> 2007 Birkhauser, pp. 113–133.</span></li>
<li id="cite_note-Aristotle.27s_syllogistic_from_the_standpoint_of_modern_formal_logic-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aristotle.27s_syllogistic_from_the_standpoint_of_modern_formal_logic_6-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan ?ukasiewicz">?ukasiewicz, Jan</a> (1957). <i>Aristotle's syllogistic from the standpoint of modern formal logic</i> (2nd ed.). Oxford University Press. p.&#160;7. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-19-824144-7" title="Special:BookSources/978-0-19-824144-7">978-0-19-824144-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Jan&amp;rft.aulast=%C5%81ukasiewicz&amp;rft.btitle=Aristotle%27s+syllogistic+from+the+standpoint+of+modern+formal+logic&amp;rft.date=1957&amp;rft.edition=2nd&amp;rft.genre=book&amp;rft.isbn=978-0-19-824144-7&amp;rft.pages=7&amp;rft.pub=Oxford+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></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">Arnauld, <i>Logic or the Art of Thinking</i> Part 2 Chapter 3.</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">Locke, 1690. <a href="/wiki/An_Essay_Concerning_Human_Understanding" title="An Essay Concerning Human Understanding">An Essay Concerning Human Understanding</a>, IV. v. 1-8)</span></li>
<li id="cite_note-Bergmann.2C_Merrie_2009-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bergmann.2C_Merrie_2009_9-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Bergmann, Merrie; Moor, James; Nelson, Jack (2009). <i>The Logic Book</i> (Fifth ed.). New York, NY: McGraw-Hill. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-07-353563-0" title="Special:BookSources/978-0-07-353563-0">978-0-07-353563-0</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Merrie&amp;rft.aulast=Bergmann&amp;rft.au=Moor%2C+James&amp;rft.au=Nelson%2C+Jack&amp;rft.btitle=The+Logic+Book&amp;rft.date=2009&amp;rft.edition=Fifth&amp;rft.genre=book&amp;rft.isbn=978-0-07-353563-0&amp;rft.place=New+York%2C+NY&amp;rft.pub=McGraw-Hill&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></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"><i>Internet Encyclopedia of Philosophy</i>, <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/val-snd/">Validity and Soundness</a></span></li>
<li id="cite_note-Introduction_to_Mathematical_Logic-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Introduction_to_Mathematical_Logic_11-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Mendelson, Elliott (1964). "Quantification Theory: Completeness Theorems". <i>Introduction to Mathematical Logic</i>. Van Nostrand. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-412-80830-7" title="Special:BookSources/0-412-80830-7">0-412-80830-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Quantification+Theory%3A+Completeness+Theorems&amp;rft.aufirst=Elliott&amp;rft.aulast=Mendelson&amp;rft.btitle=Introduction+to+Mathematical+Logic&amp;rft.date=1964&amp;rft.genre=bookitem&amp;rft.isbn=0-412-80830-7&amp;rft.pub=Van+Nostrand&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-abduction-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-abduction_12-0">^</a></b></span>
<ul>
<li><span class="reference-text">Magnani, L. "Abduction, Reason, and Science: Processes of Discovery and Explanation". <i>Kluwer Academic Plenum Publishers, New York, 2001</i>. xvii. 205 pages. Hard cover, <a href="/wiki/Special:BookSources/0306465140" class="internal mw-magiclink-isbn">ISBN 0-306-46514-0</a>.</span></li>
<li><span class="reference-text">R. Josephson, J. &amp; G. Josephson, S. "Abductive Inference: Computation, Philosophy, Technology" <i>Cambridge University Press, New York &amp; Cambridge (U.K.)</i>. viii. 306 pages. Hard cover (1994), <a href="/wiki/Special:BookSources/0521434610" class="internal mw-magiclink-isbn">ISBN 0-521-43461-0</a>, Paperback (1996), <a href="/wiki/Special:BookSources/0521575451" class="internal mw-magiclink-isbn">ISBN 0-521-57545-1</a>.</span></li>
<li><span class="reference-text">Bunt, H. &amp; Black, W. "Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics" <i>(Natural Language Processing, 1.) John Benjamins, Amsterdam &amp; Philadelphia, 2000</i>. vi. 471 pages. Hard cover, <a href="/wiki/Special:BookSources/9027249830" class="internal mw-magiclink-isbn">ISBN 90-272-4983-0</a> (Europe),</span></li>
</ul>
<span class="reference-text">1-58619-794-2 (U.S.)</span></li>
<li id="cite_note-guess-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-guess_13-0">^</a></b></span> <span class="reference-text">Peirce, C. S.</span>
<ul>
<li><span class="reference-text">"On the Logic of drawing History from Ancient Documents especially from Testimonies" (1901), <i>Collected Papers</i> v. 7, paragraph 219.</span></li>
<li><span class="reference-text">"PAP" ["Prolegomena to an Apology for Pragmatism"], MS 293 c. 1906, <i>New Elements of Mathematics</i> v. 4, pp. 319-320.</span></li>
<li><span class="reference-text">A Letter to F. A. Woods (1913), <i>Collected Papers</i> v. 8, paragraphs 385-388.</span></li>
</ul>
<span class="reference-text">(See under "<a rel="nofollow" class="external text" href="http://www.helsinki.fi/science/commens/terms/abduction.html">Abduction</a>" and "<a rel="nofollow" class="external text" href="http://www.helsinki.fi/science/commens/terms/retroduction.html">Retroduction</a>" at <i>Commens Dictionary of Peirce's Terms</i>.)</span></li>
<li id="cite_note-HL-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-HL_14-0">^</a></b></span> <span class="reference-text">Peirce, C. S. (1903), Harvard lectures on pragmatism, <i>Collected Papers</i> v. 5, <a rel="nofollow" class="external text" href="http://www.textlog.de/7664-2.html">paragraphs 188–189</a>.</span></li>
<li id="cite_note-stanford-logic-onthology-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-stanford-logic-onthology_15-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Hofweber, T. (2004). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-ontology">"Logic and Ontology"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N</a>. <i>Stanford Encyclopedia of Philosophy</i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Logic+and+Ontology&amp;rft.aufirst=T.&amp;rft.aulast=Hofweber&amp;rft.btitle=Stanford+Encyclopedia+of+Philosophy&amp;rft.date=2004&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-ontology&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-brandom-2000-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-brandom-2000_16-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Brandom, Robert (2000). <i>Articulating Reasons</i>. Cambridge, MA: Harvard University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-674-00158-3" title="Special:BookSources/0-674-00158-3">0-674-00158-3</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Robert&amp;rft.aulast=Brandom&amp;rft.btitle=Articulating+Reasons&amp;rft.date=2000&amp;rft.genre=book&amp;rft.isbn=0-674-00158-3&amp;rft.place=Cambridge%2C+MA&amp;rft.pub=Harvard+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-achievement-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-achievement_17-0">^</a></b></span> <span class="reference-text">E.g., Kline (1972, p.53) wrote "A major achievement of Aristotle was the founding of the science of logic".</span></li>
<li id="cite_note-mtu-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-mtu_18-0">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://chemistry.mtu.edu/%7Epcharles/SCIHISTORY/aristotle.html">Aristotle</a>", MTU Department of Chemistry.</span></li>
<li id="cite_note-google-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-google_19-0">^</a></b></span> <span class="reference-text">Jonathan Lear (1986). "<i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=lXI7AAAAIAAJ&amp;pg=PA34&amp;dq&amp;hl=en#v=onepage&amp;q=&amp;f=false">Aristotle and Logical Theory</a></i>". <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p.34. <a href="/wiki/Special:BookSources/0521311780" class="internal mw-magiclink-isbn">ISBN 0-521-31178-0</a></span></li>
<li id="cite_note-google1-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-google1_20-0">^</a></b></span> <span class="reference-text">Simo Knuuttila (1981). "<i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=iCCUF_OtA8AC&amp;pg=PA71&amp;dq&amp;hl=en#v=onepage&amp;q=&amp;f=false">Reforging the great chain of being: studies of the history of modal theories</a></i>". Springer Science &amp; Business. p.71. <a href="/wiki/Special:BookSources/9027711259" class="internal mw-magiclink-isbn">ISBN 90-277-1125-9</a></span></li>
<li id="cite_note-google2-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-google2_21-0">^</a></b></span> <span class="reference-text">Michael Fisher, Dov M. Gabbay, Lluís Vila (2005). "<i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ajsvo6jWNhgC&amp;pg=PA119&amp;dq&amp;hl=en#v=onepage&amp;q=&amp;f=false">Handbook of temporal reasoning in artificial intelligence</a></i>". Elsevier. p.119. <a href="/wiki/Special:BookSources/0444514937" class="internal mw-magiclink-isbn">ISBN 0-444-51493-7</a></span></li>
<li id="cite_note-google3-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-google3_22-0">^</a></b></span> <span class="reference-text">Harold Joseph Berman (1983). "<i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=9-8fIBVgCQYC&amp;pg=PA133&amp;dq&amp;hl=en#v=onepage&amp;q=&amp;f=false">Law and revolution: the formation of the Western legal tradition</a></i>". <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>. p.133. <a href="/wiki/Special:BookSources/0674517768" class="internal mw-magiclink-isbn">ISBN 0-674-51776-8</a></span></li>
<li id="cite_note-propositions-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-propositions_23-0">^</a></b></span> <span class="reference-text">The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek <a href="/wiki/Tetralemma" title="Tetralemma">tetralemma</a>, which in turn are analogous to the four <a href="/wiki/Truth_value" title="Truth value">truth values</a> of modern <a href="/wiki/Relevance_logic" title="Relevance logic">relevance logic</a> Cf. Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in <i>Philosophy East and West</i>, University of Hawaii Press).</span></li>
<li id="cite_note-Chakrabarti-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chakrabarti_24-0">^</a></b></span> <span class="reference-text"><cite class="citation journal">Kisor Kumar Chakrabarti (June 1976). "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic". <i>Philosophy and Phenomenological Research</i> (International Phenomenological Society) <b>36</b> (4): 554–563. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.2307%2F2106873">10.2307/2106873</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/2106873">2106873</a>. <q>This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Some+Comparisons+Between+Frege%27s+Logic+and+Navya-Nyaya+Logic&amp;rft.au=Kisor+Kumar+Chakrabarti&amp;rft.date=1976-06&amp;rft.genre=article&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2106873&amp;rft_id=info%3Adoi%2F10.2307%2F2106873&amp;rft.issue=4&amp;rft.jtitle=Philosophy+and+Phenomenological+Research&amp;rft.pages=554-563&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=36" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Indian_logic:_a_reader-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Indian_logic:_a_reader_25-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Jonardon Ganeri (2001). <i>Indian logic: a reader</i>. <a href="/wiki/Routledge" title="Routledge">Routledge</a>. pp.&#160;vii, 5, 7. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-7007-1306-9" title="Special:BookSources/0-7007-1306-9">0-7007-1306-9</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.au=Jonardon+Ganeri&amp;rft.btitle=Indian+logic%3A+a+reader&amp;rft.date=2001&amp;rft.genre=book&amp;rft.isbn=0-7007-1306-9&amp;rft.pages=vii%2C+5%2C+7&amp;rft.pub=Routledge&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Aristotle-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aristotle_26-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/346217/history-of-logic/65920/Aristotle">"Aristotle"</a>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Aristotle&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F346217%2Fhistory-of-logic%2F65920%2FAristotle&amp;rft.jtitle=Encyclop%C3%A6dia+Britannica&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Britannica-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Britannica_27-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.britannica.com/ebc/article-65928">"History of logic: Arabic logic"</a>. <a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.btitle=History+of+logic%3A+Arabic+logic&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.britannica.com%2Febc%2Farticle-65928&amp;rft.pub=Encyclop%C3%A6dia+Britannica&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite class="citation journal">Rescher, Nicholas (1978). <a rel="nofollow" class="external text" href="http://ojs.uwindsor.ca/ojs/leddy/index.php/informal_logic/article/view/2809">"Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge"</a>. <i>Informal Logic</i> <b>1</b> (#3).</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Dialectics%3A+A+Controversy-Oriented+Approach+to+the+Theory+of+Knowledge&amp;rft.aufirst=Nicholas&amp;rft.aulast=Rescher&amp;rft.date=1978&amp;rft.genre=article&amp;rft_id=http%3A%2F%2Fojs.uwindsor.ca%2Fojs%2Fleddy%2Findex.php%2Finformal_logic%2Farticle%2Fview%2F2809&amp;rft.issue=%233&amp;rft.jtitle=Informal+Logic&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=1" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite class="citation journal">Hetherington, Stephen (2006). <a rel="nofollow" class="external text" href="https://ndpr.nd.edu/news/25075-philosophical-dialectics-an-essay-on-metaphilosophy/">"Nicholas Rescher: Philosophical Dialectics"</a>. <i>Notre Dame Philosophical Reviews</i> (2006.07.16).</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Nicholas+Rescher%3A+Philosophical+Dialectics&amp;rft.aufirst=Stephen&amp;rft.aulast=Hetherington&amp;rft.date=2006&amp;rft.genre=article&amp;rft_id=https%3A%2F%2Fndpr.nd.edu%2Fnews%2F25075-philosophical-dialectics-an-essay-on-metaphilosophy%2F&amp;rft.issue=2006.07.16&amp;rft.jtitle=Notre+Dame+Philosophical+Reviews&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><cite class="citation book">Rescher, Nicholas (2009). Jacquette,Dale, ed. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T65moyMMioYC"><i>Reason, Method, and Value: A Reader on the Philosophy of Nicholas Rescher</i></a>. Ontos Verlag.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Nicholas&amp;rft.aulast=Rescher&amp;rft.btitle=Reason%2C+Method%2C+and+Value%3A+A+Reader+on+the+Philosophy+of+Nicholas+Rescher&amp;rft.date=2009&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT65moyMMioYC&amp;rft.pub=Ontos+Verlag&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Introduction_to_Elementary_Mathematical_Logic-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Introduction_to_Elementary_Mathematical_Logic_31-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Stolyar, Abram A. (1983). <i>Introduction to Elementary Mathematical Logic</i>. Dover Publications. p.&#160;3. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-486-64561-4" title="Special:BookSources/0-486-64561-4">0-486-64561-4</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Abram+A.&amp;rft.aulast=Stolyar&amp;rft.btitle=Introduction+to+Elementary+Mathematical+Logic&amp;rft.date=1983&amp;rft.genre=book&amp;rft.isbn=0-486-64561-4&amp;rft.pages=3&amp;rft.pub=Dover+Publications&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-The_Cambridge_Companion_to_Aristotle-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-The_Cambridge_Companion_to_Aristotle_32-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Barnes, Jonathan (1995). <i>The Cambridge Companion to Aristotle</i>. Cambridge University Press. p.&#160;27. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-42294-9" title="Special:BookSources/0-521-42294-9">0-521-42294-9</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Jonathan&amp;rft.aulast=Barnes&amp;rft.btitle=The+Cambridge+Companion+to+Aristotle&amp;rft.date=1995&amp;rft.genre=book&amp;rft.isbn=0-521-42294-9&amp;rft.pages=27&amp;rft.pub=Cambridge+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Prior_Analytics-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Prior_Analytics_33-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (1989). <i>Prior Analytics</i>. Hackett Publishing Co. p.&#160;115. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-87220-064-7" title="Special:BookSources/978-0-87220-064-7">978-0-87220-064-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.au=Aristotle&amp;rft.btitle=Prior+Analytics&amp;rft.date=1989&amp;rft.genre=book&amp;rft.isbn=978-0-87220-064-7&amp;rft.pages=115&amp;rft.pub=Hackett+Publishing+Co.&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Introduction_to_Mathematical_Logic5-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Introduction_to_Mathematical_Logic5_34-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Mendelson, Elliott (1964). "Formal Number Theory: Gödel's Incompleteness Theorem". <i>Introduction to Mathematical Logic</i>. Monterey, Calif.: Wadsworth &amp; Brooks/Cole Advanced Books &amp; Software. <a href="/wiki/OCLC" title="OCLC">OCLC</a>&#160;<a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/13580200">13580200</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Formal+Number+Theory%3A+G%C3%B6del%27s+Incompleteness+Theorem&amp;rft.aufirst=Elliott&amp;rft.aulast=Mendelson&amp;rft.btitle=Introduction+to+Mathematical+Logic&amp;rft.date=1964&amp;rft.genre=bookitem&amp;rft_id=info%3Aoclcnum%2F13580200&amp;rft.place=Monterey%2C+Calif.&amp;rft.pub=Wadsworth+%26+Brooks%2FCole+Advanced+Books+%26+Software&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Barwise (1982) divides the subject of mathematical logic into model theory, proof theory, set theory and recursion theory.</span></li>
<li id="cite_note-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity_36-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Brookshear, J. Glenn (1989). "Computability: Foundations of Recursive Function Theory". <i>Theory of computation: formal languages, automata, and complexity</i>. Redwood City, Calif.: Benjamin/Cummings Pub. Co. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-8053-0143-7" title="Special:BookSources/0-8053-0143-7">0-8053-0143-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Computability%3A+Foundations+of+Recursive+Function+Theory&amp;rft.aufirst=J.+Glenn&amp;rft.aulast=Brookshear&amp;rft.btitle=Theory+of+computation%3A+formal+languages%2C+automata%2C+and+complexity&amp;rft.date=1989&amp;rft.genre=bookitem&amp;rft.isbn=0-8053-0143-7&amp;rft.place=Redwood+City%2C+Calif.&amp;rft.pub=Benjamin%2FCummings+Pub.+Co.&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity6-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Theory_of_computation:_formal_languages.2C_automata.2C_and_complexity6_37-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Brookshear, J. Glenn (1989). "Complexity". <i>Theory of computation: formal languages, automata, and complexity</i>. Redwood City, Calif.: Benjamin/Cummings Pub. Co. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-8053-0143-7" title="Special:BookSources/0-8053-0143-7">0-8053-0143-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Complexity&amp;rft.aufirst=J.+Glenn&amp;rft.aulast=Brookshear&amp;rft.btitle=Theory+of+computation%3A+formal+languages%2C+automata%2C+and+complexity&amp;rft.date=1989&amp;rft.genre=bookitem&amp;rft.isbn=0-8053-0143-7&amp;rft.place=Redwood+City%2C+Calif.&amp;rft.pub=Benjamin%2FCummings+Pub.+Co.&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><cite id="CITEREFGoldman1986" class="citation">Goldman, Alvin I. (1986), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9XwhBw2jMqMC&amp;pg=PA293"><i>Epistemology and Cognition</i></a>, Harvard University Press, p.&#160;293, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780674258969" title="Special:BookSources/9780674258969">9780674258969</a>, <q>untrained subjects are prone to commit various sorts of fallacies and mistakes</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Alvin+I.&amp;rft.aulast=Goldman&amp;rft.btitle=Epistemology+and+Cognition&amp;rft.date=1986&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9XwhBw2jMqMC%26pg%3DPA293&amp;rft.isbn=9780674258969&amp;rft.pages=293&amp;rft.pub=Harvard+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span>.</span></li>
<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite id="CITEREFDemetriouEfklides1994" class="citation">Demetriou, A.; Efklides, A., eds. (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HkukZOU2pzIC&amp;pg=PA194"><i>Intelligence, Mind, and Reasoning: Structure and Development</i></a>, Advances in Psychology <b>106</b>, Elsevier, p.&#160;194, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780080867601" title="Special:BookSources/9780080867601">9780080867601</a></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.btitle=Intelligence%2C+Mind%2C+and+Reasoning%3A+Structure+and+Development&amp;rft.date=1994&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHkukZOU2pzIC%26pg%3DPA194&amp;rft.isbn=9780080867601&amp;rft.pages=194&amp;rft.pub=Elsevier&amp;rft.series=Advances+in+Psychology&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span>.</span></li>
<li id="cite_note-philosophical-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-philosophical_40-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Georg_Wilhelm_Friedrich_Hegel" title="Georg Wilhelm Friedrich Hegel">Hegel, G. W. F</a> (1971) [1817]. <i><a href="/wiki/Philosophy_of_Mind" class="mw-redirect" title="Philosophy of Mind">Philosophy of Mind</a></i>. Encyclopedia of the Philosophical Sciences. trans. <a href="/wiki/William_Wallace_(Scottish_philosopher)" class="mw-redirect" title="William Wallace (Scottish philosopher)">William Wallace</a>. Oxford: Clarendon Press. p.&#160;174. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-19-875014-5" title="Special:BookSources/0-19-875014-5">0-19-875014-5</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=G.+W.+F&amp;rft.aulast=Hegel&amp;rft.btitle=Philosophy+of+Mind&amp;rft.date=1971&amp;rft.genre=book&amp;rft.isbn=0-19-875014-5&amp;rft.pages=174&amp;rft.place=Oxford&amp;rft.pub=Clarendon+Press&amp;rft.series=Encyclopedia+of+the+Philosophical+Sciences&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Brenner2008-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brenner2008_41-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Joseph E. Brenner (3 August 2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jnj5E6C9UwsC&amp;pg=PA28"><i>Logic in Reality</i></a>. Springer. pp.&#160;28–30. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-4020-8374-7" title="Special:BookSources/978-1-4020-8374-7">978-1-4020-8374-7</a><span class="reference-accessdate">. Retrieved <span class="nowrap">9 April</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.au=Joseph+E.+Brenner&amp;rft.btitle=Logic+in+Reality&amp;rft.date=2008-08-03&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJnj5E6C9UwsC%26pg%3DPA28&amp;rft.isbn=978-1-4020-8374-7&amp;rft.pages=28-30&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><cite id="CITEREFZegarelli2010" class="citation">Zegarelli, Mark (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xm0-0Rs2GVoC&amp;pg=PA30"><i>Logic For Dummies</i></a>, John Wiley &amp; Sons, p.&#160;30, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9781118053072" title="Special:BookSources/9781118053072">9781118053072</a></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Mark&amp;rft.aulast=Zegarelli&amp;rft.btitle=Logic+For+Dummies&amp;rft.date=2010&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dxm0-0Rs2GVoC%26pg%3DPA30&amp;rft.isbn=9781118053072&amp;rft.pages=30&amp;rft.pub=John+Wiley+%26+Sons&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span>.</span></li>
<li id="cite_note-stanford-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-stanford_43-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Petr_H%C3%A1jek" title="Petr Hájek">Hájek, Petr</a> (2006). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-fuzzy/">"Fuzzy Logic"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Fuzzy+Logic&amp;rft.aufirst=Petr&amp;rft.aulast=H%C3%A1jek&amp;rft.btitle=Stanford+Encyclopedia+of+Philosophy&amp;rft.date=2006&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-fuzzy%2F&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Is_Logic_Empirical.3F-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-Is_Logic_Empirical.3F_44-0">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Putnam, H.</a> (1969). "Is Logic Empirical?". <i>Boston Studies in the Philosophy of Science</i> <b>5</b>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Is+Logic+Empirical%3F&amp;rft.aufirst=H.&amp;rft.aulast=Putnam&amp;rft.date=1969&amp;rft.genre=article&amp;rft.jtitle=Boston+Studies+in+the+Philosophy+of+Science&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=5" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-The_Logic_of_Quantum_Mechanics-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-The_Logic_of_Quantum_Mechanics_45-0">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, G.</a>; <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, J.</a> (1936). "The Logic of Quantum Mechanics". <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i> (Annals of Mathematics) <b>37</b> (4): 823–843. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.2307%2F1968621">10.2307/1968621</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/1968621">1968621</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=The+Logic+of+Quantum+Mechanics&amp;rft.aufirst=G.&amp;rft.aulast=Birkhoff&amp;rft.au=von+Neumann%2C+J.&amp;rft.date=1936&amp;rft.genre=article&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F1968621&amp;rft_id=info%3Adoi%2F10.2307%2F1968621&amp;rft.issue=4&amp;rft.jtitle=Annals+of+Mathematics&amp;rft.pages=823-843&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=37" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Truth_and_Other_Enigmas-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-Truth_and_Other_Enigmas_46-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Michael_Dummett" title="Michael Dummett">Dummett, M.</a> (1978). "Is Logic Empirical?". <i>Truth and Other Enigmas</i>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-674-91076-1" title="Special:BookSources/0-674-91076-1">0-674-91076-1</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Is+Logic+Empirical%3F&amp;rft.aufirst=M.&amp;rft.aulast=Dummett&amp;rft.btitle=Truth+and+Other+Enigmas&amp;rft.date=1978&amp;rft.genre=bookitem&amp;rft.isbn=0-674-91076-1&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-stanford7-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-stanford7_47-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Graham_Priest" title="Graham Priest">Priest, Graham</a> (2008). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/dialetheism">"Dialetheism"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Dialetheism&amp;rft.aufirst=Graham&amp;rft.aulast=Priest&amp;rft.btitle=Stanford+Encyclopedia+of+Philosophy&amp;rft.date=2008&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fdialetheism&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-nietzsche-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-nietzsche_48-0">^</a></b></span> <span class="reference-text">Nietzsche, 1873, <a href="/wiki/On_Truth_and_Lies_in_a_Nonmoral_Sense" title="On Truth and Lies in a Nonmoral Sense">On Truth and Lies in a Nonmoral Sense</a>.</span></li>
<li id="cite_note-nietzsche8-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-nietzsche8_49-0">^</a></b></span> <span class="reference-text">Nietzsche, 1882, <i><a href="/wiki/The_Gay_Science" title="The Gay Science">The Gay Science</a></i>.</span></li>
<li id="cite_note-nietzsche9-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-nietzsche9_50-0">^</a></b></span> <span class="reference-text">Nietzsche, 1878, <i><a href="/wiki/Human,_All_Too_Human" title="Human, All Too Human">Human, All Too Human</a></i></span></li>
<li id="cite_note-nietzsche10-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-nietzsche10_51-0">^</a></b></span> <span class="reference-text">Babette Babich, Habermas, Nietzsche, and Critical Theory</span></li>
<li id="cite_note-marxists-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-marxists_52-0">^</a></b></span> <span class="reference-text"><cite class="citation web">Georg Lukács. <a rel="nofollow" class="external text" href="http://www.marxists.org/archive/lukacs/works/destruction-reason/ch03.htm">"The Destruction of Reason by Georg Lukács 1952"</a>. Marxists.org<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-06-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.au=Georg+Luk%C3%A1cs&amp;rft.btitle=The+Destruction+of+Reason+by+Georg+Luk%C3%A1cs+1952&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.marxists.org%2Farchive%2Flukacs%2Fworks%2Fdestruction-reason%2Fch03.htm&amp;rft.pub=Marxists.org&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><cite id="CITEREFRussell1945" class="citation">Russell, Bertrand (1945), <a rel="nofollow" class="external text" href="http://www.ntslibrary.com/PDF%20Books/History%20of%20Western%20Philosophy.pdf/"><i>A History of Western Philosophy And Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day</i></a> <span style="font-size:85%;">(PDF)</span>, Simon and Schuster, p.&#160;762</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Bertrand&amp;rft.aulast=Russell&amp;rft.btitle=A+History+of+Western+Philosophy+And+Its+Connection+with+Political+and+Social+Circumstances+from+the+Earliest+Times+to+the+Present+Day&amp;rft.date=1945&amp;rft.genre=book&amp;rft_id=http%3A%2F%2Fwww.ntslibrary.com%2FPDF%2520Books%2FHistory%2520of%2520Western%2520Philosophy.pdf%2F&amp;rft.pages=762&amp;rft.pub=Simon+and+Schuster&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
</ol>
</div>
<h2><span class="mw-headline" id="Bibliography">Bibliography</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=26" title="Edit section: Bibliography">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="refbegin" style="">
<ul>
<li><a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, J.</a> (1982). <i>Handbook of Mathematical Logic</i>. Elsevier. <a href="/wiki/Special:BookSources/9780080933641" class="internal mw-magiclink-isbn">ISBN 9780080933641</a>.</li>
<li><a href="/wiki/Nuel_Belnap" title="Nuel Belnap">Belnap, N.</a> (1977). "A useful four-valued logic". In Dunn &amp; Eppstein, <i>Modern uses of multiple-valued logic</i>. Reidel: Boston.</li>
<li><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński, J. M.</a> (1959). <i>A précis of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a></i>. Translated from the French and German editions by Otto Bird. D. Reidel, Dordrecht, South Holland.</li>
<li><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński, J. M.</a> (1970). <i>A history of <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a></i>. 2nd Edition. Translated and edited from the German edition by Ivo Thomas. Chelsea Publishing, New York.</li>
<li><cite class="citation book">Brookshear, J. Glenn (1989). <i>Theory of computation: formal languages, automata, and complexity</i>. Redwood City, Calif.: Benjamin/Cummings Pub. Co. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-8053-0143-7" title="Special:BookSources/0-8053-0143-7">0-8053-0143-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=J.+Glenn&amp;rft.aulast=Brookshear&amp;rft.btitle=Theory+of+computation%3A+formal+languages%2C+automata%2C+and+complexity&amp;rft.date=1989&amp;rft.genre=book&amp;rft.isbn=0-8053-0143-7&amp;rft.place=Redwood+City%2C+Calif.&amp;rft.pub=Benjamin%2FCummings+Pub.+Co.&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li>Cohen, R.S, and Wartofsky, M.W. (1974). <i>Logical and Epistemological Studies in Contemporary Physics</i>. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. <a href="/wiki/Special:BookSources/9027703779" class="internal mw-magiclink-isbn">ISBN 90-277-0377-9</a>.</li>
<li>Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).</li>
<li><a href="/wiki/Dov_Gabbay" title="Dov Gabbay">Gabbay, D.M.</a>, and Guenthner, F. (eds., 2001–2005). <i>Handbook of Philosophical Logic</i>. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.</li>
<li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, D.</a>, and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Ackermann, W</a>, (1928). <i>Grundzüge der theoretischen Logik</i> (<i><a href="/wiki/Principles_of_Mathematical_Logic" title="Principles of Mathematical Logic">Principles of Mathematical Logic</a></i>). Springer-Verlag. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/2085765">OCLC 2085765</a></li>
<li><a href="/wiki/Susan_Haack" title="Susan Haack">Susan Haack</a> (1996). <i>Deviant Logic, Fuzzy Logic: Beyond the Formalism</i>, University of Chicago Press.</li>
<li><a href="/wiki/Wilfred_Hodges" class="mw-redirect" title="Wilfred Hodges">Hodges, W.</a> (2001). <i>Logic. An introduction to Elementary Logic</i>, Penguin Books.</li>
<li>Hofweber, T. (2004), <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-ontology/">Logic and Ontology</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Edward N. Zalta</a> (ed.).</li>
<li>Hughes, R.I.G. (1993, ed.). <i>A Philosophical Companion to First-Order Logic</i>. Hackett Publishing.</li>
<li><cite class="citation book">Kline, Morris (1972). <i>Mathematical Thought From Ancient to Modern Times</i>. Oxford University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-19-506135-7" title="Special:BookSources/0-19-506135-7">0-19-506135-7</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.aufirst=Morris&amp;rft.aulast=Kline&amp;rft.btitle=Mathematical+Thought+From+Ancient+to+Modern+Times&amp;rft.date=1972&amp;rft.genre=book&amp;rft.isbn=0-19-506135-7&amp;rft.pub=Oxford+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><a href="/wiki/William_Kneale_(logician)" class="mw-redirect" title="William Kneale (logician)">Kneale, William</a>, and Kneale, Martha, (1962). <i>The Development of Logic</i>. Oxford University Press, London, UK.</li>
<li><cite class="citation web"><a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell, Henry George</a>; <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott, Robert</a>. <a rel="nofollow" class="external text" href="http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2363716">"Logikos"</a>. <i><a href="/wiki/A_Greek-English_Lexicon" class="mw-redirect" title="A Greek-English Lexicon">A Greek-English Lexicon</a></i>. <a href="/wiki/Perseus_Project" title="Perseus Project">Perseus Project</a><span class="reference-accessdate">. Retrieved <span class="nowrap">8 May</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Logikos&amp;rft.aufirst=Henry+George&amp;rft.aulast=Liddell&amp;rft.au=Scott%2C+Robert&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.perseus.tufts.edu%2Fcgi-bin%2Fptext%3Fdoc%3DPerseus%253Atext%253A1999.04.0057%253Aentry%253D%252363716&amp;rft.jtitle=A+Greek-English+Lexicon&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li>Mendelson, Elliott, (1964). <i>Introduction to Mathematical Logic</i>. Wadsworth &amp; Brooks/Cole Advanced Books &amp; Software: Monterey, Calif. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/13580200">OCLC 13580200</a></li>
<li><cite class="citation web">Harper, Robert (2001). <a rel="nofollow" class="external text" href="http://www.etymonline.com/index.php?term=logic">"Logic"</a>. <i><a href="/wiki/Online_Etymology_Dictionary" title="Online Etymology Dictionary">Online Etymology Dictionary</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">8 May</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Logic&amp;rft.aufirst=Robert&amp;rft.aulast=Harper&amp;rft.date=2001&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.etymonline.com%2Findex.php%3Fterm%3Dlogic&amp;rft.jtitle=Online+Etymology+Dictionary&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><a href="/wiki/Barry_Smith_(ontologist)" class="mw-redirect" title="Barry Smith (ontologist)">Smith, B.</a> (1989). "Logic and the Sachverhalt". <i>The Monist</i> 72(1):52–69.</li>
<li><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead, Alfred North</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> (1910). <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>. Cambridge University Press: Cambridge, England. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1041146">OCLC 1041146</a></li>
</ul>
</div>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logic&amp;action=edit&amp;section=27" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
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<b>Logic</b>
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<ul>
<li><a class="external text" href="//tools.wmflabs.org/ftl/cgi-bin/ftl?st=wp&amp;su=Logic">Resources in your library</a></li>
<li><a class="external text" href="//tools.wmflabs.org/ftl/cgi-bin/ftl?st=wp&amp;su=Logic&amp;library=0CHOOSE0">Resources in other libraries</a></li>
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<ul>
<li><a rel="nofollow" class="external text" href="http://philpapers.org/browse/logic-and-philosophy-of-logic">Logic</a> at <a href="/wiki/PhilPapers" title="PhilPapers">PhilPapers</a></li>
<li><a rel="nofollow" class="external text" href="https://inpho.cogs.indiana.edu/taxonomy/2245">Logic</a> at the <a href="/wiki/Indiana_Philosophy_Ontology_Project" title="Indiana Philosophy Ontology Project">Indiana Philosophy Ontology Project</a></li>
<li><cite class="citation encyclopaedia"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/category/s-l-m/logic/">"Logic"</a>. <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Logic&amp;rft.btitle=Internet+Encyclopedia+of+Philosophy&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fcategory%2Fs-l-m%2Flogic%2F&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel, ed. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/l060690">"Logical calculus"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4">978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic&amp;rft.atitle=Logical+calculus&amp;rft.btitle=Encyclopedia+of+Mathematics&amp;rft.date=2001&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fl060690&amp;rft.isbn=978-1-55608-010-4&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><a rel="nofollow" class="external text" href="http://logic-law.com/index.php?title=Propositional_Logic">An Outline for Verbal Logic</a></li>
<li>Introductions and tutorials
<ul>
<li><a rel="nofollow" class="external text" href="http://www.galilean-library.org/manuscript.php?postid=43782">An Introduction to Philosophical Logic</a>, by Paul Newall, aimed at beginners.</li>
<li><a rel="nofollow" class="external text" href="http://www.fecundity.com/logic/">forall x: an introduction to formal logic</a>, by <a href="/w/index.php?title=P.D._Magnus&amp;action=edit&amp;redlink=1" class="new" title="P.D. Magnus (page does not exist)">P.D. Magnus</a>, covers sentential and quantified logic.</li>
<li><a rel="nofollow" class="external text" href="http://www.filozofia.uw.edu.pl/kpaprzycka/Publ/xLogicSelfTaught.html">Logic Self-Taught: A Workbook</a> (originally prepared for on-line logic instruction).
<ul>
<li><a href="/wiki/Nicholas_Rescher" title="Nicholas Rescher">Nicholas Rescher</a>. (1964). <i>Introduction to Logic</i>, St. Martin's Press.</li>
</ul>
</li>
</ul>
</li>
<li>Essays
<ul>
<li><a rel="nofollow" class="external text" href="http://durendal.org:8080/lcsl/">"Symbolic Logic"</a> and <a rel="nofollow" class="external text" href="http://www.gutenberg.org/etext/4763">"The Game of Logic"</a>, <a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a>, 1896.</li>
<li><a rel="nofollow" class="external text" href="http://etext.lib.virginia.edu/DicHist/analytic/anaVII.html">Math &amp; Logic: The history of formal mathematical, logical, linguistic and methodological ideas.</a> In <i>The Dictionary of the History of Ideas.</i></li>
</ul>
</li>
<li>Online Tools
<ul>
<li><a rel="nofollow" class="external text" href="http://thefirstscience.org/syllogistic-machine/">Interactive Syllogistic Machine</a> A web based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms.</li>
</ul>
</li>
<li>Reference material
<ul>
<li><a rel="nofollow" class="external text" href="http://www.earlham.edu/~peters/courses/log/transtip.htm">Translation Tips</a>, by Peter Suber, for translating from English into logical notation.</li>
<li><a rel="nofollow" class="external text" href="http://www.ontology.co/history-of-logic.htm">Ontology and History of Logic. An Introduction</a> with an annotated bibliography.</li>
</ul>
</li>
<li>Reading lists
<ul>
<li>The <a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/">London Philosophy Study Guide</a> offers many suggestions on what to read, depending on the student's familiarity with the subject:
<ul>
<li><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/L&amp;M.htm">Logic &amp; Metaphysics</a></li>
<li><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/SetTheory.htm">Set Theory and Further Logic</a></li>
<li><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/MathLogic.htm">Mathematical Logic</a></li>
</ul>
</li>
</ul>
</li>
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<li><a href="/wiki/Antinomy" title="Antinomy">Antinomy</a></li>
<li><a href="/wiki/A_priori_and_a_posteriori" title="A priori and a posteriori"><i>A priori</i> and <i>a posteriori</i></a></li>
<li><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>
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<li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>
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<li><a href="/wiki/List_of_logicians" title="List of logicians">Logicians</a></li>
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<li><a href="/wiki/List_of_fallacies" title="List of fallacies">Fallacies</a></li>
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<li><a href="/wiki/Wikipedia:WikiProject_Logic" title="Wikipedia:WikiProject Logic">WikiProject</a>&#160;(<a href="/wiki/Wikipedia_talk:WikiProject_Logic" title="Wikipedia talk:WikiProject Logic">talk</a>)</li>
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<li><a href="/wiki/Philosophy_of_design" title="Philosophy of design">Design</a></li>
<li><a href="/wiki/Philosophy_of_music" title="Philosophy of music">Music</a></li>
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<li><a href="/wiki/Philosophical_anthropology" title="Philosophical anthropology">Human nature</a></li>
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<li><a href="/wiki/Philosophy_and_economics" title="Philosophy and economics">Economics</a></li>
<li><a href="/wiki/Justice" title="Justice">Justice</a></li>
<li><a href="/wiki/Jurisprudence" title="Jurisprudence">Law</a></li>
<li><a href="/wiki/Political_philosophy" title="Political philosophy">Politics</a></li>
<li><a href="/wiki/Social_philosophy" title="Social philosophy">Society</a></li>
</ul>
</li>
<li><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Space and time</a></li>
<li><a href="/wiki/Philosophy_of_sport" title="Philosophy of sport">Sport</a></li>
<li><a href="/wiki/Philosophy_of_technology" title="Philosophy of technology">Technology</a>
<ul>
<li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Artificial intelligence</a></li>
<li><a href="/wiki/Philosophy_of_computer_science" title="Philosophy of computer science">Computer science</a></li>
<li><a href="/wiki/Philosophy_of_engineering" title="Philosophy of engineering">Engineering</a></li>
<li><a href="/wiki/Philosophy_of_information" title="Philosophy of information">Information</a></li>
</ul>
</li>
<li><a href="/wiki/Philosophy_of_war" title="Philosophy of war">War</a></li>
</ul>
</div>
</td>
</tr>
</table>
</td>
</tr>
</table>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks collapsible collapsed navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="col" class="navbox-title" colspan="2" style=";"><span style="float:left;width:6em">&#160;</span>
<div style="font-size:114%"><a href="/wiki/List_of_philosophies" title="List of philosophies">Schools of thought</a></div>
</th>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/History_of_philosophy" class="mw-redirect" title="History of philosophy">By era</a></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Ancient_philosophy" title="Ancient philosophy">Ancient</a></li>
<li><a href="/wiki/Western_philosophy" title="Western philosophy">Western</a>
<ul>
<li><a href="/wiki/Medieval_philosophy" title="Medieval philosophy">Medieval</a></li>
<li><a href="/wiki/Renaissance_philosophy" title="Renaissance philosophy">Renaissance</a></li>
<li><a href="/wiki/Early_modern_philosophy" title="Early modern philosophy">Early modern</a></li>
<li><a href="/wiki/Modern_philosophy" title="Modern philosophy">Modern</a></li>
<li><a href="/wiki/Contemporary_philosophy" title="Contemporary philosophy">Contemporary</a></li>
</ul>
</li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Ancient_philosophy" title="Ancient philosophy">Ancient</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Chinese_philosophy" title="Chinese philosophy">Chinese</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Agriculturalism" title="Agriculturalism">Agriculturalism</a></li>
<li><a href="/wiki/Confucianism" title="Confucianism">Confucianism</a></li>
<li><a href="/wiki/Legalism_(Chinese_philosophy)" title="Legalism (Chinese philosophy)">Legalism</a></li>
<li><a href="/wiki/School_of_Names" title="School of Names">Logicians</a></li>
<li><a href="/wiki/Mohism" title="Mohism">Mohism</a></li>
<li><a href="/wiki/School_of_Naturalists" title="School of Naturalists">Chinese naturalism</a></li>
<li><a href="/wiki/Xuanxue" title="Xuanxue">Neotaoism</a></li>
<li><a href="/wiki/Taoism" title="Taoism">Taoism</a></li>
<li><a href="/wiki/Yangism" title="Yangism">Yangism</a></li>
<li><a href="/wiki/Zen" title="Zen">Zen</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><span style="font-size:90%;"><a href="/wiki/Ancient_Greek_philosophy" title="Ancient Greek philosophy">Greco-</a><a href="/wiki/Hellenistic_philosophy" title="Hellenistic philosophy">Roman</a></span></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Aristotelianism" title="Aristotelianism">Aristotelianism</a></li>
<li><a href="/wiki/Atomism" title="Atomism">Atomism</a></li>
<li><a href="/wiki/Cynicism_(philosophy)" title="Cynicism (philosophy)">Cynicism</a></li>
<li><a href="/wiki/Cyrenaics" title="Cyrenaics">Cyrenaics</a></li>
<li><a href="/wiki/Eleatics" title="Eleatics">Eleatics</a></li>
<li><a href="/wiki/Eretrian_school" title="Eretrian school">Eretrian school</a></li>
<li><a href="/wiki/Epicureanism" title="Epicureanism">Epicureanism</a></li>
<li><a href="/wiki/Hermeneutics" title="Hermeneutics">Hermeneutics</a></li>
<li><a href="/wiki/Ionian_School_(philosophy)" title="Ionian School (philosophy)">Ionian</a>
<ul>
<li><a href="/wiki/Ephesian_school" title="Ephesian school">Ephesian</a></li>
<li><a href="/wiki/Milesian_school" title="Milesian school">Milesian</a></li>
</ul>
</li>
<li><a href="/wiki/Megarian_school" title="Megarian school">Megarian school</a></li>
<li><a href="/wiki/Neoplatonism" title="Neoplatonism">Neoplatonism</a></li>
<li><a href="/wiki/Peripatetic_school" title="Peripatetic school">Peripatetic</a></li>
<li><a href="/wiki/Platonism" title="Platonism">Platonism</a></li>
<li><a href="/wiki/Pluralist_school" title="Pluralist school">Pluralism</a></li>
<li><a href="/wiki/Pre-Socratic_philosophy" title="Pre-Socratic philosophy">Presocratic</a></li>
<li><a href="/wiki/Pyrrhonism" title="Pyrrhonism">Pyrrhonism</a></li>
<li><a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreanism</a></li>
<li><a href="/wiki/Neopythagoreanism" title="Neopythagoreanism">Neopythagoreanism</a></li>
<li><a href="/wiki/Sophism" title="Sophism">Sophism</a></li>
<li><a href="/wiki/Stoicism" title="Stoicism">Stoicism</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Indian_philosophy" title="Indian philosophy">Indian</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Buddhist_philosophy" title="Buddhist philosophy">Buddhist</a></li>
<li><a href="/wiki/C%C4%81rv%C4%81ka" class="mw-redirect" title="C?rv?ka">C?rv?ka</a></li>
<li><a href="/wiki/Hindu_philosophy" title="Hindu philosophy">Hindu</a></li>
<li><a href="/wiki/Jain_philosophy" title="Jain philosophy">Jain</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Iranian_philosophy" title="Iranian philosophy">Persian</a></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Mazdak#Mazdakism" title="Mazdak">Mazdakism</a></li>
<li><a href="/wiki/Zoroastrianism" title="Zoroastrianism">Zoroastrianism</a></li>
<li><a href="/wiki/Zurvanism" title="Zurvanism">Zurvanism</a></li>
</ul>
</div>
</td>
</tr>
</table>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Medieval_philosophy" title="Medieval philosophy">Medieval</a></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/European_philosophy" class="mw-redirect" title="European philosophy">European</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Christian_philosophy" title="Christian philosophy">Christian philosophy</a></li>
<li><a href="/wiki/Scholasticism" title="Scholasticism">Scholasticism</a></li>
<li><a href="/wiki/Thomism" title="Thomism">Thomism</a></li>
<li><a href="/wiki/Renaissance_humanism" title="Renaissance humanism">Renaissance humanism</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">East Asian</th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Korean_Confucianism" title="Korean Confucianism">Korean Confucianism</a></li>
<li><a href="/wiki/Edo_Neo-Confucianism" title="Edo Neo-Confucianism">Edo Neo-Confucianism</a></li>
<li><a href="/wiki/Neo-Confucianism" title="Neo-Confucianism">Neo-Confucianism</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">Indian</th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Dvaita" title="Dvaita">Dvaita</a></li>
<li><a href="/wiki/Navya-Ny%C4%81ya" title="Navya-Ny?ya">Navya-Ny?ya</a></li>
<li><a href="/wiki/Vishishtadvaita" title="Vishishtadvaita">Vishishtadvaita</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Islamic_philosophy" title="Islamic philosophy">Islamic</a></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Averroism" title="Averroism">Averroism</a></li>
<li><a href="/wiki/Avicenna#Avicennian_philosophy" title="Avicenna">Avicennism</a></li>
<li><a href="/wiki/Illuminationism#Persian_school_of_Illuminationism" title="Illuminationism">Persian Illuminationism</a></li>
<li><a href="/wiki/Ilm_al-Kalam" class="mw-redirect" title="Ilm al-Kalam">Ilm al-Kalam</a></li>
<li><a href="/wiki/Sufi_philosophy" title="Sufi philosophy">Sufi</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Jewish_philosophy" title="Jewish philosophy">Jewish</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Judeo-Islamic_philosophies_(800%E2%80%931400)" title="Judeo-Islamic philosophies (800–1400)">Judeo-Islamic</a></li>
</ul>
</div>
</td>
</tr>
</table>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Modern_philosophy" title="Modern philosophy">Modern</a></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">People</th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Cartesianism" title="Cartesianism">Cartesianism</a></li>
<li><a href="/wiki/Kantianism" title="Kantianism">Kantianism</a></li>
<li><a href="/wiki/Neo-Kantianism" title="Neo-Kantianism">Neo-Kantianism</a></li>
<li><a href="/wiki/Hegelianism" title="Hegelianism">Hegelianism</a></li>
<li><a href="/wiki/Marxist_philosophy" title="Marxist philosophy">Marxism</a></li>
<li><a href="/wiki/Spinozism" title="Spinozism">Spinozism</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><span style="font-size:90%;"><a href="/wiki/Idea" title="Idea">Ideal</a>&#160;/ <a href="/wiki/Matter_(philosophy)" title="Matter (philosophy)">Material</a></span></th>
<td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Determinism" title="Determinism">Determinism</a></li>
<li><a href="/wiki/Dualism" title="Dualism">Dualism</a></li>
<li><a href="/wiki/Empiricism" title="Empiricism">Empiricism</a></li>
<li><a href="/wiki/Idealism" title="Idealism">Idealism</a>
<ul>
<li><a href="/wiki/Absolute_idealism" title="Absolute idealism">Absolute</a></li>
<li><a href="/wiki/British_idealism" title="British idealism">British</a></li>
<li><a href="/wiki/German_idealism" title="German idealism">German</a></li>
<li><a href="/wiki/Objective_idealism" title="Objective idealism">Objective</a></li>
<li><a href="/wiki/Subjective_idealism" title="Subjective idealism">Subjective</a></li>
<li><a href="/wiki/Transcendental_idealism" title="Transcendental idealism">Transcendental</a></li>
</ul>
</li>
<li><a href="/wiki/Classical_realism" class="mw-redirect" title="Classical realism">Classical realism</a></li>
<li><a href="/wiki/Materialism" title="Materialism">Materialism</a></li>
<li><a href="/wiki/Monism" title="Monism">Monism</a></li>
<li><a href="/wiki/Naturalism_(philosophy)" title="Naturalism (philosophy)">Naturalism</a></li>
<li><a href="/wiki/Pragmatism" title="Pragmatism">Pragmatism</a></li>
<li><a href="/wiki/Reductionism" title="Reductionism">Reductionism</a></li>
<li><a href="/wiki/Rationalism" title="Rationalism">Rationalism</a></li>
<li><a href="/wiki/Utilitarianism" title="Utilitarianism">Utilitarianism</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">Other</th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Anarchism" title="Anarchism">Anarchism</a></li>
<li><a href="/wiki/Collectivism" title="Collectivism">Collectivism</a></li>
<li><a href="/wiki/New_Confucianism" title="New Confucianism">New Confucianism</a></li>
<li><a href="/wiki/Conservatism" title="Conservatism">Conservatism</a></li>
<li><a href="/wiki/Existentialism" title="Existentialism">Existentialism</a></li>
<li><a href="/wiki/Foundationalism" title="Foundationalism">Foundationalism</a></li>
<li><a href="/wiki/Historicism" title="Historicism">Historicism</a></li>
<li><a href="/wiki/Holism" title="Holism">Holism</a></li>
<li><a href="/wiki/Humanism" title="Humanism">Humanism</a></li>
<li><a href="/wiki/Individualism" title="Individualism">Individualism</a></li>
<li><a href="/wiki/Kokugaku" title="Kokugaku">Kokugaku</a></li>
<li><a href="/wiki/Classical_liberalism" title="Classical liberalism">Liberalism</a></li>
<li><a href="/wiki/Modernism" title="Modernism">Modernism</a></li>
<li><a href="/wiki/Natural_Law" class="mw-redirect" title="Natural Law">Natural Law</a></li>
<li><a href="/wiki/Nihilism" title="Nihilism">Nihilism</a></li>
<li><a href="/wiki/Phenomenology_(philosophy)" title="Phenomenology (philosophy)">Phenomenology</a></li>
<li><a href="/wiki/Positivism" title="Positivism">Positivism</a></li>
<li><a href="/wiki/Neo-Scholasticism" title="Neo-Scholasticism">Neo-Scholasticism</a></li>
<li><a href="/wiki/Social_contract" title="Social contract">Social contract</a></li>
<li><a href="/wiki/Socialism" title="Socialism">Socialism</a></li>
<li><a href="/wiki/Transcendentalism" title="Transcendentalism">Transcendentalism</a></li>
</ul>
</div>
</td>
</tr>
</table>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
<tr>
<th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Contemporary_philosophy" title="Contemporary philosophy">Contemporary</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em"></div>
<table class="nowraplinks navbox-subgroup" style="border-spacing:0">
<tr>
<th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Analytic_philosophy" title="Analytic philosophy">Analytic</a></th>
<td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px">
<div style="padding:0em 0.25em">
<ul>
<li><a href="/wiki/Applied_ethics" title="Applied ethics">Applied ethics</a></li>
<li><a href="/wiki/Analytical_feminism" title="Analytical feminism">Analytic feminism</a></li>
<li><a href="/wiki/Analytical_Marxism" title="Analytical Marxism">Analytical Marxism</a></li>
<li><a href="/wiki/Communitarianism" title="Communitarianism">Communitarianism</a></li>
<li><a href="/wiki/Consequentialism" title="Consequentialism">Consequentialism</a></li>
<li><a href="/wiki/Critical_rationalism" title="Critical rationalism">Critical rationalism</a></li>
<li><a href="/wiki/Experimental_philosophy" title="Experimental philosophy">Experimental philosophy</a></li>
<li><a href="/wiki/Falsifiability" title="Falsifiability">Falsificationism</a></li>
<li><a href="/wiki/Foundationalism" title="Foundationalism">Foundationalism</a>&#160;/ <a href="/wiki/Coherentism" title="Coherentism">Coherentism</a></li>
<li><a href="/wiki/Generative_linguistics" class="mw-redirect" title="Generative linguistics">Generative linguistics</a></li>
<li><a href="/wiki/Internalism_and_externalism" title="Internalism and externalism">Internalism and Externalism</a></li>
<li><a href="/wiki/Logical_positivism" title="Logical positivism">Logical positivism</a></li>
<li><a href="/wiki/Legal_positivism" title="Legal positivism">Legal positivism</a></li>
<li><a href="/wiki/Normative_ethics" title="Normative ethics">Normative ethics</a></li>
<li><a href="/wiki/Meta-ethics" title="Meta-ethics">Meta-ethics</a></li>
<li><a href="/wiki/Moral_realism" title="Moral realism">Moral realism</a></li>
<li><a href="/wiki/Virtue_ethics#Contemporary_.27aretaic_turn.27" title="Virtue ethics">Neo-Aristotelian</a></li>
<li><a href="/wiki/Naturalized_epistemology" title="Naturalized epistemology">Quinean naturalism</a></li>
<li><a href="/wiki/Ordinary_language_philosophy" title="Ordinary language philosophy">Ordinary language philosophy</a></li>
<li><a href="/wiki/Postanalytic_philosophy" title="Postanalytic philosophy">Postanalytic philosophy</a></li>
<li><a href="/wiki/Quietism_(philosophy)" title="Quietism (philosophy)">Quietism</a></li>
<li><a href="/wiki/John_Rawls" title="John Rawls">Rawlsian</a></li>
<li><a href="/wiki/Reformed_epistemology" title="Reformed epistemology">Reformed epistemology</a></li>
<li><a href="/wiki/Systemics" title="Systemics">Systemics</a></li>
<li><a href="/wiki/Scientism" title="Scientism">Scientism</a></li>
<li><a href="/wiki/Scientific_realism" title="Scientific realism">Scientific realism</a></li>
<li><a href="/wiki/Scientific_skepticism" title="Scientific skepticism">Scientific skepticism</a></li>
<li><a href="/wiki/Utilitarianism#Twentieth-century_developments" title="Utilitarianism">Contemporary utilitarianism</a></li>
<li><a href="/wiki/Vienna_Circle" title="Vienna Circle">Vienna Circle</a></li>
<li><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgensteinian</a></li>
</ul>
</div>
</td>
</tr>
<tr style="height:2px">
<td colspan="2"></td>
</tr>
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						<li class="interlanguage-link interwiki-af"><a href="//af.wikipedia.org/wiki/Logika" title="Logika – Afrikaans" lang="af" hreflang="af">Afrikaans</a></li><li class="interlanguage-link interwiki-als"><a href="//als.wikipedia.org/wiki/Logik" title="Logik – Alemannisch" lang="als" hreflang="als">Alemannisch</a></li><li class="interlanguage-link interwiki-am"><a href="//am.wikipedia.org/wiki/%E1%88%B5%E1%8A%90_%E1%8A%A0%E1%88%9D%E1%8A%AD%E1%8A%95%E1%8B%AE" title="ስ? አ?ክንዮ – Amharic" lang="am" hreflang="am">አማርኛ</a></li><li class="interlanguage-link interwiki-ar"><a href="//ar.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Arabic" lang="ar" hreflang="ar">العربية</a></li><li class="interlanguage-link interwiki-an"><a href="//an.wikipedia.org/wiki/Lochica" title="Lochica – Aragonese" lang="an" hreflang="an">Aragonés</a></li><li class="interlanguage-link interwiki-as"><a href="//as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%B0%E0%A7%8D%E0%A6%95%E0%A6%B6%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="তৰ?কশাস?ত?ৰ – Assamese" lang="as" hreflang="as">অসমীয়া</a></li><li class="interlanguage-link interwiki-az"><a href="//az.wikipedia.org/wiki/M%C9%99ntiq" title="Məntiq – Azerbaijani" lang="az" hreflang="az">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn"><a href="//bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%81%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="য?ক?তি – Bengali" lang="bn" hreflang="bn">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan"><a href="//zh-min-nan.wikipedia.org/wiki/L%C3%B4-chek" title="Lô-chek – Chinese (Min Nan)" lang="zh-min-nan" hreflang="zh-min-nan">Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba"><a href="//ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Bashkir" lang="ba" hreflang="ba">Башҡорт?а</a></li><li class="interlanguage-link interwiki-be"><a href="//be.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Логіка – Belarusian" lang="be" hreflang="be">Белару?ка?</a></li><li class="interlanguage-link interwiki-be-x-old"><a href="//be-x-old.wikipedia.org/wiki/%D0%9B%D1%91%D0%B3%D1%96%D0%BA%D0%B0" title="Лёгіка – белару?ка? 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German" lang="de" hreflang="de">Deutsch</a></li><li class="interlanguage-link interwiki-et"><a href="//et.wikipedia.org/wiki/Loogika" title="Loogika – Estonian" lang="et" hreflang="et">Eesti</a></li><li class="interlanguage-link interwiki-el"><a href="//el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE" title="Λογική – Greek" lang="el" hreflang="el">Ελληνικά</a></li><li class="interlanguage-link interwiki-es"><a href="//es.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Spanish" lang="es" hreflang="es">Español</a></li><li class="interlanguage-link interwiki-eo"><a href="//eo.wikipedia.org/wiki/Logiko" title="Logiko – Esperanto" lang="eo" hreflang="eo">Esperanto</a></li><li class="interlanguage-link interwiki-ext"><a href="//ext.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Extremaduran" lang="ext" hreflang="ext">Estremeñu</a></li><li class="interlanguage-link interwiki-eu"><a href="//eu.wikipedia.org/wiki/Logika" title="Logika – Basque" lang="eu" hreflang="eu">Euskara</a></li><li class="interlanguage-link interwiki-fa"><a href="//fa.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Persian" lang="fa" hreflang="fa">?ارسی</a></li><li class="interlanguage-link interwiki-hif"><a href="//hif.wikipedia.org/wiki/Logic" title="Logic – Fiji Hindi" lang="hif" hreflang="hif">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fr"><a href="//fr.wikipedia.org/wiki/Logique" title="Logique – French" lang="fr" hreflang="fr">Français</a></li><li class="interlanguage-link interwiki-fy"><a href="//fy.wikipedia.org/wiki/Logika" title="Logika – Western Frisian" lang="fy" hreflang="fy">Frysk</a></li><li class="interlanguage-link interwiki-ga"><a href="//ga.wikipedia.org/wiki/Loighic" title="Loighic – Irish" lang="ga" hreflang="ga">Gaeilge</a></li><li class="interlanguage-link interwiki-gl"><a href="//gl.wikipedia.org/wiki/L%C3%B3xica" title="Lóxica – Galician" lang="gl" hreflang="gl">Galego</a></li><li class="interlanguage-link interwiki-gan"><a href="//gan.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E5%AD%B8" title="?輯學 – Gan Chinese" lang="gan" hreflang="gan">贛語</a></li><li class="interlanguage-link interwiki-ki"><a href="//ki.wikipedia.org/wiki/Logic" title="Logic – Kikuyu" lang="ki" hreflang="ki">Gĩkũyũ</a></li><li class="interlanguage-link interwiki-ko"><a href="//ko.wikipedia.org/wiki/%EB%85%BC%EB%A6%AC%ED%95%99" title="논리학 – Korean" lang="ko" hreflang="ko">한국어</a></li><li class="interlanguage-link interwiki-hy"><a href="//hy.wikipedia.org/wiki/%D5%8F%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="?րամաբանություն – Armenian" lang="hy" hreflang="hy">Հայերեն</a></li><li class="interlanguage-link interwiki-hi"><a href="//hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="तर?कशास?त?र – Hindi" lang="hi" hreflang="hi">हिन?दी</a></li><li class="interlanguage-link interwiki-hr"><a href="//hr.wikipedia.org/wiki/Logika" title="Logika – Croatian" lang="hr" hreflang="hr">Hrvatski</a></li><li class="interlanguage-link interwiki-io"><a href="//io.wikipedia.org/wiki/Logiko" title="Logiko – Ido" lang="io" hreflang="io">Ido</a></li><li class="interlanguage-link interwiki-ig"><a href="//ig.wikipedia.org/wiki/%C3%93m%C3%A1r%C3%AD" title="Ómárí – Igbo" lang="ig" hreflang="ig">Igbo</a></li><li class="interlanguage-link interwiki-ilo"><a href="//ilo.wikipedia.org/wiki/Lohika" title="Lohika – Iloko" lang="ilo" hreflang="ilo">Ilokano</a></li><li class="interlanguage-link interwiki-id"><a href="//id.wikipedia.org/wiki/Logika" title="Logika – Indonesian" lang="id" hreflang="id">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia"><a href="//ia.wikipedia.org/wiki/Logica" title="Logica – Interlingua" lang="ia" hreflang="ia">Interlingua</a></li><li class="interlanguage-link interwiki-ie"><a href="//ie.wikipedia.org/wiki/Logica" title="Logica – Interlingue" lang="ie" hreflang="ie">Interlingue</a></li><li class="interlanguage-link interwiki-xh"><a href="//xh.wikipedia.org/wiki/I-logic" title="I-logic – Xhosa" lang="xh" hreflang="xh">IsiXhosa</a></li><li class="interlanguage-link interwiki-is"><a href="//is.wikipedia.org/wiki/R%C3%B6kfr%C3%A6%C3%B0i" title="Rökfræði – Icelandic" lang="is" hreflang="is">?slenska</a></li><li class="interlanguage-link interwiki-it"><a href="//it.wikipedia.org/wiki/Logica" title="Logica – Italian" lang="it" hreflang="it">Italiano</a></li><li class="interlanguage-link interwiki-he"><a href="//he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94" title="לוגיקה – Hebrew" lang="he" hreflang="he">עברית</a></li><li class="interlanguage-link interwiki-jv"><a href="//jv.wikipedia.org/wiki/Logika" title="Logika – Javanese" lang="jv" hreflang="jv">Basa Jawa</a></li><li class="interlanguage-link interwiki-ka"><a href="//ka.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%99%E1%83%90" title="ლ?გიკ? – Georgian" lang="ka" hreflang="ka">ქ?რთული</a></li><li class="interlanguage-link interwiki-kk"><a href="//kk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Kazakh" lang="kk" hreflang="kk">Қазақша</a></li><li class="interlanguage-link interwiki-sw"><a href="//sw.wikipedia.org/wiki/Mantiki" title="Mantiki – Swahili" lang="sw" hreflang="sw">Kiswahili</a></li><li class="interlanguage-link interwiki-ku"><a href="//ku.wikipedia.org/wiki/Mentiq" title="Mentiq – Kurdish" lang="ku" hreflang="ku">Kurdî</a></li><li class="interlanguage-link interwiki-ky"><a href="//ky.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Kyrgyz" lang="ky" hreflang="ky">Кыргызча</a></li><li class="interlanguage-link interwiki-lbe"><a href="//lbe.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D0%B8%D0%BA%D1%8C" title="Мантикь – лакку" lang="lbe" hreflang="lbe">Лакку</a></li><li class="interlanguage-link interwiki-la"><a href="//la.wikipedia.org/wiki/Logica" title="Logica – Latin" lang="la" hreflang="la">Latina</a></li><li class="interlanguage-link interwiki-lv"><a href="//lv.wikipedia.org/wiki/Lo%C4%A3ika" title="Loģika – Latvian" lang="lv" hreflang="lv">Latviešu</a></li><li class="interlanguage-link interwiki-lb"><a href="//lb.wikipedia.org/wiki/Logik" title="Logik – Luxembourgish" lang="lb" hreflang="lb">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt"><a href="//lt.wikipedia.org/wiki/Logika" title="Logika – Lithuanian" lang="lt" hreflang="lt">Lietuvių</a></li><li class="interlanguage-link interwiki-jbo"><a href="//jbo.wikipedia.org/wiki/logji" title="logji – Lojban" lang="jbo" hreflang="jbo">La .lojban.</a></li><li class="interlanguage-link interwiki-hu"><a href="//hu.wikipedia.org/wiki/Logika" title="Logika – Hungarian" lang="hu" hreflang="hu">Magyar</a></li><li class="interlanguage-link interwiki-mk"><a href="//mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Macedonian" lang="mk" hreflang="mk">Македон?ки</a></li><li class="interlanguage-link interwiki-ml"><a href="//ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%BC%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="തർക?കശാസ?ത?രം – Malayalam" lang="ml" hreflang="ml">മലയാളം</a></li><li class="interlanguage-link interwiki-arz"><a href="//arz.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Egyptian Arabic" lang="arz" hreflang="arz">مصرى</a></li><li class="interlanguage-link interwiki-ms"><a href="//ms.wikipedia.org/wiki/Logik" title="Logik – Malay" lang="ms" hreflang="ms">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mwl"><a href="//mwl.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Mirandese" lang="mwl" hreflang="mwl">Mirandés</a></li><li class="interlanguage-link interwiki-mn"><a href="//mn.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA" title="Логик – Mongolian" lang="mn" hreflang="mn">Монгол</a></li><li class="interlanguage-link interwiki-my"><a href="//my.wikipedia.org/wiki/%E1%80%9A%E1%80%AF%E1%80%90%E1%80%B9%E1%80%90%E1%80%AD%E1%80%97%E1%80%B1%E1%80%92" title="ယု?္?ိဗေဒ – Burmese" lang="my" hreflang="my">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl"><a href="//nl.wikipedia.org/wiki/Logica" title="Logica – Dutch" lang="nl" hreflang="nl">Nederlands</a></li><li class="interlanguage-link interwiki-new"><a href="//new.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95" title="तर?क – Newari" lang="new" hreflang="new">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja"><a href="//ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E5%AD%A6" title="論?学 – Japanese" lang="ja" hreflang="ja">日本語</a></li><li class="interlanguage-link interwiki-ce"><a href="//ce.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Chechen" lang="ce" hreflang="ce">?охчийн</a></li><li class="interlanguage-link interwiki-no"><a href="//no.wikipedia.org/wiki/Logikk" title="Logikk – Norwegian" lang="no" hreflang="no">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn"><a href="//nn.wikipedia.org/wiki/Logikk" title="Logikk – Norwegian Nynorsk" lang="nn" hreflang="nn">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov"><a href="//nov.wikipedia.org/wiki/Logike" title="Logike – Novial" lang="nov" hreflang="nov">Novial</a></li><li class="interlanguage-link interwiki-oc"><a href="//oc.wikipedia.org/wiki/Logica" title="Logica – Occitan" lang="oc" hreflang="oc">Occitan</a></li><li class="interlanguage-link interwiki-uz"><a href="//uz.wikipedia.org/wiki/Mantiq" title="Mantiq – Uzbek" lang="uz" hreflang="uz">Oʻzbekcha/ўзбекча</a></li><li class="interlanguage-link interwiki-pa"><a href="//pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%B0%E0%A8%95_%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B0" title="ਤਰਕ ਸ਼ਾਸਤਰ – Punjabi" lang="pa" hreflang="pa">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb"><a href="//pnb.wikipedia.org/wiki/%D8%AA%D9%8F%DA%A9" title="ت?ک – Western Punjabi" lang="pnb" hreflang="pnb">پنجابی</a></li><li class="interlanguage-link interwiki-ps"><a href="//ps.wikipedia.org/wiki/%D8%B3%D9%88%D9%84%D9%BE%D9%88%D9%87%D9%86%D9%87" title="سولپوهنه – Pashto" lang="ps" hreflang="ps">پښتو</a></li><li class="interlanguage-link interwiki-jam"><a href="//jam.wikipedia.org/wiki/Lajik" title="Lajik – Jamaican Creole English" lang="jam" hreflang="jam">Patois</a></li><li class="interlanguage-link interwiki-pms"><a href="//pms.wikipedia.org/wiki/L%C3%B2gica" title="Lògica – Piedmontese" lang="pms" hreflang="pms">Piemontèis</a></li><li class="interlanguage-link interwiki-tpi"><a href="//tpi.wikipedia.org/wiki/Lajik" title="Lajik – Tok Pisin" lang="tpi" hreflang="tpi">Tok Pisin</a></li><li class="interlanguage-link interwiki-nds"><a href="//nds.wikipedia.org/wiki/Logik" title="Logik – Low German" lang="nds" hreflang="nds">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl"><a href="//pl.wikipedia.org/wiki/Logika" title="Logika – Polish" lang="pl" hreflang="pl">Polski</a></li><li class="interlanguage-link interwiki-pt"><a href="//pt.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Portuguese" lang="pt" hreflang="pt">Português</a></li><li class="interlanguage-link interwiki-ro"><a href="//ro.wikipedia.org/wiki/Logic%C4%83" title="Logică – Romanian" lang="ro" hreflang="ro">Română</a></li><li class="interlanguage-link interwiki-rue"><a href="//rue.wikipedia.org/wiki/%D0%9B%D0%BE%D2%91%D1%96%D0%BA%D0%B0" title="Лоґіка – Rusyn" lang="rue" hreflang="rue">Ру?инь?кый</a></li><li class="interlanguage-link interwiki-ru"><a href="//ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Russian" lang="ru" hreflang="ru">Ру??кий</a></li><li class="interlanguage-link interwiki-sah"><a href="//sah.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Sakha" lang="sah" hreflang="sah">Саха тыла</a></li><li class="interlanguage-link interwiki-sc"><a href="//sc.wikipedia.org/wiki/L%C3%B2gica" title="Lògica – Sardinian" lang="sc" hreflang="sc">Sardu</a></li><li class="interlanguage-link interwiki-sco"><a href="//sco.wikipedia.org/wiki/Logic" title="Logic – Scots" lang="sco" hreflang="sco">Scots</a></li><li class="interlanguage-link interwiki-sq"><a href="//sq.wikipedia.org/wiki/Logjika" title="Logjika – Albanian" lang="sq" hreflang="sq">Shqip</a></li><li class="interlanguage-link interwiki-scn"><a href="//scn.wikipedia.org/wiki/L%C3%B2ggica" title="Lòggica – Sicilian" lang="scn" hreflang="scn">Sicilianu</a></li><li class="interlanguage-link interwiki-simple"><a href="//simple.wikipedia.org/wiki/Logic" title="Logic – Simple English" lang="simple" hreflang="simple">Simple English</a></li><li class="interlanguage-link interwiki-sd"><a href="//sd.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Sindhi" lang="sd" hreflang="sd">سنڌي</a></li><li class="interlanguage-link interwiki-sk"><a href="//sk.wikipedia.org/wiki/Logika" title="Logika – Slovak" lang="sk" hreflang="sk">Sloven?ina</a></li><li class="interlanguage-link interwiki-sl"><a href="//sl.wikipedia.org/wiki/Logika" title="Logika – Slovenian" lang="sl" hreflang="sl">Slovenš?ina</a></li><li class="interlanguage-link interwiki-ckb"><a href="//ckb.wikipedia.org/wiki/%D9%84%DB%86%DA%98%DB%8C%DA%A9" title="لۆژیک – Central Kurdish" lang="ckb" hreflang="ckb">کوردیی ناوەندی</a></li><li class="interlanguage-link interwiki-sr"><a href="//sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Serbian" lang="sr" hreflang="sr">Срп?ки / srpski</a></li><li class="interlanguage-link interwiki-sh"><a href="//sh.wikipedia.org/wiki/Logika" title="Logika – Serbo-Croatian" lang="sh" hreflang="sh">Srpskohrvatski / ?рп?кохрват?ки</a></li><li class="interlanguage-link interwiki-su"><a href="//su.wikipedia.org/wiki/Logika" title="Logika – Sundanese" lang="su" hreflang="su">Basa Sunda</a></li><li class="interlanguage-link interwiki-fi"><a href="//fi.wikipedia.org/wiki/Logiikka" title="Logiikka – Finnish" lang="fi" hreflang="fi">Suomi</a></li><li class="interlanguage-link interwiki-sv"><a href="//sv.wikipedia.org/wiki/Logik" title="Logik – Swedish" lang="sv" hreflang="sv">Svenska</a></li><li class="interlanguage-link interwiki-tl"><a href="//tl.wikipedia.org/wiki/Lohika" title="Lohika – Tagalog" lang="tl" hreflang="tl">Tagalog</a></li><li class="interlanguage-link interwiki-ta"><a href="//ta.wikipedia.org/wiki/%E0%AE%8F%E0%AE%B0%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="?ரணம? – Tamil" lang="ta" hreflang="ta">தமிழ?</a></li><li class="interlanguage-link interwiki-tt"><a href="//tt.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D1%8B%D0%B9%D0%BA" title="Мантыйк – Tatar" lang="tt" hreflang="tt">Татарча/tatarça</a></li><li class="interlanguage-link interwiki-th"><a href="//th.wikipedia.org/wiki/%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="ตรร?ศาสตร์ – Thai" lang="th" hreflang="th">ไทย</a></li><li class="interlanguage-link interwiki-tg"><a href="//tg.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D0%B8%D2%9B" title="Мантиқ – Tajik" lang="tg" hreflang="tg">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr"><a href="//tr.wikipedia.org/wiki/Mant%C4%B1k" title="Mantık – Turkish" lang="tr" hreflang="tr">Türkçe</a></li><li class="interlanguage-link interwiki-tk"><a href="//tk.wikipedia.org/wiki/Logika" title="Logika – Turkmen" lang="tk" hreflang="tk">Türkmençe</a></li><li class="interlanguage-link interwiki-tyv"><a href="//tyv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BB%D0%B8%D0%B3%D0%BB%D0%B5%D0%BB" title="Билиглел – Tuvinian" lang="tyv" hreflang="tyv">Тыва дыл</a></li><li class="interlanguage-link interwiki-uk"><a href="//uk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Логіка – Ukrainian" lang="uk" hreflang="uk">Україн?ька</a></li><li class="interlanguage-link interwiki-ur"><a href="//ur.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Urdu" lang="ur" hreflang="ur">اردو</a></li><li class="interlanguage-link interwiki-vi"><a href="//vi.wikipedia.org/wiki/Logic" title="Logic – Vietnamese" lang="vi" hreflang="vi">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro"><a href="//fiu-vro.wikipedia.org/wiki/Loogiga" title="Loogiga – Võro" lang="fiu-vro" hreflang="fiu-vro">Võro</a></li><li class="interlanguage-link interwiki-wa"><a href="//wa.wikipedia.org/wiki/Lodjike" title="Lodjike – Walloon" lang="wa" hreflang="wa">Walon</a></li><li class="interlanguage-link interwiki-zh-classical"><a href="//zh-classical.wikipedia.org/wiki/%E7%90%86%E5%89%87" title="?則 – Classical Chinese" lang="zh-classical" hreflang="zh-classical">文言</a></li><li class="interlanguage-link interwiki-war"><a href="//war.wikipedia.org/wiki/Lohika" title="Lohika – Waray" lang="war" hreflang="war">Winaray</a></li><li class="interlanguage-link interwiki-yi"><a href="//yi.wikipedia.org/wiki/%D7%9C%D7%90%D7%92%D7%99%D7%A7" title="ל?גיק – Yiddish" lang="yi" hreflang="yi">ייִדיש</a></li><li class="interlanguage-link interwiki-yo"><a href="//yo.wikipedia.org/wiki/%E1%BB%8Cgb%E1%BB%8D%CC%81n" title="Ọgb??n – Yoruba" lang="yo" hreflang="yo">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue"><a href="//zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E5%AD%B8" title="?輯學 – Cantonese" lang="zh-yue" hreflang="zh-yue">粵語</a></li><li class="interlanguage-link interwiki-bat-smg"><a href="//bat-smg.wikipedia.org/wiki/Luog%C4%97ka" title="Luogėka – Samogitian" lang="bat-smg" hreflang="bat-smg">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh"><a href="//zh.wikipedia.org/wiki/%E9%80%BB%E8%BE%91" title="逻辑 – Chinese" lang="zh" hreflang="zh">中文</a></li><li class="uls-p-lang-dummy"><a href="#"></a></li>					</ul>
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