# 14.3: Quantifiers in logical form

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Our analysis of all as denoting a subset relation, no as meaning ‘empty intersection’, and some as meaning ‘non-empty intersection’, is reflected in the logical forms we proposed in Chapter 4 for sentences involving these words. These logical forms are repeated here in (10).

(10) a. All men snore. ∀x[MAN(x) → SNORE(x)]
b. No women snore. ¬∃x[WOMAN(x) ∧ SNORE(x)]
c. Some man snores. ∃x[MAN(x) ∧ SNORE(x)]

Now we are in a position to understand why these forms work as translations of the English quantifier words. The use of material implication (→) in (10a) follows from the definition of the subset relation which we presented in Chapter 13, repeated here in (11a). The use of logical ∧ ‘and’ in (10b–c) follows from the definition of set intersection presented in Chapter 13, repeated here in (11b).

(11) a. (A ⊆ B) ↔ ∀x[(x∈A) → (x∈B)] [subset]
(JMAN K ⊆J SNORE K ) ↔ ∀x[(x∈J MANK ) → (x∈J SNORE K )]

b. ∀x[x ∈ (A∩B) ↔ ((x∈A) ∧ (x∈B))] [intersection]
(JMANK ∩ JSNOREK ≠ ⌀) ↔ ∃x[(x∈J MAN K ) ∧ (x∈J SNORE K )]

Many other quantifier meanings can also be expressed using the basic predicate logic notation. For example, the NP four men could be translated as shown in (12):

(12) Four men snore.
∃w∃x∃y∃z[w≠x≠y≠z ∧ MAN(w) ∧ MAN(x) ∧ MAN(y) ∧ MAN(z) ∧ SNORE(w) ∧ SNORE(x) ∧ SNORE(y) ∧ SNORE(z)]

As we can see even in this simple example, the standard predicate logic notation is a somewhat clumsy tool for this task. Moreover, it turns out that there are some quantifier meanings which cannot be expressed at all using the predicate logic we have introduced thus far. For example, the interpretation for most suggested in (5f) is that the cardinality of the intersection of the two sets is greater than half of the cardinality of the first set. The basic problem here is that the logical predicates we have been using thus far represent properties of individual entities. This type of logic is called first-order logic. However, the cardinality of a set is not a property of any individual, but rather a property of the set as a whole. What we would need in order to express quantifier meanings like most is some version of second-order logic, which deals with properties of sets of individuals.

For example, we could define the denotation set of a NP like most men to be the set of all properties which are true of most men. The sentence Most men snore would be true just in case the property of snoring is a member of ((most men)).4 However, the mathematical formalism of this approach is more complex than we can handle in the present book. Rather than trying to work out all the technical details, we will proceed from here on with a more descriptive approach.

One convenient way of expressing propositions which contain quantifier meanings like most is called the restricted qantifier notation. This notation consists of three parts: the quantifier operator, the restriction, and the nuclear scope. In example (13a), the operator is most; the restriction is the open proposition “STUDENT(x)”; and the nuclear scope is the open proposition “BRILLIANT(x)”. This same format can be used for other quantifiers as well, as illustrated in (13b– c).

(13) a. Most students are brilliant. [most x: STUDENT(x)] BRILLIANT(x)
(operator = “most”; restriction = “STUDENT(x)”; scope = “BRILLIANT(x)”)
b. No women snore. [no x: WOMAN(x)] SNORE(x)
c. All brave men are lonely. [all x: MAN(x) ∧ BRAVE(x)] LONELY(x)

In contrast to the standard logical notation, using this restricted quantifier notation allows us to adopt a uniform procedure for interpreting sentences which contain quantifying determiners:

• the quantifying determiner itself specifies the operator;
• the remainder of the NP which contains the quantifying determiner specifies the material in the restriction;
• the rest of the sentence specifies the material in the nuclear scope.

For example, the quantifying determiner in (13c) is all; this determines the operator. The remainder of the NP which contains the quantifying determiner is brave men; this specifies the material in the restriction (MAN(x) ∧ BRAVE(x)). The rest of the sentence (are lonely) specifies the material in the nuclear scope (LONELY(x)). Some additional examples are provided in (14).

(14) a. Most men who snore are libertarians.
[most x: MAN(x) ∧ SNORE(x)] LIBERTARIAN(x)

b. Few strict Baptists drink or smoke.
[few x: BAPTIST(x) ∧ STRICT(x)] DRINK(x) ∨ SMOKE(x)

Of course, translations in this format do not tell us what the quantifying determiners actually mean; the meaning of each quantifier needs to be defined separately, as illustrated in (15):

(15) a. [all x: P(x)] Q(x) ↔ ((P)) ⊆ ((Q))
b. [no x: P(x)] Q(x) ↔ ((P)) ∩ ((Q)) = ⌀
c. [four x: P(x)] Q(x) ↔ | ((P)) ∩ ((Q)) | = 4
d. [most x: P(x)] Q(x) ↔ | ((P)) ∩ ((Q)) | > ½ | ((P)) |

As these definitions show, a quantifying determiner names a relation between two sets: one defined by the predicate(s) in the restriction (represented by P in the formulae in 15), and the other defined by the predicate(s) in the scope (represented by Q). Interpretations for the examples in (13) are shown in (16). Use these examples to study how the content of the restriction and scope of the logical form in restricted quantifier notation get inserted into the set theoretic interpretation.

(16) a. Most students are brilliant.
[most x: STUDENT(x)] BRILLIANT(x)
| ((STUDENT)) ∩ ((BRILLIANT)) | > ½ | ((STUDENT)) |

b. No women snore.
[no x: WOMAN(x)] SNORE(x)
(WOMAN)) ∩ ((SNORE)) = ⌀

c. All brave men are lonely.
[all x: MAN(x) ∧ BRAVE(x)] LONELY(x)
(((MAN)) ∩ ((BRAVE))) ⊆ ((LONELY))

This same procedure applies whether the quantified NP is a subject, object, or oblique argument. Some examples of quantified object NPs are given in (17).

(17) a. John loves all pretty girls.
[all x: GIRL(x) ∧ PRETTY(x)] LOVE(j,x)
(((GIRL)) ∩ ((PRETTY)) ) ⊆ {x: <j,x> ∈ ((LOVE)) }

b. Susan has married a cowboy who teases her.
[an x: COWBOY(x) ∧ TEASE(x,s)] MARRY(s,x)
(((COWBOY)) ∩ {x: <x,s> ∈ ((TEASE)) ) ∩ {y: <s,y> ∈ ((MARRY)) } ≠ ⌀

At least for the moment, we will provisionally treat the articles the and a(n) as quantifying determiners. We will discuss the definite article below in §14.4. For now we will treat the indefinite article as an existential quantifier, as illustrated in (17b). (Note that this applies to indefinite articles occurring in argument NPs, not predicate NPs. We suggested in Chapter 13 that indefinite articles occurring in predicate NPs typically do not contribute any independent meaning.)

Compound words such as someone, everyone, no one, something, nothing, anything, everywhere, etc. include a quantifier root plus another root that restricts the quantification to a general class (people, things, places, etc.). It is often helpful to include this “classifier” meaning as a predicate within the restriction of the quantifier, as illustrated in (18).

(18) a. Everyone loves Snoopy. [all x: PERSON(x)] LOVE(x,s)

b. Columbus discovered something. [some x: THING(x)] DISCOVER(c,x)

c. Nowhere on Earth is safe. [no x: PLACE(x) ∧ ON(x,e)] SAFE(x)

4 This analysis, under which quantified NPs denote sets of sets, is called the Generalized Quantifier approach. The meanings of the quantified NPs themselves are referred to as Generalized Quantifiers, which leads to a certain amount of ambiguity in the use of the word quantifier. Sometimes it is used to refer to the whole NP, and sometimes just to the quantifying determiner.

This page titled 14.3: Quantifiers in logical form is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Paul Kroeger (Language Library Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.