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Quantifiers (Group 5)

Overview

Members

Short description of the topic

Quantifiers are words that precede and modify nouns; they indicate quantity.

In logic quantification quantifiers bind variables which means that a quantifier connects parts of a sentence which classify a domain of discourse.

There are different types of quantifiers which can be divided in:

Universal quantifiers (Logical quantifier)
Existential quantifiers (Logical quantifier)
Restricted quantifiers

Individual variables are written as "x" and can refer to any individual.

The Universal Quantifier ∀:

The Universal quantifier formalizes the notion that something is true for everything, or every relevant thing.
Symbol ∀ denotes given any or for all.
The sequence "∀x" is read as "For any value of x", "For all values of x" or "Whatever x may be".

The Existential Quantifier ∃:

An existential sentence states the existence of at least one thing of the domain.
Symbol ∃ stands for noun phrases with a/an, some or there exists sentences.
The sequence "∃x" is read as "There is an x" or "There is at least one thing x".

The Restricted Quantifier:

Restricted quantifiers point out a proportion of a set, not the proportion of everything there is.
Sentences which contain restricted quantifiers are written with square brackets.
Examples: some – several – many – most – a few – a number – one/two/three – no – no one – someone – all – every

Scopal Ambiguity:

Scopal ambiguity arises when a sentence contains two or more quantifiers. If this is the case the sentence can be understood in different ways. The difference in meaning can be clarified by expressing the sentence with the corresponding logical formulae. Each of these logical forms represents a particular meaning and thus cannot be ambiguous.

For further explanation and exercises please have a look at our Presentation and our Podcast.

Examples

The Universal Quantifier:

Every dog is barking.

∀x (DOG (x) → BARK (x))

"For every thing x, if x is a dog then x is barking."

The Existential Quantifier ∃:

Some birds were singing.

∃x (BIRD (x) & SING (x))

"There is at least one thing x such that x is a bird and x sings."

The Restricted Quantifier:

Several cars crashed.

[Several x: CAR (x)] CRASH (x)

Scopal Ambiguity:

Some students heard both concerts.

[Some x: STUDENT (x)] [Both y: CONCERT (y)] HEAR (x, y)

"There exist some students such that each of them heard both concerts."

OR

[Both y: CONCERT (y)] [Some x: STUDENT (x)] HEAR (x, y)

"Both concerts were such that each, individually, got heard by some students (but not necessarily the same ones)."

References and Links

References

Kearns, Kate (2000): Semantics. Basingstoke: Macmillan.
Swan, Michael (2005): Practical English Usage. 3rd edition. Oxford: Oxford University Press.
| Definition of Quantifiers, Determiners and Articles
Explanation of Scopal Ambiguity: http://www.philosophyetc.net/2004/08/scopal-ambiguity.html

Links

Definitions in the Basic Glossary:

Quantifiers

Our E-Learning Objects

Our Wiki Pages

Definitions in the Basic Glossary:

Our Podcasts

Podcast on Scopal Ambiguity:

Our Materials for an Interactive Whiteboard

Presentation on Quantifiers: Quantifiers.notebook

Our exercises

The following excercises provide links to their respective solutions. The answer to the question you just worked on is shown on the very top of the new website that pops up.

Due to long loading times, it might be useful to open the links via right-click, then choose "Open in new tab". In this way you can easily go back to the excercises without loading the group page again. However, at the bottom of each solution, there is a link that leads you back here.

Depending on the size of your PC-screen or the extent to which you zoomed in or out of your browser, you may see more than one answer per exercise. In your own interest, please try not to look at or remember those, if that should be the case.

Now, have fun with some excercises on quantifiers!

For further explanation and exercises please have a look at our Presentation and our Podcast.