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Finite partially-ordered quantification

  • Wilbur John Walkoe (a1)

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In [3] Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification. One formulation of this sort of language is the closure of a first-order language under the formation rule that is a formula whenever φ is a formula and Q, which is to be thought of as a quantifier-prefix, is a system of partial order whose universe is a set of quantifiers. Although he introduced this idea in a discussion of infinitary logic, Henkin went on to discuss its application to finitary languages, and he concluded his discussion with a theorem of Ehrenfeucht that the incorporation of an extremely simple partially-ordered quantifier-prefix (the quantifiers x, y, v, and w, with the ordering {〈x, v〉, 〈y, w〉}) into any first-order language with identity gives a language capable of expressing the infinitary quantifier 0x.

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[1]Erdös, P. and Rado, R., A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427489.
[2]Erdös, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers, Acta mathematica, vol. 16 (1965), pp. 93196.
[3]Henkin, L., Some remarks on infinitely long formulas, Infinitistic methods, Proceedings of the Symposium on the Foundations of Mathematics, Warsaw, 1959, pp. 167183.

Finite partially-ordered quantification

  • Wilbur John Walkoe (a1)

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