Finite partially-ordered quantification

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Abstract
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 Qφ 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 ∃zκ0x.

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