Lara's Term Paper

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

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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.”

The Existential Quantifier ∃

• The existential quantifier is used to mean that the statement is true of at least one entity in the domain.
• symbolized by the operator ∃
• stands for expressions with a/an (one), some and there is
• The sequence “∃x” is read as
  • “There is an x”
  • “There is at least one thing x”

Negation:

• The special case of the determiner no(ne) is analyzed with ∃ and negation.
• The sequence “~∃x” (same as “¬∃x”) is read as
  • “There is no x”

Glossary entries

Links to additionally created glossary entries

Online excercises

Links to additionally created online exercises