Generalized quantifier

A version of type theory is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of types recursively as follows:

1. e and t are types.
2. If a and b are both types, then so is ${\displaystyle \langle a,b\rangle }$
3. Nothing is a type, except what can be constructed on the basis of lines 1 and 2 above.

Given this definition, we have the simple types e and t, but also a countable infinity of complex types, some of which include:

$${\displaystyle \langle e,t\rangle$$ ;\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 }

• Expressions of type e denote elements of the universe of discourse, the set of entities the discourse is about. This set is usually written as ${\displaystyle D_{e}}$. Examples of type e expressions include John and he.
• Expressions of type t denote a truth value, usually rendered as the set ${\displaystyle \{0,1\}}$, where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of type t are sentences or propositions.
• Expressions of type ${\displaystyle \langle e,t\rangle }$ denote functions from the set of entities to the set of truth values. This set of functions is rendered as ${\displaystyle D_{t}^{D_{e}}}$. Such functions are characteristic functions of sets. 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 sets rather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type are predicates, nouns and some kinds of adjectives.
• In general, expressions of complex types ${\displaystyle \langle a,b\rangle }$ denote functions from the set of entities of type ${\displaystyle a}$ to the set of entities of type ${\displaystyle b}$, a construct we can write as follows: ${\displaystyle D_{b}^{D_{a}}}$.

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

• Type(boy) = ${\displaystyle \langle e,t\rangle }$
• Type(sleeps) = ${\displaystyle \langle e,t\rangle }$
• Type(every) = ${\displaystyle \langle \langle e,t\rangle ,\langle \langle e,t\rangle ,t\rangle \rangle }$
• Type(every boy) = ${\displaystyle \langle \langle e,t\rangle ,t\rangle }$

and so we can see that the generalized quantifier in our example is of type ${\displaystyle \langle \langle e,t\rangle ,t\rangle }$

Thus, every denotes a function from a set 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 A,B, every(A)(B)= 1 if and only if ${\displaystyle A\subseteq B}$.

A useful way to write complex functions is the lambda calculus. For example, one can write the meaning of sleeps as the following lambda expression, which is a function from an individual x to the proposition that x sleeps.

$${\displaystyle \lambda x.\mathrm {sleep} '(x)}$$

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 x is a variable that ranges over elements of ${\displaystyle D_{e}}$, then the following lambda term denotes the identity function on individuals:

$${\displaystyle \lambda x.x}$$

We can now write the meaning of every with the following lambda term, where X,Y are variables of type ${\displaystyle \langle e,t\rangle }$:

$${\displaystyle \lambda X.\lambda Y.X\subseteq Y}$$

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

$${\displaystyle (\lambda X.\lambda Y.X\subseteq Y)(B)(S)}$$

By β-reduction,

$${\displaystyle (\lambda Y.B\subseteq Y)(S)}$$

and

$${\displaystyle B\subseteq S}$$

The expression every is a determiner. Combined with a noun, it yields a generalized quantifier of type ${\displaystyle \langle \langle e,t\rangle ,t\rangle }$.

• Scope (formal semantics)
• Lindström quantifier
• Branching quantifier

• Stanley Peters; Dag Westerståhl (2006). Quantifiers in language and logic. Clarendon Press. ISBN 978-0-19-929125-0.
• Antonio Badia (2009). Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages. Springer. ISBN 978-0-387-09563-9.
• Wągiel M (2021). Subatomic quantification (pdf). Berlin: Language Science Press. doi:10.5281/zenodo.5106382. ISBN 978-3-98554-011-2.

• Dag Westerståhl, 2011. 'Generalized Quantifiers'. Stanford Encyclopedia of Philosophy.