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The diversity of quantifier prefixes1

  • H. Jerome Keisler (a1) and Wilbur Walkoe (a2)

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The Arithmetical Hierarchy Theorem of Kleene [1] states that in the complete theory of the standard model of arithmetic there is for each positive integer r a Σr0 formula which is not equivalent to any Πr0 formula, and a Πr0 formula which is not equivalent to any Πr0 formula. A Πr0 formula is a formula of the form

where φ has only bounded quantifiers; Πr0 formulas are defined dually.

The Linear Prefix Theorem in [3] is an analogous result for predicate logic. Consider the first order predicate logic L with identity symbol, countably many n-placed relation symbols for each n, and no constant or function symbols. A prefix is a finite sequence

of quantifier symbols ∃ and ∀, for example ∀∃∀∀∀∃. By a Q formula we mean a formula of L of the form

where v1, …, vr are distinct variables and φ has no quantifiers. A sentence is a formula with no free variables. The Linear Prefix Theorem is as follows.

Linear Prefix Theorem. Let Q and q be two different prefixes of the same length r. Then there is a Q sentence which is not logically equivalent to any q sentence.

Moreover, for each s there is a Q formula with s free variables which is not logically equivalent to any q formula with s free variables.

For example, there is an ∀∃∀∀∀∃ sentence which is not logically equivalent to any ∀∃∃∀∀∃ sentence, and vice versa. Recall that in arithmetic two consecutive ∃'s or ∀'s can be collapsed; for instance all ∀∃∀∀∀∃ and ∀∃∃∀∀∃ formulas are logically equivalent to Π40 formulas. But the Linear Prefix Theorem shows that in predicate logic the number of quantifiers in each block, as well as the number of blocks, counts.

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1

This research was supported in part by National Science Foundation grant GP-27633.

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[1]Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 4173.
[2]Ramsey, F. P., On a problem in formal logic, Proceedings of the London Mathematical Society, Series 2, vol. 30 (1928), pp. 338384.
[3]Walkoe, W., Finite partially-ordered quantification, this Journal, vol. 35 (1970), pp. 535555.

The diversity of quantifier prefixes1

  • H. Jerome Keisler (a1) and Wilbur Walkoe (a2)

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