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== Definition == | == Definition == | ||
A ‘quantifier’ is a word like many, few, little, several, which is used in a noun phrase to indicate the quantity of something. | A ‘quantifier’ is a word like ''many'', ''few'', ''little'', ''several'', which is used in a noun phrase to indicate the quantity of something. | ||
In logic, quantifiers are operators binding variables, which means that a quantifier connects parts of a sentence that classify a domain of discourse. | In logic, quantifiers are operators binding variables, which means that a quantifier connects parts of a sentence that classify a domain of discourse. | ||
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* Restricted quantifiers | * Restricted quantifiers | ||
In predicate logic the two fundamental quantifiers are the logical quantifiers (also called generalized quantifiers), which are the universal quantifier and the existential quantifier. | In predicate logic the two fundamental quantifiers are the '''logical quantifiers''' (also called '''generalized quantifiers'''), which are the universal quantifier and the existential quantifier. | ||
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=== The Universal Quantifier ∀ === | === The Universal Quantifier ∀ === | ||
* The universal quantifier is used to mean that the statement is true for every entity in the domain in question. | * The universal quantifier is used to mean that the statement is true for '''every entity''' in the domain in question. | ||
* symbolized by the operator ∀ | * symbolized by the '''operator ∀''' | ||
* conveyed by such expressions as all, every and each | * conveyed by such expressions as '''''all''''', '''''every''''' and '''''each''''' | ||
* The sequence “∀x” is read as | * The sequence “∀x” is read as | ||
** “For every thing x” | ** ''“For every thing x”'' | ||
** “For any value of x” | ** ''“For any value of x”'' | ||
** “For all values of x” | ** ''“For all values of x”'' | ||
** “Whatever x may be” | ** ''“Whatever x may be”'' | ||
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Every dog is barking. | '''Every dog is barking.''' | ||
∀x (DOG (x) → BARK (x)) | ∀x (DOG (x) → BARK (x)) | ||
“For every thing x, if x is a dog then x is barking.” | ''“For every thing x, if x is a dog then x is barking.”'' | ||
All students were tired. | '''All students were tired.''' | ||
∀x (STUDENT (x) → TIRED (x)) | ∀x (STUDENT (x) → TIRED (x)) | ||
“For every thing x, if x is a student then x is tired.” | ''“For every thing x, if x is a student then x is tired.”'' | ||
NOCH EIN BEISPIEL | <span style="color:#00CC00">NOCH EIN BEISPIEL</span> | ||
NOCH EIN BEISPIEL | <span style="color:#00CC00">NOCH EIN BEISPIEL</span> | ||
'''Negation:''' | '''Negation:''' | ||
No trains are late. | '''No trains are late.''' | ||
Wherever you put an x, this statement does not hold: | Wherever you put an x, this statement does not hold: | ||
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∀x ~ (TRAIN (x) → LATE (x)) | ∀x ~ (TRAIN (x) → LATE (x)) | ||
“It is not the case that for all x, if x is a train then x is late.” | ''“It is not the case that for all x, if x is a train then x is late.”'' | ||
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~ ∃x (TRAIN (x) & LATE (x)) | ~ ∃x (TRAIN (x) & LATE (x)) | ||
“There is no x such that x is a train and x is late.” | ''“There is no x such that x is a train and x is late.”'' | ||
= Glossary entries = | = Glossary entries = | ||
Revision as of 20:08, 23 June 2016
Introduction to Quantifiers
The following e-learning material provides an introduction to the topic of quantifiers by integrating examples, illustrations as well as different types of interactive exercises of varying difficulty and their solutions. It intends to cover the central ideas of the concept of quantifiers without too much formal logic and therefore provides a basis for further study.
Author: Lara
(Back to the Group Overview or Hauptseminar: "New Media in Teaching Semantics")
Wiki pages
Definition
A ‘quantifier’ is a word like many, few, little, several, which is used in a noun phrase to indicate the quantity of something.
In logic, quantifiers are operators binding variables, which means that a quantifier connects parts of a sentence that classify a domain of discourse.
Individual variables are written as “x” and can refer to any individual.
Different Types of Quantifiers
There are different types of quantifiers which can be divided in:
- Logical quantifiers
- Universal quantifier
- Existential quantifier
- Restricted quantifiers
In predicate logic the two fundamental quantifiers are the logical quantifiers (also called generalized quantifiers), which are the universal quantifier and the existential quantifier.
Logical Quantifiers (Generalized Quantifiers)
T A B E L L E
The Universal Quantifier ∀
- The universal quantifier is used to mean that the statement is true for every entity in the domain in question.
- symbolized by the operator ∀
- conveyed by such expressions as all, every and each
- The sequence “∀x” is read as
- “For every thing x”
- “For any value of x”
- “For all values of x”
- “Whatever x may be”
EXAMPLES
The following examples show quantified sentences translated into predicate logic formulae.
Every dog is barking.
∀x (DOG (x) → BARK (x))
“For every thing x, if x is a dog then x is barking.”
All students were tired.
∀x (STUDENT (x) → TIRED (x))
“For every thing x, if x is a student then x is tired.”
NOCH EIN BEISPIEL
NOCH EIN BEISPIEL
Negation:
No trains are late.
Wherever you put an x, this statement does not hold:
∀x ~ (TRAIN (x) → LATE (x))
“It is not the case that for all x, if x is a train then x is late.”
Or equivalently, we can use the existential quantifier (see below) and say:
There is no x for which this statement would hold:
~ ∃x (TRAIN (x) & LATE (x))
“There is no x such that x is a train and x is late.”
Glossary entries
Links to additionally created glossary entries
Online excercises
Links to additionally created online exercises