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On the semantics of the Henkin quantifier

Published online by Cambridge University Press:  12 March 2014

Michał Krynicki  and
Alistair H. Lachlan
Michał Krynicki
Affiliation:
Institute of Mathematics, University of Warsaw, Poland
Alistair H. Lachlan
Affiliation:
Simon Fraser University, British Columbia, Canada
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Extract

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH was

Here (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this is

In [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 2 , June 1979 , pp. 184 - 200
Copyright
Copyright © Association for Symbolic Logic 1979

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References

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