Constructive Generalized Quantifiers Revisited

Abstract

This paper proposes a proof-theoretic definition for generalized quantifiers (GQs). Sundholm first proposed a proof-theoretic definition of GQs in the framework of constructive type theory. However, that definition is associated with three problems: the proportion problem, absence of strong interpretation and lack of definitional uniformity. This paper presents an alternative definition for “most” based on polymorphic dependent type theory and shows strong potential to serve as an alternative to the traditional model-theoretic approach.

Appendix

Definition 1

(Construction of the Natural Numbers). Sundholm [18] presented the construction of natural numbers with the $M_r$ sequence proposed by Aczel [1]. First put

$$ \begin{array}{l} f(a)=R(a,eq(N,0,0),(x,y)(y+eq(N,0,0))):U.\\ (a:N, U \ \text{ is } \text{ the } \text{ universe } \text{ of } \text{ small } \text{ types }\!, + \ \text{ is } \text{ a } \text{ disjoint } \text{ sum }\!.) \end{array} $$

and

$$ \begin{array}{l} f(0)=eq(N,0,0):U\\ f(s(a))=f(a)+eq(N,0,0):U \ (a:N). \end{array} $$

Finally, put

$$ M(a)=R(a,\bot ,(x,y)f(x)):U \ (a:N) $$

then

$$ \begin{array}{l} M(0)= \bot :U\\ M(1)=eq(N,0,0):U\\ M(s(s(a)))=M(s(a))+eq(N,0,0):U \ (a:N). \end{array} $$

Definition 2

(Primitive Recursive Functions).

$$ \begin{array}{lll} sg(0)&{}=&{}1\\ sg(s(a))&{}=&{}0 \end{array} \qquad \left\{ \begin{array}{lll} rem(0/2)&{}= &{}0\\ rem(s(a)/2)&{}= &{}sg(rem(a/2)) \end{array} \right. \nonumber $$

$$ \begin{array}{lll} \bigl [ 0/2 \bigr ] &{}= &{}0 \\ \bigl [ s(a)/2 \bigr ] &{}=&{} \bigl [ a/2 \bigr ] +rem(a/2) \end{array} $$

Definition 3

( $| \_ |$ ) .

Definition 4

( $\ge$ ) .

Definition 5

(Surjection).

Definition 6

(Bijection).

Definition 7

(Finite).