## Journal of Symbolic Logic

### Minimal predicates, fixed-points, and definability

Johan van Benthem

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 22 July 2005

https://projecteuclid.org/euclid.jsl/1122038910

Digital Object Identifier
doi:10.2178/jsl/1122038910

Mathematical Reviews number (MathSciNet)
MR2155262

Zentralblatt MATH identifier
1089.03010

#### Citation

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

#### References

• P. Aczel An introduction to inductive definitions, Handbook of mathematical logic (J. Barwise, editor), North-Holland, Amsterdam,1977, pp. 739--782.
• J. Barwise and J. van Benthem Interpolation, preservation, and pebble games, Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 881--903.
• J. van Benthem Modal logic and classical logic, Bibliopolis, Naples,1983.
• R. Benton A simple incomplete extension of T, Journal of Philosophical Logic, vol. 31 (2002), no. 6, pp. 527--541.
• P. Blackburn, M. de Rijke, and Y. Venema Modal logic, Cambridge University Press, Cambridge,2001.
• C. C. Chang and H. J. Keisler Model theory, North-Holland, Amsterdam,1973.
• H.C. Doets From logic to logic programming, MIT Press, Cambridge, MA,1994.
• H-D Ebbinghaus and J. Flum Finite model theory, Springer-Verlag, Berlin,1999.
• D. Gabbay and H.J. Ohlbach Quantifier elimination in second-order predicate logic, South African Computer Journal, vol. 7 (1992), pp. 35--43.
• V. Goranko and D. Vakarelov Elementary canonical formulas I. Extending Sahlqvist's theorem, Department of Mathematics, Rand Afrikaans University, Johannesburg,2003, and Faculty of Mathematics and Computer Science, Kliment Ohridski University, Sofia. Submitted for publication.
• J.M. Le Bars The 0-1 law fails for frame satisfiability of propositional modal logic, Proceedings of Logic in Computer Science,2002.
• B. Mahr and J. Makovsky Characterizing specification languages which admit initial semantics, Proceedings of the 8th CAAP, Springer, Berlin,1983.
• A. I. Mal$'$cev The metamathematics of algebraic systems, North-Holland, Amsterdam,1971.
• J. McCarthy Circumscription---a form of nonmonotonic reasoning, Artificial Intelligence, vol. 13 (1980), pp. 27--39.
• Y. N. Moschovakis Elementary induction on abstract structures, North-Holland, Amsterdam,1974.
• A. Nonnengart and A. Szałas A fixed-point approach to second-order quantifier elimination with applications to modal correspondence theory, Logic at work (E. Orlowska, editor), Physica-Verlag, Heidelberg,1999, pp. 89--108.
• M-A Papalaskari and S. Weinstein Minimal consequence in sentential logic, Journal of Logic Programming, vol. 9 (1990), pp. 19--31.
• C. Stirling Bisimulation, modal logic and model checking games, Logic Journal of the Interest Group in Pure and Applied Logics, vol. 7 (1999), no. 1, pp. 103--124.
• J. M. Weinstein First-order properties preserved by direct products, Ph.D. thesis, University of Wisconsin, Madison,1965.