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September 2005 Minimal predicates, fixed-points, and definability
Johan van Benthem
J. Symbolic Logic 70(3): 696-712 (September 2005). DOI: 10.2178/jsl/1122038910

Abstract

Minimal predicates P satisfying a given first-order description φ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ φ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

Citation

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Johan van Benthem. "Minimal predicates, fixed-points, and definability." J. Symbolic Logic 70 (3) 696 - 712, September 2005. https://doi.org/10.2178/jsl/1122038910

Information

Published: September 2005
First available in Project Euclid: 22 July 2005

zbMATH: 1089.03010
MathSciNet: MR2155262
Digital Object Identifier: 10.2178/jsl/1122038910

Rights: Copyright © 2005 Association for Symbolic Logic

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Vol.70 • No. 3 • September 2005
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