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Relativised quantification: Some canonical varieties of sequence-set algebras

Published online by Cambridge University Press:  12 March 2014

Hajnal Andréka ,
Robert Goldblatt  and
István Németi
Hajnal Andréka
Affiliation:
Mathematical Institute of The Hungarian Academy of Sciences, P. O. Box 127, H-1364 Budapest, Hungary, E-mail: andreka@math-inst.hu
Robert Goldblatt
Affiliation:
School of Mathematical and Computing Sciences, Victoria University, P. O. Box 600, Wellington, New Zealand, E-mail: Rob.Goldblatt@vuw.ac.nz
István Németi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, P. O. Box 127, H-1364 Budapest, Hungary, E-mail: nemeti@math-inst.hu
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Extract

This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.

Standard first-order quantificational logic is modelled algebraically by cylindric algebras. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U for a language that is based on the set {υκ: κ < α} of variables, each formula φ is represented by the set

of all those α-length sequences x = 〈xκ: κ < α〉 that satisfy φ in U. Such a sequence provides a value-assignment to the variables (υκ is assigned value xκ), but it may also be viewed geometrically as a point in the α-dimensional Cartesian spaceαU of all α-length sequences whose terms come from the underlying set U of U. Then existential quantification is represented by the operation of cylindrification. To explain this, define a binary relation Tκ on sequences by putting xTκy if and only if x and y differ at most at their κth coordinate, i.e.,

Then for any set X ⊆ αU, the set

is the “cylinder” generated by translation of X parallel to the κth coordinate axis in αU. Given the standard semantics for the existential quantifier ∃υκ as

it is evident that

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 163 - 184
Copyright
Copyright © Association for Symbolic Logic 1998

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