Journal of Symbolic Logic

Minimal predicates, fixed-points, and definability

Johan van Benthem

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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.

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J. Symbolic Logic, Volume 70, Issue 3 (2005), 696-712.

First available in Project Euclid: 22 July 2005

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van Benthem, Johan. Minimal predicates, fixed-points, and definability. J. Symbolic Logic 70 (2005), no. 3, 696--712. doi:10.2178/jsl/1122038910.

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