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class="vector-toc-numb">2</span> <span>Syntax and fragments</span> </div> </a> <ul id="toc-Syntax_and_fragments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semantics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Semantics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Semantics</span> </div> </a> <ul id="toc-Semantics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressive_power" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressive_power"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Expressive power</span> </div> </a> <ul id="toc-Expressive_power-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deductive_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Deductive_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Deductive systems</span> </div> </a> <ul id="toc-Deductive_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-reducibility_to_first-order_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-reducibility_to_first-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Non-reducibility to first-order logic</span> </div> </a> <ul id="toc-Non-reducibility_to_first-order_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metalogical_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Metalogical_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Metalogical results</span> </div> </a> <ul id="toc-Metalogical_results-sublist" class="vector-toc-list"> </ul> </li> <li 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class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> 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href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B5%CF%85%CF%84%CE%AD%CF%81%CE%BF%CF%85_%CE%B2%CE%B1%CE%B8%CE%BC%CE%BF%CF%8D" title="Λογική δευτέρου βαθμού – Greek" lang="el" hreflang="el" data-title="Λογική δευτέρου βαθμού" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%B3gica_de_segundo_orden" title="Lógica de segundo orden – Spanish" lang="es" hreflang="es" data-title="Lógica de segundo orden" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logique_d%27ordre_sup%C3%A9rieur#Logique_du_second_ordre" title="Logique d'ordre supérieur – French" lang="fr" hreflang="fr" data-title="Logique d'ordre 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Form of logic that allows quantification over predicates</div> <p>In <a href="/wiki/Logic" title="Logic">logic</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>second-order logic</b> is an extension of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, which itself is an extension of <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Second-order logic is in turn extended by <a href="/wiki/Higher-order_logic" title="Higher-order logic">higher-order logic</a> and <a href="/wiki/Type_theory" title="Type theory">type theory</a>. </p><p>First-order logic <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifies</a> only variables that range over individuals (elements of the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a>); second-order logic, in addition, quantifies over <a href="/wiki/Finitary_relation" title="Finitary relation">relations</a>. For example, the second-order <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>P</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mi>x</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f89919d74e98ae54d99a30b88594a38b0cde49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.14ex; height:2.843ex;" alt="{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}"></span> says that for every <a href="/wiki/Well-formed_formula" title="Well-formed formula">formula</a> <i>P</i>, and every individual <i>x</i>, either <i>Px</i> is true or not(<i>Px</i>) is true (this is the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>). Second-order logic also includes quantification over <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, and other variables (see section <a href="#Syntax_and_fragments">below</a>). Both first-order and second-order logic use the idea of a <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kindl-Treppe_Aug2021d.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Kindl-Treppe_Aug2021d.jpg/220px-Kindl-Treppe_Aug2021d.jpg" decoding="async" width="220" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Kindl-Treppe_Aug2021d.jpg/330px-Kindl-Treppe_Aug2021d.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Kindl-Treppe_Aug2021d.jpg/440px-Kindl-Treppe_Aug2021d.jpg 2x" data-file-width="4000" data-file-height="2420" /></a><figcaption><a href="/wiki/Graffiti" title="Graffiti">Graffiti</a> in <a href="/wiki/Neuk%C3%B6lln" title="Neukölln">Neukölln</a> (Berlin) showing the simplest second-order sentence admitting nontrivial models, “<span class="texhtml">∃<i>φ</i> <i>φ</i></span>”.</figcaption></figure> <p>First-order logic can quantify over individuals, but not over properties. That is, we can take an <a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic sentence</a> like Cube(<i>b</i>) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier:<sup id="cite_ref-mcohen_2-0" class="reference"><a href="#cite_note-mcohen-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="texhtml">∃<i>x</i> Cube(<i>x</i>)</span></dd></dl> <p>However, we cannot do the same with the predicate. That is, the following expression: </p> <dl><dd><span class="texhtml">∃P P(<i>b</i>)</span></dd></dl> <p>is not a sentence of first-order logic, but this is a legitimate sentence of second-order logic. Here, <i>P</i> is a <a href="/wiki/Predicate_variable" title="Predicate variable">predicate variable</a> and is semantically a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of individuals.<sup id="cite_ref-mcohen_2-1" class="reference"><a href="#cite_note-mcohen-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>As a result, second-order logic has greater expressive power than first-order logic. For example, there is no way in first-order logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in second-order logic as: </p> <dl><dd><span class="texhtml">∃P ∀<i>x</i> (Px ↔ (Cube(<i>x</i>) ∨ Tet(<i>x</i>))).</span></dd></dl> <p>We can then assert properties of this set. For instance, the following says that the set of all cubes and tetrahedrons does not contain any dodecahedrons: </p> <dl><dd><span class="texhtml">∀P (∀<i>x</i> (Px ↔ (Cube(<i>x</i>) ∨ Tet(<i>x</i>))) → ¬ ∃<i>x</i> (Px ∧ Dodec(<i>x</i>))).</span></dd></dl> <p>Second-order quantification is especially useful because it gives the ability to express <a href="/wiki/Reachability" title="Reachability">reachability</a> properties. For example, if Parent(<i>x</i>, <i>y</i>) denotes that <i>x</i> is a parent of <i>y</i>, then first-order logic cannot express the property that <i>x</i> is an <a href="/wiki/Ancestor" title="Ancestor">ancestor</a> of <i>y</i>. In second-order logic we can express this by saying that every set of people containing <i>y</i> and closed under the Parent relation contains <i>x</i>: </p> <dl><dd><span class="texhtml">∀P ((P<i>y</i> ∧ ∀<i>a</i> ∀<i>b</i> ((P<i>b</i> ∧ Parent(<i>a</i>, <i>b</i>)) → P<i>a</i>)) → P<i>x</i>).</span></dd></dl> <p>It is notable that while we have variables for predicates in second-order-logic, we don't have variables for properties of predicates. We cannot say, for example, that there is a property Shape(<i>P</i>) that is true for the predicates <i>P</i> Cube, Tet, and Dodec. This would require <a href="/wiki/Higher-order_logic" title="Higher-order logic">third-order logic</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Syntax_and_fragments">Syntax and fragments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=2" title="Edit section: Syntax and fragments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="fragments"></span> The syntax of second-order logic tells which expressions are well formed <a href="/wiki/Formula_(mathematical_logic)" class="mw-redirect" title="Formula (mathematical logic)">formulas</a>. In addition to the <a href="/wiki/First-order_logic#Formation_rules" title="First-order logic">syntax of first-order logic</a>, second-order logic includes many new <i>sorts</i> (sometimes called <i>types</i>) of variables. These are: </p> <ul><li>A sort of variables that range over sets of individuals. If <i>S</i> is a variable of this sort and <i>t</i> is a first-order term then the expression <i>t</i> ∈ <i>S</i> (also written <i>S</i>(<i>t</i>), or <i>St</i> to save parentheses) is an <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formula</a>. Sets of individuals can also be viewed as <a href="/wiki/Finitary_relation" title="Finitary relation">unary relations</a> on the domain.</li> <li>For each natural number <i>k</i> there is a sort of variables that ranges over all <i>k</i>-ary relations on the individuals. If <i>R</i> is such a <i>k</i>-ary relation variable and <i>t</i><sub>1</sub>,...,<i>t</i><sub><i>k</i></sub> are first-order terms then the expression <i>R</i>(<i>t</i><sub>1</sub>,...,<i>t</i><sub><i>k</i></sub>) is an atomic formula.</li> <li>For each natural number <i>k</i> there is a sort of variables that ranges over all functions taking <i>k</i> elements of the domain and returning a single element of the domain. If <i>f</i> is such a <i>k</i>-ary function variable and <i>t</i><sub>1</sub>,...,<i>t</i><sub><i>k</i></sub> are first-order terms then the expression <i>f</i>(<i>t</i><sub>1</sub>,...,<i>t</i><sub><i>k</i></sub>) is a first-order term.</li></ul> <p>Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A <i>sentence</i> in second-order logic, as in first-order logic, is a well-formed formula with no free variables (of any sort). </p><p>It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an <i>n</i>-ary function variable can be represented by a relation variable of arity <i>n</i>+1 and an appropriate formula for the uniqueness of the "result" in the <i>n</i>+1 argument of the relation. (Shapiro 2000, p. 63) </p><p><b><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic second-order logic</a></b> (MSO) is a restriction of second-order logic in which only quantification over unary relations (i.e. sets) is allowed. Quantification over functions, owing to the equivalence to relations as described above, is thus also not allowed. The second-order logic without these restrictions is sometimes called <i>full second-order logic</i> to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of <a href="/wiki/Courcelle%27s_theorem" title="Courcelle's theorem">Courcelle's theorem</a>, an algorithmic meta-theorem in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>. The MSO theory of the complete infinite binary tree (<a href="/wiki/S2S_(mathematics)" title="S2S (mathematics)">S2S</a>) is decidable. By contrast, full second order logic over any infinite set (or MSO logic over for example (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>,+)) can interpret the true <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>. </p><p>Just as in first-order logic, second-order logic may include <a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical symbols</a> in a particular second-order language. These are restricted, however, in that all terms that they form must be either first-order terms (which can be substituted for a first-order variable) or second-order terms (which can be substituted for a second-order variable of an appropriate sort). </p><p>A formula in second-order logic is said to be of first-order (and sometimes denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b645fb2363708f79c8d1147284d039d3b6bf5dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{0}^{1}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae152f64cb76d9503f9a85e10d72902114b12c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{0}^{1}}"></span>) if its quantifiers (which may be universal or existential) range only over variables of first order, although it may have free variables of second order. A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253bf5a8d60183537e1b309e490ed3cd7583f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}}"></span> (existential second-order) formula is one additionally having some existential quantifiers over second order variables, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>…</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be5dbccd7a0697ad22e2cac7ee2bc7a53caf970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.725ex; height:2.509ex;" alt="{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is a first-order formula. The fragment of second-order logic consisting only of existential second-order formulas is called <b>existential second-order logic</b> and abbreviated as ESO, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253bf5a8d60183537e1b309e490ed3cd7583f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}}"></span>, or even as ∃SO. The fragment of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd076fb78a7a5ee340920a3d846ddf67455f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}"></span> formulas is defined dually, it is called universal second-order logic. More expressive fragments are defined for any <i>k</i> > 0 by mutual recursion: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{k+1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{k+1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f66287c2aa4ba6021f5f488dabcf463dd08646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.867ex; height:3.343ex;" alt="{\displaystyle \Sigma _{k+1}^{1}}"></span> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>…</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be5dbccd7a0697ad22e2cac7ee2bc7a53caf970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.725ex; height:2.509ex;" alt="{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{k}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{k}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a2d83c1310b84947e624450841a37d8c917e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.832ex; height:3.176ex;" alt="{\displaystyle \Pi _{k}^{1}}"></span> formula, and similar, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{k+1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{k+1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb267aff8a0bf704c8f6151e1e6da6a3395264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.932ex; height:3.343ex;" alt="{\displaystyle \Pi _{k+1}^{1}}"></span> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall R_{0}\ldots \forall R_{m}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>…</mo> <mi mathvariant="normal">∀</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall R_{0}\ldots \forall R_{m}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8312d6cb497c0a198bb9bb84aa5badb803051941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.725ex; height:2.509ex;" alt="{\displaystyle \forall R_{0}\ldots \forall R_{m}\phi }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{k}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{k}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c99b11238de6803a86afd1a986fdcdb26edfa6f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.767ex; height:3.176ex;" alt="{\displaystyle \Sigma _{k}^{1}}"></span> formula. (See <a href="/wiki/Analytical_hierarchy" title="Analytical hierarchy">analytical hierarchy</a> for the analogous construction of <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Semantics">Semantics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=3" title="Edit section: Semantics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The semantics of second-order logic establish the meaning of each sentence. Unlike first-order logic, which has only one standard semantics, there are two different semantics that are commonly used for second-order logic: <b>standard semantics</b> and <b>Henkin semantics</b>. In each of these semantics, the interpretations of the first-order quantifiers and the logical connectives are the same as in first-order logic. Only the ranges of quantifiers over second-order variables differ in the two types of semantics <a href="#Vaananen2001">(Väänänen 2001)</a>. </p><p>In standard semantics, also called full semantics, the quantifiers range over <i>all</i> sets or functions of the appropriate sort. A model with this condition is called a full model, and these are the same as models in which the range of the second-order quantifiers is the powerset of the model's first-order part. (<a href="#Vaananen2001">2001</a>) Thus once the domain of the first-order variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give second-order logic its expressive power, and they will be assumed for the remainder of this article. </p><p><a href="/wiki/Leon_Henkin" title="Leon Henkin">Leon Henkin</a> (1950) defined an alternative kind of semantics for second-order and higher-order theories, in which the meaning of the higher-order domains is partly determined by an explicit axiomatisation, drawing on <a href="/wiki/Type_theory" title="Type theory">type theory</a>, of the properties of the sets or functions ranged over. Henkin semantics is a kind of many-sorted first-order semantics, where there are a class of models of the axioms, instead of the semantics being fixed to just the standard model as in the standard semantics. A model in Henkin semantics will provide a set of sets or set of functions as the interpretation of higher-order domains, which may be a proper subset of all sets or functions of that sort. For his axiomatisation, Henkin proved that <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a> and <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a>, which hold for first-order logic, carry over to second-order logic with Henkin semantics. Since also the <a href="/wiki/Skolem%E2%80%93L%C3%B6wenheim_theorem" class="mw-redirect" title="Skolem–Löwenheim theorem">Skolem–Löwenheim theorems</a> hold for Henkin semantics, <a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's theorem</a> imports that Henkin models are just <i>disguised first-order models</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>For theories such as second-order arithmetic, the existence of non-standard interpretations of higher-order domains isn't just a deficiency of the particular axiomatisation derived from type theory that Henkin used, but a necessary consequence of <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a>: Henkin's axioms can't be supplemented further to ensure the standard interpretation is the only possible model. Henkin semantics are commonly used in the study of <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>. </p><p><a href="/wiki/Jouko_V%C3%A4%C3%A4n%C3%A4nen" title="Jouko Väänänen">Jouko Väänänen</a> (<a href="#Vaananen2001">2001</a>) argued that the distinction between Henkin semantics and full semantics for second-order logic is analogous to the distinction between provability in <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a> and truth in <i><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">V</a></i>, in that the former obeys model-theoretic properties like the Lowenheim-Skolem theorem and compactness, and the latter has categoricity phenomena. For example, "we cannot meaningfully ask whether the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> as defined in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ZFC} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ZFC} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d84f49f277751cb9713a10ad040a86a10bfa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle \mathrm {ZFC} }"></span> is the real <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> . But if we reformalize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ZFC} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ZFC} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d84f49f277751cb9713a10ad040a86a10bfa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle \mathrm {ZFC} }"></span> inside <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ZFC} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ZFC} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d84f49f277751cb9713a10ad040a86a10bfa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle \mathrm {ZFC} }"></span>, then we can note that the reformalized <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ZFC} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ZFC} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d84f49f277751cb9713a10ad040a86a10bfa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle \mathrm {ZFC} }"></span> ... has countable models and hence cannot be categorical." </p> <div class="mw-heading mw-heading2"><h2 id="Expressive_power">Expressive power</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=4" title="Edit section: Expressive power"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a>, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀<i>x</i> ∃<i>y</i> (<i>x</i> + <i>y</i> = 0) but one needs second-order logic to assert the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">least-upper-bound</a> property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>. If the domain is the set of all real numbers, the following second-order sentence (split over two lines) expresses the least upper bound property: </p> <dl><dd><span class="texhtml">(∀ A) ([<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:#800000">(∃ <i>w</i>) (<i>w</i> ∈ A)</span> ∧ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#008000">(∃ <i>z</i>)(∀ <i>u</i>)(<i>u</i> ∈ A → <i>u</i> ≤ <i>z</i>)</span>]</span> <dl><dd><span class="texhtml">→ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#800080">(∃ <i>x</i>)(∀ <i>y</i>)([(∀ <i>w</i>)(<i>w</i> ∈ A → <i>w</i> ≤ <i>x</i>)] ∧ [(∀ <i>u</i>)(<i>u</i> ∈ A → <i>u</i> ≤ <i>y</i>)] → <i>x</i> ≤ <i>y</i>)</span>)</span></dd></dl></dd></dl> <p>This formula is a direct formalization of "every <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#800000">nonempty</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#008000">bounded</span> set A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#800080">has a least upper bound</span>." It can be shown that any <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> that satisfies this property is isomorphic to the real number field. On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every <a href="/wiki/Real-closed_field" class="mw-redirect" title="Real-closed field">real-closed field</a> satisfies the same first-order sentences in the signature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle +,\cdot ,\leq \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>≤</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle +,\cdot ,\leq \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa318d8311504ee83f21e5e87445b111854ceb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.14ex; height:2.843ex;" alt="{\displaystyle \langle +,\cdot ,\leq \rangle }"></span> as the real numbers.) </p><p>In second-order logic, it is possible to write formal sentences that say "the domain is <a href="/wiki/Finite_set" title="Finite set">finite</a>" or "the domain is of <a href="/wiki/Countable_set" title="Countable set">countable</a> <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>." To say that the domain is finite, use the sentence that says that every <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> function from the domain to itself is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>. To say that the domain has countable cardinality, use the sentence that says that there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between every two infinite subsets of the domain. It follows from the <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a> and the <a href="/wiki/Upward_L%C3%B6wenheim%E2%80%93Skolem_theorem" class="mw-redirect" title="Upward Löwenheim–Skolem theorem">upward Löwenheim–Skolem theorem</a> that it is not possible to characterize finiteness or countability, respectively, in first-order logic. </p><p>Certain fragments of second-order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic that allow non-linear ordering of quantifier dependencies, like first-order logic extended with <a href="/wiki/Henkin_quantifier" class="mw-redirect" title="Henkin quantifier">Henkin quantifiers</a>, <a href="/wiki/Hintikka" class="mw-redirect" title="Hintikka">Hintikka</a> and Sandu's <a href="/wiki/Independence-friendly_logic" title="Independence-friendly logic">independence-friendly logic</a>, and Väänänen's <a href="/wiki/Dependence_logic" title="Dependence logic">dependence logic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Deductive_systems">Deductive systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=5" title="Edit section: Deductive systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a> for a logic is a set of <a href="/wiki/Inference_rules" class="mw-redirect" title="Inference rules">inference rules</a> and logical axioms that determine which sequences of formulas constitute valid proofs. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is <a href="/wiki/Soundness" title="Soundness">sound</a>, which means any sentence they can be used to prove is logically valid in the appropriate semantics. </p><p>The weakest deductive system that can be used consists of a standard deductive system for first-order logic (such as <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a>) augmented with substitution rules for second-order terms.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This deductive system is commonly used in the study of <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>. </p><p>The deductive systems considered by Shapiro (1991) and Henkin (1950) add to the augmented first-order deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard second-order semantics. They are sound for Henkin semantics restricted to Henkin models satisfying the comprehension and choice axioms.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-reducibility_to_first-order_logic">Non-reducibility to first-order logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=6" title="Edit section: Non-reducibility to first-order logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One might attempt to reduce the second-order theory of the real numbers, with full second-order semantics, to the first-order theory in the following way. First expand the domain from the set of all real numbers to a two-sorted domain, with the second sort containing all <i>sets of</i> real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the <a href="/wiki/Power_set" title="Power set">power set</a> of the real numbers. </p><p>But notice that the domain was asserted to include <i>all</i> sets of real numbers. That requirement cannot be reduced to a first-order sentence, as the <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a> shows. That theorem implies that there is some <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> subset of the real numbers, whose members we will call <i>internal numbers</i>, and some countably infinite collection of sets of internal numbers, whose members we will call "internal sets", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences as are satisfied by the domain of real numbers and sets of real numbers. In particular, it satisfies a sort of least-upper-bound axiom that says, in effect: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">Every nonempty <i>internal</i> set that has an <i>internal</i> upper bound has a least <i>internal</i> upper bound.</div> <p>Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all <i>internal</i> sets implies that it is not the set of <i>all</i> subsets of the set of all <i>internal</i> numbers (since <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a> implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to <a href="/wiki/Skolem%27s_paradox" title="Skolem's paradox">Skolem's paradox</a>. </p><p>Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean</a> <a href="/wiki/Real_number" title="Real number">complete ordered field</a>, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models. </p><p>There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> holds and that has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle. </p><p>Additional limitations of second-order logic are described in the next section. </p> <div class="mw-heading mw-heading2"><h2 id="Metalogical_results">Metalogical results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=7" title="Edit section: Metalogical results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is a corollary of <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a> that there is no deductive system (that is, no notion of <i>provability</i>) for second-order formulas that simultaneously satisfies these three desired attributes:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li>(<a href="/wiki/Soundness" title="Soundness">Soundness</a>) Every provable second-order sentence is universally valid, i.e., true in all domains under standard semantics.</li> <li>(<a href="/wiki/Completeness_(logic)" title="Completeness (logic)">Completeness</a>) Every universally valid second-order formula, under standard semantics, is provable.</li> <li>(<a href="/wiki/Decidability_(logic)" title="Decidability (logic)">Effectiveness</a>) There is a <a href="/wiki/Automated_proof_checking" class="mw-redirect" title="Automated proof checking">proof-checking</a> algorithm that can correctly decide whether a given sequence of symbols is a proof or not.</li></ul> <p>This corollary is sometimes expressed by saying that second-order logic does not admit a complete <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>. In this respect second-order logic with standard semantics differs from first-order logic; <a href="/wiki/W._V._Quine" class="mw-redirect" title="W. V. Quine">Quine</a> (1970, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S_NhnP0izA4C&pg=PA90">pp. 90–91</a>) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not <i>logic</i>, properly speaking. </p><p>As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with <a href="/wiki/Higher-order_logic" title="Higher-order logic">Henkin semantics</a>, and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles. </p><p>The <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a> and the <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a> do not hold for full models of second-order logic. They do hold however for Henkin models.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: xi">: xi </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History_and_disputed_value">History and disputed value</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=8" title="Edit section: History and disputed value"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Predicate logic was introduced to the mathematical community by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">C. S. Peirce</a>, who coined the term <i>second-order logic</i> and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, who published his work several years prior to Peirce but whose works remained less known until <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> made them famous. Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various ways—to what is now called <a href="/wiki/First-order_predicate_calculus" class="mw-redirect" title="First-order predicate calculus">first-order logic</a>—eliminated this problem: sets and properties cannot be quantified over in first-order logic alone. The now-standard hierarchy of orders of logics dates from this time. </p><p>It was found that <a href="/wiki/Set_theory" title="Set theory">set theory</a> could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">completeness</a>, but nothing so bad as Russell's paradox), and this was done (see <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>), as sets are vital for <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. <a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a>, <a href="/wiki/Mereology" title="Mereology">mereology</a>, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a> and <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Skolem</a>'s adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.<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. (January 2010)">citation needed</span></a></i>]</sup> </p><p>This rejection was actively advanced by some logicians, most notably <a href="/wiki/W._V._Quine" class="mw-redirect" title="W. V. Quine">W. V. Quine</a>. Quine advanced the view<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. (January 2010)">citation needed</span></a></i>]</sup> that in predicate-language sentences like <i>Fx</i> the "<i>x</i>" is to be thought of as a variable or name denoting an object and hence can be quantified over, as in "For all things, it is the case that . . ." but the "<i>F</i>" is to be thought of as an <i>abbreviation</i> for an incomplete sentence, not the name of an object (not even of an <a href="/wiki/Abstract_object" class="mw-redirect" title="Abstract object">abstract object</a> like a property). For example, it might mean " . . . is a dog." But it makes no sense to think we can quantify over something like this. (Such a position is quite consistent with Frege's own arguments on the <a href="/wiki/Concept_and_object" title="Concept and object">concept-object</a> distinction). So to use a predicate as a variable is to have it occupy the place of a name, which only individual variables should occupy. This reasoning has been rejected by <a href="/wiki/George_Boolos" title="George Boolos">George Boolos</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 2020)">citation needed</span></a></i>]</sup> </p><p>In recent years<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (October 2017)">when?</span></a></i>]</sup> second-order logic has made something of a recovery, buoyed by Boolos' interpretation of second-order quantification as <a href="/wiki/Plural_quantification" title="Plural quantification">plural quantification</a> over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed <a href="/wiki/Nonfirstorderizability" title="Nonfirstorderizability">nonfirstorderizability</a> of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else", which he argues can only be expressed by the full force of second-order quantification. However, <a href="/wiki/Generalized_quantifier" title="Generalized quantifier">generalized quantification</a> and <a href="/wiki/Branching_quantification" class="mw-redirect" title="Branching quantification">partially ordered</a> (or branching) quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and these do not appeal to second-order quantification. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_computational_complexity">Relation to computational complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=9" title="Edit section: Relation to computational complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Applications_to_complexity"></span><span class="anchor" id="relation_to_computational_complexity"></span> The expressive power of various forms of second-order logic on finite structures is intimately tied to <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>. The field of <a href="/wiki/Descriptive_complexity" class="mw-redirect" title="Descriptive complexity">descriptive complexity</a> studies which computational <a href="/wiki/Complexity_class" title="Complexity class">complexity classes</a> can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string <i>w</i> = <i>w</i><sub>1</sub>···<i>w<sub>n</sub></i> in a finite alphabet <i>A</i> can be represented by a finite structure with domain <i>D</i> = {1,...,<i>n</i>}, unary predicates <i>P<sub>a</sub></i> for each <i>a</i> ∈ <i>A</i>, satisfied by those indices <i>i</i> such that <i>w<sub>i</sub></i> = <i>a</i>, and additional predicates that serve to uniquely identify which index is which (typically, one takes the graph of the successor function on <i>D</i> or the order relation <, possibly with other arithmetic predicates). Conversely, the <a href="/wiki/Cayley_table" title="Cayley table">Cayley tables</a> of any finite structure (over a finite <a href="/wiki/Signature_(logic)" title="Signature (logic)">signature</a>) can be encoded by a finite string. </p><p>This identification leads to the following characterizations of variants of second-order logic over finite structures: </p> <ul><li>REG (the <a href="/wiki/Regular_language" title="Regular language">regular languages</a>) is the set of languages definable by monadic, second-order formulas (<a href="/wiki/B%C3%BCchi-Elgot-Trakhtenbrot_theorem" title="Büchi-Elgot-Trakhtenbrot theorem">Büchi-Elgot-Trakhtenbrot theorem</a>, 1960)</li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a> is the set of languages definable by existential, second-order formulas (<a href="/wiki/Fagin%27s_theorem" title="Fagin's theorem">Fagin's theorem</a>, 1974).</li> <li><a href="/wiki/Co-NP" title="Co-NP">co-NP</a> is the set of languages definable by universal, second-order formulas.</li> <li><a href="/wiki/PH_(complexity)" title="PH (complexity)">PH</a> is the set of languages definable by second-order formulas.</li> <li><a href="/wiki/PSPACE" title="PSPACE">PSPACE</a> is the set of languages definable by second-order formulas with an added <a href="/wiki/Transitive_closure" title="Transitive closure">transitive closure</a> operator.</li> <li><a href="/wiki/EXPTIME" title="EXPTIME">EXPTIME</a> is the set of languages definable by second-order formulas with an added <a href="/wiki/Least_fixed_point" title="Least fixed point">least fixed point</a> operator.</li></ul> <p>Relationships among these classes directly impact the relative expressiveness of the logics over finite structures; for example, if <b>PH</b> = <b>PSPACE</b>, then adding a transitive closure operator to second-order logic would not make it any more expressive over finite structures. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></span></li></ul> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order logic</a></li> <li><a href="/wiki/L%C3%B6wenheim_number" title="Löwenheim number">Löwenheim number</a></li> <li><a href="/wiki/Omega_language" title="Omega language">Omega language</a></li> <li><a href="/wiki/Second-order_propositional_logic" title="Second-order propositional logic">Second-order propositional logic</a></li> <li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic second-order logic</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Shapiro (1991) and Hinman (2005) give complete introductions to the subject, with full definitions.</span> </li> <li id="cite_note-mcohen-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mcohen_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mcohen_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Professor Marc Cohen lecture notes <a rel="nofollow" class="external free" href="https://faculty.washington.edu/smcohen/120/SecondOrder.pdf">https://faculty.washington.edu/smcohen/120/SecondOrder.pdf</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVäänänen2021" class="citation cs2">Väänänen, Jouko (2021), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2021/entries/logic-higher-order/">"Second-order and Higher-order Logic"</a>, in Zalta, Edward N. (ed.), <i>The Stanford Encyclopedia of Philosophy</i> (Fall 2021 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">2022-05-03</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Second-order+and+Higher-order+Logic&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Fall+2021&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2021&rft.aulast=V%C3%A4%C3%A4n%C3%A4nen&rft.aufirst=Jouko&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2021%2Fentries%2Flogic-higher-order%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">*<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson,_Elliot2009" class="citation book cs1">Mendelson, Elliot (2009). <i>Introduction to Mathematical Logic</i> (hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman and Hall/CRC. p. 387. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-876-5" title="Special:BookSources/978-1-58488-876-5"><bdi>978-1-58488-876-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.place=Boca+Raton&rft.series=Discrete+Mathematics+and+Its+Applications&rft.pages=387&rft.edition=5th&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2009&rft.isbn=978-1-58488-876-5&rft.au=Mendelson%2C+Elliot&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></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">Such a system is used without comment by Hinman (2005).</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">These are the models originally studied by Henkin (1950).</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">The proof of this corollary is that a sound, complete, and effective deduction system for standard semantics could be used to produce a <a href="/wiki/Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a> completion of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, which Gödel's theorem shows cannot exist.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="/wiki/Mar%C3%ADa_Manzano" title="María Manzano">Manzano, M.</a>, <i>Model Theory</i>, trans. Ruy J. G. B. de Queiroz (<a href="/wiki/Oxford" title="Oxford">Oxford</a>: <a href="/wiki/Oxford_University_Press#Clarendon_Press" title="Oxford University Press">Clarendon Press</a>, 1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Nc5uXx057JwC&pg=PR11">p. xi</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrews,_Peter2002" class="citation book cs1">Andrews, Peter (2002). <i>An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof</i> (2nd ed.). Kluwer Academic Publishers.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mathematical+Logic+and+Type+Theory%3A+To+Truth+Through+Proof&rft.edition=2nd&rft.pub=Kluwer+Academic+Publishers&rft.date=2002&rft.au=Andrews%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoolos,_George1984" class="citation journal cs1">Boolos, George (1984). "To Be Is To Be a Value of a Variable (or to Be Some Values of Some Variables)". <i>Journal of Philosophy</i>. <b>81</b> (8): 430–50. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2026308">10.2307/2026308</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2026308">2026308</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Philosophy&rft.atitle=To+Be+Is+To+Be+a+Value+of+a+Variable+%28or+to+Be+Some+Values+of+Some+Variables%29&rft.volume=81&rft.issue=8&rft.pages=430-50&rft.date=1984&rft_id=info%3Adoi%2F10.2307%2F2026308&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2026308%23id-name%3DJSTOR&rft.au=Boolos%2C+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span>. Reprinted in Boolos, <i>Logic, Logic and Logic</i>, 1998.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenkin,_L.1950" class="citation journal cs1">Henkin, L. (1950). "Completeness in the theory of types". <i>Journal of Symbolic Logic</i>. <b>15</b> (2): 81–91. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2266967">10.2307/2266967</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2266967">2266967</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:36309665">36309665</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Completeness+in+the+theory+of+types&rft.volume=15&rft.issue=2&rft.pages=81-91&rft.date=1950&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A36309665%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2266967%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2266967&rft.au=Henkin%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinman,_P.2005" class="citation book cs1">Hinman, P. (2005). <i>Fundamentals of Mathematical Logic</i>. A K Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-262-0" title="Special:BookSources/1-56881-262-0"><bdi>1-56881-262-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Mathematical+Logic&rft.pub=A+K+Peters&rft.date=2005&rft.isbn=1-56881-262-0&rft.au=Hinman%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPutnam,_Hilary1982" class="citation journal cs1"><a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Putnam, Hilary</a> (1982). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2882%2990123-9">"Peirce the Logician"</a>. <i>Historia Mathematica</i>. <b>9</b> (3): 290–301. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2882%2990123-9">10.1016/0315-0860(82)90123-9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Peirce+the+Logician&rft.volume=9&rft.issue=3&rft.pages=290-301&rft.date=1982&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2882%2990123-9&rft.au=Putnam%2C+Hilary&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252882%252990123-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span>. Reprinted in Putnam, Hilary (1990), <i>Realism with a Human Face</i>, <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7JFTo-ZXROMC&pg=PT252">pp. 252–260</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFW._V._Quine1970" class="citation book cs1">W. V. Quine (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S_NhnP0izA4C"><i>Philosophy of Logic</i></a>. <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674665637" title="Special:BookSources/9780674665637"><bdi>9780674665637</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Logic&rft.pub=Prentice+Hall&rft.date=1970&rft.isbn=9780674665637&rft.au=W.+V.+Quine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DS_NhnP0izA4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRossberg,_M.2004" class="citation conference cs1">Rossberg, M. (2004). <a rel="nofollow" class="external text" href="http://logic.amu.edu.pl/images/4/46/Completenessrossberg.pdf">"First-Order Logic, Second-Order Logic, and Completeness"</a> <span class="cs1-format">(PDF)</span>. In V. Hendricks; et al. (eds.). <i>First-order logic revisited</i>. Berlin: Logos-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=First-Order+Logic%2C+Second-Order+Logic%2C+and+Completeness&rft.btitle=First-order+logic+revisited&rft.place=Berlin&rft.pub=Logos-Verlag&rft.date=2004&rft.au=Rossberg%2C+M.&rft_id=http%3A%2F%2Flogic.amu.edu.pl%2Fimages%2F4%2F46%2FCompletenessrossberg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShapiro,_S.2000" class="citation book cs1"><a href="/wiki/Stewart_Shapiro" title="Stewart Shapiro">Shapiro, S.</a> (2000) [1991]. <i><span></span></i>Foundations without Foundationalism: A Case for Second-Order Logic<i><span></span></i>. Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-825029-0" title="Special:BookSources/0-19-825029-0"><bdi>0-19-825029-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+without+Foundationalism%3A+A+Case+for+Second-Order+Logic&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=2000&rft.isbn=0-19-825029-0&rft.au=Shapiro%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="Vaananen2001" class="citation journal cs1">Väänänen, J. (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231008112652/http://www.math.helsinki.fi/logic/people/jouko.vaananen/VaaSec.pdf">"Second-Order Logic and Foundations of Mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of Symbolic Logic</i>. <b>7</b> (4): 504–520. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.5579">10.1.1.25.5579</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2687796">10.2307/2687796</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2687796">2687796</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7465054">7465054</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Symbolic+Logic&rft.atitle=Second-Order+Logic+and+Foundations+of+Mathematics&rft.volume=7&rft.issue=4&rft.pages=504-520&rft.date=2001&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.25.5579%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7465054%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2687796%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2687796&rft.aulast=V%C3%A4%C3%A4n%C3%A4nen&rft.aufirst=J.&rft_id=https%3A%2F%2Fweb.archive.org%2Fweb%2F20231008112652%2Fhttp%3A%2F%2Fwww.math.helsinki.fi%2Flogic%2Fpeople%2Fjouko.vaananen%2FVaaSec.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second-order_logic&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrädelKolaitisLibkinMaarten2007" class="citation book cs1">Grädel, Erich; Kolaitis, Phokion G.; <a href="/wiki/Leonid_Libkin" title="Leonid Libkin">Libkin, Leonid</a>; Maarten, Marx; <a href="/wiki/Joel_Spencer" title="Joel Spencer">Spencer, Joel</a>; <a href="/wiki/Moshe_Y._Vardi" class="mw-redirect" title="Moshe Y. Vardi">Vardi, Moshe Y.</a>; Venema, Yde; Weinstein, Scott (2007). <i>Finite model theory and its applications</i>. Texts in Theoretical Computer Science. An EATCS Series. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-00428-8" title="Special:BookSources/978-3-540-00428-8"><bdi>978-3-540-00428-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1133.03001">1133.03001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+model+theory+and+its+applications&rft.place=Berlin&rft.series=Texts+in+Theoretical+Computer+Science.+An+EATCS+Series&rft.pub=Springer-Verlag&rft.date=2007&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1133.03001%23id-name%3DZbl&rft.isbn=978-3-540-00428-8&rft.aulast=Gr%C3%A4del&rft.aufirst=Erich&rft.au=Kolaitis%2C+Phokion+G.&rft.au=Libkin%2C+Leonid&rft.au=Maarten%2C+Marx&rft.au=Spencer%2C+Joel&rft.au=Vardi%2C+Moshe+Y.&rft.au=Venema%2C+Yde&rft.au=Weinstein%2C+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVäänänen" class="citation book cs1">Väänänen, Jouko. Edward N. Zalta (ed.). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2021/entries/logic-higher-order/"><i>Second-order and Higher-order Logic</i></a>. The Stanford Encyclopedia of Philosophy (Fall 2021 Edition).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Second-order+and+Higher-order+Logic&rft.series=The+Stanford+Encyclopedia+of+Philosophy+%28Fall+2021+Edition%29&rft.aulast=V%C3%A4%C3%A4n%C3%A4nen&rft.aufirst=Jouko&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2021%2Fentries%2Flogic-higher-order%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond-order+logic" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a class="mw-selflink selflink">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a 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