Abstract
We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.
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Alechina, N., & Gurevich, Y. (1997). Syntax vs. semantics on finite structures. Structures in Logic and Computer Science, 1997, 14–33.
Andréka H., Németi I. and van Benthem J. (1998). Modal logics and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3): 217–274
Barwise J. (1975). Admissible sets and structures. Springer, New York
Barwise, J., Feferman, S. (Eds.). (1985). Model-theoretic logics. New York: Springer.
Barwise J. and Perry J. (1983). Situations and attitudes. Bradford Books/The MIT Press, Cambridge, Mass
Barwise J. and van Benthem J. (1999). Interpolation, preservation pebble games. Journal of Symbolic Logic 64: 881–903
Beth E.W. (1953). On Padoa’s method in the theory of definition. Indagationes Mathematicae 15: 330–339
Bolzano, B. (1837). Wissenschaftslehre, Seidelsche Buchhandlung, Sulzbach. English translation, 1973. In J. Berg & B. Terrell (Eds.), Theory of science. Dordrecht: Reidel.
Bradfield J. and Stirling C. (2006). Modal μ-calculi. In: Blackburn, P. and Wolter, F. (eds) Handbook of modal logic, pp 721–756. Elsevier, Amsterdam
Craig W. (1957). Linear reasoning. A new form of the Herbrand-Gentzen theorem. Journal of Symbolic Logic 22(3): 250–268
d’Agostino, G. (1998). Modal logic and non-well founded set theory: Bisimulation, translation, and interpolation. Dissertation, Institute for Logic, Language and Computation, University of Amsterdam.
d’Agostino G. and Hollenberg M. (2000). Logical questions concerning the μ-calculus. Journal of Symbolic Logic 65: 310–332
d’Agostino G. and Lenzi G. (2005). An axiomatization of bisimulation quantifiers via the μ-calculus. Theoretical Computer Science 338(1–3): 64–95
Ebbinghaus H.-D. and Flum J. (1995). Finite model theory. Springer, Berlin
Fagin R. (1976). Probabilities on finite models. Journal of Symbolic Logic 41: 50–58
Feferman S. and Kreisel G. (1968). Persistent and invariant formulas for outer extensions. Compositio Mathematicae 20: 29–52
Glebskii Y.V., Kogan D.I., Liogonkii M.I. and Talanov V.A. (1969). Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics 5: 142–154
Hendriks, L. (1996).Computations in propositional logic. Dissertation DS-1996-01, Institute for Logic, Language and Computation, University of Amsterdam.
Hollenberg, M. (1998).Logic and bisimulation. Ph.D. thesis, University of Utrecht.
Hoogland, E., Marx, M., & Otto, M. (1999). Beth definability for the guarded fragment. In Proceedings of LPA R’99 (pp. 273–285). Berlin: Springer.
Lindström P. (1966). First-order predicate logic with generalized quantifiers. Theoria 32: 165–171
Lindström P. (1969). On extensions of elementary logic. Theoria 35: 1–11
Mason I. (1985). Undecidability of the meta-theory of the propositional calculus. Journal of Symbolic Logic 50: 451–457
Pitts A. (1992). On an interpretation of second order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57: 33–52
Schwichtenberg H. and Troelstra A. (1996). Basic proof theory. Cambridge University Press, Cambridge, UK
ten Cate, B. (2005).Model theory for extended modal languages. Ph.D. thesis, ILLC, Amsterdam.
Benthem J. (1985). The variety of logical consequence, according to Bolzano. Studia Logica 44(4): 389–403
van Benthem, J. (1997). Dynamic bits and pieces. Report LP-97-01, Institute for Logic, Language and Computation, University of Amsterdam.
van Benthem, J. (2003). Is there still logic in Bolzano’s key? In E. Morscher (Ed.), Bernard Bolzano’s Leistungen in Logik, Mathematik und Physik (pp. 11–34). Sankt Augustin: Academia Verlag.
Benthem J. (2005). Minimal predicates, fixed-points and definability. Journal of Symbolic Logic 70(3): 696–712
Benthem J. (2007). Inference in action. Publications de l’Institut Mathématique 82(96): 3–16
van Benthem, J., ten Cate, B., & Väänänen, J. (2007). Lindström theorems for fragments of first-order logic. In Proceedings LICS 2007.
Visser, A. (1996). Uniform interpolation and layered bisimulation. In Gödel ’96 Brno (pp. 139–164). Berlin: Springer.
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van Benthem, J. The many faces of interpolation. Synthese 164, 451–460 (2008). https://doi.org/10.1007/s11229-008-9351-5
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DOI: https://doi.org/10.1007/s11229-008-9351-5