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Finite Partially-Ordered Quantifiers

✍ Scribed by Herbert B. Enderton

Publisher
John Wiley and Sons
Year
1970
Tongue
English
Weight
264 KB
Volume
16
Category
Article
ISSN
0044-3050

No coin nor oath required. For personal study only.

✦ Synopsis

This is to be read "For every x there is a y and for every u there is a v (depending only on u) such that y ( x , y, u , w) ." The precise meaning of this can be given in terms of SKOLEM functions; the above formula is semantically equivalent to the second-order formula

Such partially-ordered quantifiers may seem strange to those accustomed to linear quantifiers. We suggest the following explanation for linear expressions : Formal languages were patterned after natural languages. Natural languages are spoken. We speak in real time, and real time progresses linearly. Consequently formal languages were constructed with linear expressions. But formal languages are not spoken (at least not eaily). So there is no reason to be influenced by the linearity of time into being narrow-minded about formulas. And linearity is the ultimate in narrowness.

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## Abstract Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hi

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ON THE CONCEPT OF FORMALIZATION AND PARTIALLY ORDERED QUANTIFIERS.\* \* I would like to thank an anonymous referee for very useful comments on an earlier version of this paper and Krister Segerberg, Erik Stenius and in particular Risto Hilpinen and Patrick Sibelius for discussing the topic of this p