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---
title: Quantification
TARGET DECK: Obsidian::STEM
FILE TAGS: logic::quantification
tags:
- logic
- quantification
---
## Overview
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A **quantifier** refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are $\exists$ and $\forall$, though others (such as the counting quantifier) are also used.
%%ANKI
Basic
What are the most common first-order logic quantifiers?
Back: $\exists$ and $\forall$
Reference: Gries, David.
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END%%
%%ANKI
Basic
What term refers to operators like $\exists$ and $\forall$?
Back: Quantifiers.
Reference: Gries, David.
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END%%
* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property.
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%%ANKI
Basic
What symbol denotes existential quantification?
Back: $\exists$
Reference: Gries, David.
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END%%
%%ANKI
Basic
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How many members in the domain of discourse must satisfy a property in existential quantification?
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Back: At least one.
Reference: Gries, David.
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END%%
%%ANKI
Basic
$\exists x : S, P(x)$ is shorthand for what?
Back: $\exists x, x \in S \land P(x)$
Reference: Gries, David.
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END%%
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%%ANKI
Basic
What term refers to $S$ in $\exists x : S, P(x)$?
Back: The domain of discourse.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction* , 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf ](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf ).
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END%%
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%%ANKI
Basic
What is the identity element of $\lor$?
Back: $F$
Reference: Gries, David.
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END%%
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* **Universal quantification** ($\forall$) asserts that every member of a set satisfies a property.
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%%ANKI
Basic
What symbol denotes universal quantification?
Back: $\forall$
Reference: Gries, David.
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END%%
%%ANKI
Basic
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How many members in the domain of discourse must satisfy a property in universal quantification?
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Back: All of them.
Reference: Gries, David.
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END%%
%%ANKI
Basic
$\forall x : S, P(x)$ is shorthand for what?
Back: $\forall x, x \in S \Rightarrow P(x)$
Reference: Gries, David.
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END%%
%%ANKI
Basic
What is the identity element of $\land$?
Back: $T$
Reference: Gries, David.
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END%%
%%ANKI
Cloze
{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}.
Reference: Gries, David.
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END%%
%%ANKI
Basic
How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification?
Back: $\neg \exists x : S, \neg P(x)$
Reference: Gries, David.
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END%%
%%ANKI
How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification?
Back: $\neg \forall x : S, \neg P(x)$
Reference: Gries, David.
END%%
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* **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property.
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%%ANKI
Basic
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What symbol denotes counting quantification (of *exactly* $k$ members)?
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Back: $\exists^{=k}$
Reference: Gries, David.
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END%%
%%ANKI
Basic
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What symbol denotes counting quantification (of *at least* $k$ members)?
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Back: $\exists^{\geq k}$
Reference: Gries, David.
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END%%
%%ANKI
Basic
How is $\exists x : S, P(x)$ written in terms of counting quantification?
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Back: $\exists^{\geq 1}\, x : S, P(x)$
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Reference: Gries, David.
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END%%
%%ANKI
Basic
How is $\forall x : S, P(x)$ written in terms of counting quantification?
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Back: Assuming $S$ has $k$ members, $\exists^{= k}\, x : S, P(x)$
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Reference: Gries, David.
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END%%
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%%ANKI
Cloze
Propositional logical operator: $\forall x, \forall y, P(x, y)$ {$\Leftrightarrow$} $\forall y, \forall x, P(x, y)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Cloze
Propositional logical operator: $\forall x, \exists y, P(x, y)$ {$\Leftarrow$} {$\exists y, \forall x, P(x, y)$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Cloze
Propositional logical operator: $\exists x, \forall y, P(x, y)$ {$\Rightarrow$} $\forall y, \exists x, P(x, y)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!-- ID: 1718327739978 -->
END%%
%%ANKI
Cloze
Propositional logical operator: $\exists x, \exists y, P(x, y)$ {$\Leftrightarrow$} $\exists y, \exists x, P(x, y)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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%%ANKI
Basic
When does $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$ hold true?
Back: Always.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
When does $\forall x, \exists y, P(x, y) \Rightarrow \exists y, \forall x, P(x, y)$ hold true?
Back: When there exists a $y$ that $P(x, y)$ holds for over all quantified $x$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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## Identifiers
Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free** . A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix** ), and the second consisting of no quantifiers (the **matrix** ).
%%ANKI
Basic
Prenex normal form consists of what two parts?
Back: The prefix and the matrix.
Reference: Gries, David.
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END%%
%%ANKI
Basic
How is the prefix of a formula in PNF formatted?
Back: As only quantifiers and bound variables.
Reference: Gries, David.
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END%%
%%ANKI
Basic
How is the matrix of a formula in PNF formatted?
Back: Without quantifiers.
Reference: Gries, David.
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END%%
%%ANKI
Basic
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Which identifiers in the following are bound? $\exists x, P(x) \land P(y)$
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Back: Just $x$.
Reference: Gries, David.
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END%%
%%ANKI
Basic
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Which identifiers in the following are free? $\exists x, P(x) \land P(y)$
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Back: Just $y$.
Reference: Gries, David.
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END%%
%%ANKI
Basic
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How is the following rewritten in PNF? $(\exists x, P(x)) \land (\exists y, Q(y))$
Back: $\exists x \;y, P(x) \land Q(y)$
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Reference: Gries, David.
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END%%
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%%ANKI
Basic
How is the following rewritten in PNF? $(\exists x, P(x)) \land (\forall y, Q(y))$
Back: N/A.
Reference: Gries, David.
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END%%
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## Bibliography
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* Gries, David.
* Oscar Levin, *Discrete Mathematics: An Open Introduction* , 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf ](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf ).
