Modular Quantifiers

Abstract

Let 0 ≤ r < q. We will define a new kind of quantifier $\exists^{(q,r)}$. Informally, $\exists^{(q,r)} x\phi$ means “the number of positions x that satisfy Φ is congruent to r modulo q”. Formally, let w be a V-structure over A, and let $x \notin \nu$. We obtain $\left | w \right |$ different $\left ( \nu \cup \left \{ x \right \} \right )$-structures w’ by adjoining x to the second component of a letter of w. We define

$$w \vDash \exists ^{(q,r)} x\phi$$

if and only if the number of these w’ for which

$$w' \vDash \phi$$

is congruent to r modulo q.