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Minimal Predicates. Fixed-Points, and Definability

Johan Van Benthem
The Journal of Symbolic Logic
Vol. 70, No. 3 (Sep., 2005), pp. 696-712
Stable URL: http://www.jstor.org/stable/27588390
Page Count: 17
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Minimal Predicates. Fixed-Points, and Definability
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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 quarantees 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.

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