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<title>Generalized quantifier</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized quantifier</span></span>

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<p>In <a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">formal semantics</a>, a <b>generalized quantifier</b> (<b>GQ</b>) is an expression that denotes a <a href="Set_of_sets" class="mw-redirect" title="Set of sets">set of sets</a>. This is the standard semantics assigned to <a href="Quantifier_(logic)" title="Quantifier (logic)">quantified</a> <a href="Noun_phrase" title="Noun phrase">noun phrases</a>. For example, the generalized quantifier <i>every boy</i> denotes the set of sets of which every boy is a member:

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<annotation encoding="application/x-tex">{\displaystyle \{X\mid \forall x(x{\text{ is a boy}}\to x\in X)\}}</annotation>

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</p><p>This treatment of quantifiers has been essential in achieving a <a href="Compositionality" class="mw-redirect" title="Compositionality">compositional</a> <a href="Semantics" title="Semantics">semantics</a> for sentences containing quantifiers.<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><sup id="cite_ref-Barwise_2-0" class="reference"><a href="#cite_note-Barwise-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Type_theory">Type theory</h2></div>

<p>A version of <a href="Type_theory" title="Type theory">type theory</a> is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of types <a href="Recursion" title="Recursion">recursively</a> as follows:

</p>

<ol><li><i>e</i> and <i>t</i> are types.</li>

<li>If <i>a</i> and <i>b</i> are both types, then so is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle a,b\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle a,b\rangle }</annotation>

</semantics>

</math></span><img src="./59f9c6fc93d2e4195b610bc8a9ff366be4485387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle }" loading="lazy"></span></li>

<li>Nothing is a type, except what can be constructed on the basis of lines 1 and 2 above.</li></ol>

<p>Given this definition, we have the simple types <i>e</i> and <i>t</i>, but also a <a href="Countable" class="mw-redirect" title="Countable">countable</a> <a href="Infinity" title="Infinity">infinity</a> of complex types, some of which include:

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<annotation encoding="application/x-tex">{\displaystyle \langle e,t\rangle ;\qquad \langle t,t\rangle ;\qquad \langle \langle e,t\rangle ,t\rangle ;\qquad \langle e,\langle e,t\rangle \rangle ;\qquad \langle \langle e,t\rangle ,\langle \langle e,t\rangle ,t\rangle \rangle ;\qquad \ldots }</annotation>

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</math></span></span>

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<ul><li>Expressions of type <i>e</i> denote elements of the <a href="Universe_of_discourse" class="mw-redirect" title="Universe of discourse">universe of discourse</a>, the set of entities the discourse is about. This set is usually written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{e}}">

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

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</math></span><img src="./37e785a9fc0aea966a290a076cd916bff798900a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.923ex; height:2.509ex;" alt="{\displaystyle D_{e}}" loading="lazy"></span>. Examples of type <i>e</i> expressions include <i>John</i> and <i>he</i>.</li>

<li>Expressions of type <i>t</i> denote a <a href="Truth_value" title="Truth value">truth value</a>, usually rendered as the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation>

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</math></span><img src="./28de5781698336d21c9c560fb1cbb3fb406923eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}" loading="lazy"></span>, where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of type <i>t</i> are <i>sentences</i> or <i>propositions</i>.</li>

<li>Expressions of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e,t\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle e,t\rangle }</annotation>

</semantics>

</math></span><img src="./6f4f00dd73ef1abc01cc09df9d45b3296c102342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle \langle e,t\rangle }" loading="lazy"></span> denote <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> from the set of entities to the set of truth values. This set of functions is rendered as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{t}^{D_{e}}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{t}^{D_{e}}}</annotation>

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</math></span><img src="./5ebb6fa1dee9f5acbf9d262930448266c741a26b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.303ex; height:3.509ex;" alt="{\displaystyle D_{t}^{D_{e}}}" loading="lazy"></span>. Such functions are <a href="Indicator_function" title="Indicator function">characteristic functions</a> of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a>. They map every individual that is an element of the set to "true", and everything else to "false." It is common to say that they denote <i>sets</i> rather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type are <a href="Predicate_(grammar)" title="Predicate (grammar)">predicates</a>, <a href="Noun" title="Noun">nouns</a> and some kinds of <a href="Adjective" title="Adjective">adjectives</a>.</li>

<li>In general, expressions of complex types <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle a,b\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle a,b\rangle }</annotation>

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</math></span><img src="./59f9c6fc93d2e4195b610bc8a9ff366be4485387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle }" loading="lazy"></span> denote functions from the set of entities of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> to the set of entities of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span>, a construct we can write as follows: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{b}^{D_{a}}}">

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle D_{b}^{D_{a}}}</annotation>

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</math></span><img src="./b73fd8f0095b1f1774aad25c98683b3086f3553e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.387ex; height:3.509ex;" alt="{\displaystyle D_{b}^{D_{a}}}" loading="lazy"></span>.</li></ul>

<p>We can now assign types to the words in our sentence above (Every boy sleeps) as follows.

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \langle e,t\rangle }</annotation>

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</math></span><img src="./6f4f00dd73ef1abc01cc09df9d45b3296c102342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle \langle e,t\rangle }" loading="lazy"></span></li>

<li>Type(sleeps) = <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e,t\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle e,t\rangle }</annotation>

</semantics>

</math></span><img src="./6f4f00dd73ef1abc01cc09df9d45b3296c102342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle \langle e,t\rangle }" loading="lazy"></span></li>

<li>Type(every) = <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \langle e,t\rangle ,\langle \langle e,t\rangle ,t\rangle \rangle }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \langle e,t\rangle ,\langle \langle e,t\rangle ,t\rangle \rangle }</annotation>

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</math></span><img src="./8972ac28a639137fc7a32f33ee1ad76cf2bf206d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.059ex; height:2.843ex;" alt="{\displaystyle \langle \langle e,t\rangle ,\langle \langle e,t\rangle ,t\rangle \rangle }" loading="lazy"></span></li>

<li>Type(every boy) = <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \langle e,t\rangle ,t\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \langle e,t\rangle ,t\rangle }</annotation>

</semantics>

</math></span><img src="./8167e020f844db879b27b7ad854fa7b7447eec11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.449ex; height:2.843ex;" alt="{\displaystyle \langle \langle e,t\rangle ,t\rangle }" loading="lazy"></span></li></ul>

<p>and so we can see that the generalized quantifier in our example is of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \langle e,t\rangle ,t\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \langle e,t\rangle ,t\rangle }</annotation>

</semantics>

</math></span><img src="./8167e020f844db879b27b7ad854fa7b7447eec11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.449ex; height:2.843ex;" alt="{\displaystyle \langle \langle e,t\rangle ,t\rangle }" loading="lazy"></span>

</p><p>Thus, every denotes a function from a <i>set</i> to a function from a set to a truth value. Put differently, it denotes a function from a set to a set of sets. It is that function which for any two sets <i>A,B</i>, <i>every</i>(<i>A</i>)(<i>B</i>)= 1 if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}">

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

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</math></span><img src="./b09068bd2f7ba899aeb883ebe670b2ad07b0c851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Typed_lambda_calculus">Typed lambda calculus</h2></div>

<p>A useful way to write complex functions is the <a href="Lambda_calculus" title="Lambda calculus">lambda calculus</a>. For example, one can write the meaning of <i>sleeps</i> as the following lambda expression, which is a function from an individual <i>x</i> to the proposition that <i>x sleeps</i>.

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \lambda x.\mathrm {sleep} '(x)}</annotation>

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</math></span></span>

Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing that follows the period. If <i>x</i> is a variable that ranges over elements of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{e}}">

<msub>

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

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</math></span><img src="./37e785a9fc0aea966a290a076cd916bff798900a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.923ex; height:2.509ex;" alt="{\displaystyle D_{e}}" loading="lazy"></span>, then the following lambda term denotes the <a href="Identity_function" title="Identity function">identity function</a> on individuals:

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

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</math></span></span>

</p><p>We can now write the meaning of <i>every</i> with the following lambda term, where <i>X,Y</i> are variables of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e,t\rangle }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \langle e,t\rangle }</annotation>

</semantics>

</math></span><img src="./6f4f00dd73ef1abc01cc09df9d45b3296c102342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle \langle e,t\rangle }" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle \lambda X.\lambda Y.X\subseteq Y}</annotation>

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</math></span></span>

</p><p>If we abbreviate the meaning of <i>boy</i> and <i>sleeps</i> as "<i>B</i>" and "<i>S</i>", respectively, we have that the sentence <i>every boy sleeps</i> now means the following:

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<annotation encoding="application/x-tex">{\displaystyle (\lambda X.\lambda Y.X\subseteq Y)(B)(S)}</annotation>

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</math></span></span>

By <a href="Lambda_calculus#β-reduction" title="Lambda calculus">β-reduction</a>,

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<annotation encoding="application/x-tex">{\displaystyle (\lambda Y.B\subseteq Y)(S)}</annotation>

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</math></span></span>

and

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<annotation encoding="application/x-tex">{\displaystyle B\subseteq S}</annotation>

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</math></span></span>

</p><p>The expression <i>every</i> is a <a href="Determiner_(linguistics)" class="mw-redirect" title="Determiner (linguistics)">determiner</a>. Combined with a <a href="Noun" title="Noun">noun</a>, it yields a <i>generalized quantifier</i> of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \langle e,t\rangle ,t\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \langle e,t\rangle ,t\rangle }</annotation>

</semantics>

</math></span><img src="./8167e020f844db879b27b7ad854fa7b7447eec11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.449ex; height:2.843ex;" alt="{\displaystyle \langle \langle e,t\rangle ,t\rangle }" loading="lazy"></span>.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Monotonicity">Monotonicity</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Monotone_increasing_GQs">Monotone increasing GQs</h4></div>

<p>A <i>generalized quantifier</i> GQ is said to be <a href="Monotone_increasing" class="mw-redirect" title="Monotone increasing">monotone increasing</a> (also called upward entailing) if, for every pair of sets <i>X</i> and <i>Y</i>, the following holds:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subseteq Y}</annotation>

</semantics>

</math></span><img src="./894a17f6bbae8bb911fb08785e051b353e832ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle X\subseteq Y}" loading="lazy"></span>, then GQ(<i>X</i>) <a href="Entailment" class="mw-redirect" title="Entailment">entails</a> GQ(<i>Y</i>).</dd></dl>

<p>The GQ <i>every boy</i> is monotone increasing. For example, the set of things that <i>run fast</i> is a subset of the set of things that <i>run</i>. Therefore, the first sentence below <a href="Entailment" class="mw-redirect" title="Entailment">entails</a> the second:

</p>

<ol><li>Every boy runs fast.</li>

<li>Every boy runs.</li></ol>

<div class="mw-heading mw-heading4"><h4 id="Monotone_decreasing_GQs">Monotone decreasing GQs</h4></div>

<p>A GQ is said to be <a href="Monotone_decreasing" class="mw-redirect" title="Monotone decreasing">monotone decreasing</a> (also called <a href="Downward_entailing" title="Downward entailing">downward entailing</a>) if, for every pair of sets <i>X</i> and <i>Y</i>, the following holds:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subseteq Y}</annotation>

</semantics>

</math></span><img src="./894a17f6bbae8bb911fb08785e051b353e832ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle X\subseteq Y}" loading="lazy"></span>, then GQ(<i>Y</i>) entails GQ(<i>X</i>).</dd></dl>

<p>An example of a monotone decreasing GQ is <i>no boy</i>. For this GQ we have that the first sentence below entails the second.

</p>

<p>The lambda term for the <a href="Determiner_(linguistics)" class="mw-redirect" title="Determiner (linguistics)">determiner</a> <i>no</i> is the following. It says that the two sets have an empty <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>.

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<annotation encoding="application/x-tex">{\displaystyle \lambda X.\lambda Y.X\cap Y=\emptyset }</annotation>

</semantics>

</math></span></span>

Monotone decreasing GQs are among the expressions that can license a <a href="Negative_polarity_item" class="mw-redirect" title="Negative polarity item">negative polarity item</a>, such as <i>any</i>. Monotone increasing GQs do not license negative polarity items.

</p>

<ol><li>Good: No boy has <b>any</b> money.</li>

<li>Bad: *Every boy has <b>any</b> money.</li></ol>

<div class="mw-heading mw-heading4"><h4 id="Non-monotone_GQs">Non-monotone GQs</h4></div>

<p>A GQ is said to be <i>non-monotone</i> if it is neither monotone increasing nor monotone decreasing. An example of such a GQ is <i>exactly three boys</i>. Neither of the following sentences entails the other.

</p>

<ol><li>Exactly three students ran.</li>

<li>Exactly three students ran fast.</li></ol>

<p>The first sentence does not entail the second. The fact that the number of students that ran is exactly three does not entail that each of these students <i>ran fast</i>, so the number of students that did that can be smaller than 3. Conversely, the second sentence does not entail the first. The sentence <i>exactly three students ran fast</i> can be true, even though the number of students who merely ran (i.e. not so fast) is greater than 3.

</p><p>The lambda term for the (complex) <a href="Determiner_(linguistics)" class="mw-redirect" title="Determiner (linguistics)">determiner</a> <i>exactly three</i> is the following. It says that the <a href="Cardinality" title="Cardinality">cardinality</a> of the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> between the two sets equals 3.

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda X.\lambda Y.|X\cap Y|=3}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Conservativity">Conservativity</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Conservativity" title="Conservativity">Conservativity</a></div>

<p>A determiner D is said to be <i>conservative</i> if the following equivalence holds:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(A)(B)\leftrightarrow D(A)(A\cap B)}</annotation>

</semantics>

</math></span></span>

For example, the following two sentences are equivalent.

</p>

<ol><li>Every boy sleeps.</li>

<li>Every boy is a boy who sleeps.</li></ol>

<p>It has been proposed that <i>all</i> determiners—in every natural language—are conservative.<sup id="cite_ref-Barwise_2-1" class="reference"><a href="#cite_note-Barwise-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The expression <i>only</i> is not conservative. The following two sentences are not equivalent. But it is, in fact, not common to analyze <i>only</i> as a <a href="Determiner_(linguistics)" class="mw-redirect" title="Determiner (linguistics)">determiner</a>. Rather, it is standardly treated as a focus-sensitive <a href="Adverb" title="Adverb">adverb</a>.

</p>

<ol><li>Only boys sleep.</li>

<li>Only boys are boys who sleep.</li></ol>

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

<li><a href="Lindstr%C3%B6m_quantifier" title="Lindström quantifier">Lindström quantifier</a></li>

<li><a href="Branching_quantifier" title="Branching quantifier">Branching quantifier</a></li></ul>

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

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</style><cite id="CITEREFMontague1974" class="citation book cs1"><a href="Richard_Montague" title="Richard Montague">Montague, Richard</a> (1974). "The proper treatment of quantification in English". In Kulas, J.; Fetzer, J.H.; Rankin, T.L. (eds.). <a rel="nofollow" class="external text" href="http://www.blackwellpublishing.com/content/BPL_Images/Content_store/Sample_chapter/9780631215417/Portner.pdf"><i>Philosophy, Language, and Artificial Intelligence</i></a> <span class="cs1-format">(PDF)</span>. Studies in Cognitive Systems. Vol. 2. Springer, Dordrecht. pp. <span class="nowrap">141–</span>162. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-009-2727-8_7">10.1007/978-94-009-2727-8_7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-010-7726-2</bdi>.</cite></span>

</li>

<li id="cite_note-Barwise-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Barwise_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Barwise_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBarwiseCooper1981" class="citation journal cs1"><a href="Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>; Cooper, Robin (1981). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007%2FBF00350139">"Generalized quantifiers and natural language"</a></span>. <i>Linguistics and Philosophy</i>. <b>4</b> (2): <span class="nowrap">159–</span>219. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00350139">10.1007/BF00350139</a>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFStanley_PetersDag_Westerståhl2006" class="citation book cs1">Stanley Peters; Dag Westerståhl (2006). <i>Quantifiers in language and logic</i>. Clarendon Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-929125-0</bdi>.</cite></li>

<li><cite id="CITEREFAntonio_Badia2009" class="citation book cs1">Antonio Badia (2009). <i>Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-09563-9</bdi>.</cite></li>

<li><cite id="CITEREFWągiel2021" class="citation book cs1">Wągiel M (2021). <a rel="nofollow" class="external text" href="http://langsci-press.org/catalog/book/317"><i>Subatomic quantification</i></a> <span class="cs1-format">(pdf)</span>. Berlin: Language Science Press. <a href="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/10.5281%2Fzenodo.5106382">10.5281/zenodo.5106382</a></span>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-98554-011-2</bdi>.</cite></li></ul>

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

<ul><li>Dag Westerståhl, 2011. '<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/generalized-quantifiers/">Generalized Quantifiers</a>'. <a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</li></ul>

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</style><div id="Formal_semantics_(natural_language)78" style="font-size:114%;margin:0 4em"><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Central concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li><a href="Linguistic_modality" class="mw-redirect" title="Linguistic modality">Modality</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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