# Definitions From [[Väänänen - Models and Games]] **Definition 10.1** A *weak (generalized) quantifier* is a mapping $Q$ which maps every non-empty set $A$ to a subset of $P(A)$. A weak (generalized) quantifier on a domain $A$ is any subset of $P(A)$. **Definition 10.8** A weak quantifier $Q$ is *(upwards) monotone* if $X ∈ Q(A)$ and $X ⊆ Y ⊆ A$ imply $Y ∈ Q(A)$. **Definition 10.18** If a weak (generalized) quantifier is bijection closed we drop “weak” and call it just a *(generalized) quantifier*. If a weak (generalized) quantifier on a domain is permutation closed we drop “weak” and call it just a *(generalized) quantifier*. ### From MTL chapter 2 section 4.2 pg 53 Fix a signature $\boldsymbol{\sigma}=\{S\}$ where $S$ is $l$-ary relation symbol, and $\mathfrak{R}$ a class of $\boldsymbol{\sigma}$-structures. The generalized quantifier $Q_{\mathfrak{R}}$ is defined by: $\mathfrak{A}\vDash Q_{\mathfrak{R}}\bar{x}\varphi(\bar{x},\bar{b})$ $\iff$ there is a $\boldsymbol{\sigma}$-structure $\mathfrak{C}=(A,S^{\mathfrak{C}})\in \mathfrak{R}$ such that $S^{\mathfrak{C}}= \{ \bar{a}\in A^{l} \mid \mathfrak{A}\vDash \varphi(\bar{a},\bar{b})\}$ $\iff$ $(A,\{ \bar{a}\in A^{l} \mid \mathfrak{A}\vDash \varphi(\bar{a},\bar{b})\}) \in \mathfrak{R}$ This generalizes to quantifiers of more complicated types. # Examples - [[Magidor-Malitz quantifier]] - [[Well-ordering quantifier]] - [[Härtig quantifier]] - [[Cofinality quantifier]] - [[Cardinality quantifier]] - [[Digital-garden/Definitions and Facts/Logic/Chromatic quantifier]]