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Published online by Cambridge University Press: 01 June 2011
Introduction
First-order logic is not able to express “there exists infinitely many x such that …” nor “there exists uncountably many x such that …”. Also, if we restrict ourselves to finite models, first-order logic is not able to express “there exists an even number of x such that …”. These are examples of new logical operations called generalized quantifiers. There are many others, such as the Magidor–Malitz quantifiers, cofinality quantifiers, stationary logic, and so on. We can extend first-order logic by adding such new quantifiers. In the case of “there exists infinitely many x such that …” the resulting logic is not axiomatizable, but in the case of “there exists uncountably many x such that …” the new logic is indeed axiomatizable. The proof of the Completeness Theorem for this quantifier is non-trivial going well beyond the Completeness Theorem of first-order logic.
Generalized Quantifiers
Generalized quantifiers occur everywhere in our language. Here are some examples:
Two thirds voted for John
Exactly half remains.
Most wanted to leave.
Some but not all liked it.
Between 10% and 20% were students.
Hardly anybody touched the cake.
The number of white balls is even.
There are infinitely many primes.
There are uncountably many reals.
These are instances of generalized quantifiers in natural language. The mathematical study of quantifiers provides an exact framework in which such quantifiers can be investigated.
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