Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T12:31:11.770Z Has data issue: false hasContentIssue false
  • English
  • Français

Barwise: Abstract Model Theory and Generalized Quantifiers

Published online by Cambridge University Press:  15 January 2014

Jouko Väänänen
Jouko Väänänen*
Affiliation:
Department of Mathematics, University of Helsinki, FinlandE-mail: , jouko.vaananen@helsinki.fi, URL: www.math.helsinki.?/~logic/people/juoko.vaananen/
Get access Rights & Permissions [Opens in a new window]

Extract

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties are

κ-compactness. Any set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model. Löwenheim-Skolem Theorem down to κ. If a sentence of the logic has a model, it has a model of cardinality at most κ. Interpolation Property. If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.

Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languages Lκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 1 , March 2004 , pp. 37 - 53
Copyright
Copyright © Association for Symbolic Logic 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Barwise, Jon, Infinitary logic and admissible sets, The Journal of Symbolic Logic, vol. 34 (1969), no. 2, pp. 226–252.CrossRefGoogle Scholar
[2] Barwise, Jon, Absolute logics and L∞ω , Annals of Mathematical Logic, vol. 4 (1972), pp. 309–340.CrossRefGoogle Scholar
[3] Barwise, Jon, Axioms for abstract model theory, Annals of Mathematical Logic, vol. 7 (1974), pp. 221–265.Google Scholar
[4] Barwise, Jon, Some applications of Henkin quantifiers, Israel Journal of Mathematics, vol. 25 (1976), no. 1–2, pp. 47–63.CrossRefGoogle Scholar
[5] Barwise, Jon, On branching quantifiers in English, Journal of Philosophical Logic, vol. 8 (1979), no. 1, pp. 47–80.Google Scholar
[6] Barwise, Jon, The role of the omitting types theorem in infinitary logic, Archiv für mathematische Logik und Grundlagenforschung, vol. 21 (1981), no. 1–2, pp. 55–68.Google Scholar
[7] Barwise, Jon and Cooper, Robin, Generalized quantifier and natural language, Linguistics and Philosophy, vol. 4 (1981), pp. 159–219.CrossRefGoogle Scholar
[8] Barwise, Jon and Feferman, Solomon (editors), Model-theoretic logics, Perspectives in Mathematical Logic, Springer-Verlag, New York, 1985.Google Scholar
[9] Barwise, Jon, Kaufmann, Matt, and Makkai, Michael, Stationary logic, Annals of Mathematical Logic, vol. 13 (1978), no. 2, pp. 171–224.CrossRefGoogle Scholar
[10] Barwise, Jon, A correction to: “Stationary logic”, Annals of Mathematical Logic, vol. 20 (1981), no. 2, pp. 231–232.Google Scholar
[11] Burgess, John P., Descriptive set theory and infinitary languages, Zbornik Radova, vol. 2 (1977), no. 10, pp. 9–30, Set theory, foundations of mathematics (Proc. Sympos., Belgrade, 1977).Google Scholar
[12] Chang, Chen-Chung, A note on the two cardinal problem, Proceedings of the American Mathematical Society, vol. 16 (1965), pp. 1148–1155.CrossRefGoogle Scholar
[13] Feferman, Solomon, Two notes on abstract model theory. II. Languages for which the set of valid sentences is semi-invariantly implicitly definable, Fundamenta Mathematicæ, vol. 89 (1975), no. 2, pp. 111–130.Google Scholar
[14] Fraïssé, Roland, Étude de certains opérateurs dans les classes de relations, définis à partir d'isomorphismes resteints, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 59–75.Google Scholar
[15] Friedman, Harvey, One hundred and two problems in mathematical logic, The Journal of Symbolic Logic, vol. 40 (1975), pp. 113–129.Google Scholar
[16] Härtig, Klaus, Über einen Quantifikator mit zwei Wirkungsbereichen, Colloq. Found. Math., Math. machines andappl. (Tihany, 1962) (Budapest), Akad. Kiadó, 1965, pp. 31–36.Google Scholar
[17] Hella, Lauri and Luosto, Kerkko, Finite generation problem an n-ary quantifiers, Quantifiers, logics, models and computation (Krynicki, M., Mostowski, M., and Szczerba, L., editors), vol. 248, Kluwer Academic Publishers, 1995, pp. 63–104.CrossRefGoogle Scholar
[18] Hutchinson, John E., Model theory via set theory, Israel Journal of Mathematics, vol. 24 (1976), no. 3-4, pp. 286–304.CrossRefGoogle Scholar
[19] Karp, Carol R., Finite-quantifier equavalence, Theory of models (Proc. 1963 internat. sympos. Berkeley) (Amsterdam), North-Holland, 1965, pp. 407–412.Google Scholar
[20] Keisler, H. Jerome, Logic with the quantifier “there exist uncountable many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93.CrossRefGoogle Scholar
[21] Keisler, H. Jerome and Knight, Julia, Barwise: Infinitary logic and admissible sets, this Bulletin, vol. 10, no. 1, pp. 4–36, this issue.Google Scholar
[22] Lindström, Per, First order predicate logic with generalized quantifiers, Theoria, vol. 32 (1966), pp. 186–195.Google Scholar
[23] Lindström, Per, On extensions of elementary logic, Theoria, vol. 35 (1969), pp. 1–11.Google Scholar
[24] Lindström, Per, Prologue, Quantifiers, logics, models and computation (Krynicki, M., Mostowski, M., and Szczerba, L., editors), vol. 248, Kluwer Academic Publishers, 1995, pp. 21–24.Google Scholar
[25] Lopez-Escobar, E. G. K., An addition to: “On defining well-orderings”, Fundamenta Mathematicæ, vol. 59 (1966), pp. 299–300.Google Scholar
[26] Lopez-Escobar, E. G. K., On defining well-orderings, Fundamenta Mathematicæ, vol. 59 (1966), pp. 13–21.Google Scholar
[27] Makowsky, Johann A., Securable quantifiers, k-unions and admissible sets, Logic colloquium '73 (Bristol, 1973) (Amsterdam), Studies in Logic and the Foundations of Mathematics, vol. 80, North-Holland, 1975, pp. 409–428.Google Scholar
[28] Makowsky, Johann A. and Shelah, Saharon, The theorems of Beth and Craig in abstract model theory. II. Compact logics, Archiv für mathematische Logik und Grundlagenforschung, vol. 21 (1981), no. 1–2, pp. 13–35.CrossRefGoogle Scholar
[29] Mostowski, Andrzej, On a generalization of quantifiers, Fundamenta Mathematicæ, vol. 44 (1957), pp. 12–36.Google Scholar
[30] Mostowski, Andrzej, Craig's interpolation theorem in some extended systems of logic, Logic, methodology and philosophy ofscience III (Proc. third internat. congr., Amsterdam, 1967) (Amsterdam), North-Holland, 1968, pp. 87–103.Google Scholar
[31] Oikkonen, Juha, Hierarchies of model-theoretic definability—an approach to second-order logics, Essays on mathematical and philosophical logic (Proc. fourth Scandinavian logic sympos. and first Soviet-Finnish logic conf., Jyväskylä, 1976), Synthese Library, vol. 122, Reidel, Dordrecht, 1979, pp. 197–225.Google Scholar
[32] Rescher, Nicholas, Quantifiers in many-valued logic, Logique et Analyse (N.S.), vol. 7 (1964), pp. 181–184.Google Scholar
[33] Shelah, Saharon, Generalized quantifiers and compact logic, Transactions of the American Mathematical Society, vol. 204 (1975), pp. 342–364.Google Scholar
[34] Shelah, Saharon, Remarks in abstract model theory, Annals of Pure and Applied Logic, vol. 29 (1985), no. 3, pp. 255–288.Google Scholar
[35] Shelah, Saharon,The pair (ℵ n , ℵ0) may fail ℵ0-compactness, Proceedings of the logic colloquium 2001, Lecture Notes in Logic, Association for Symbolic Logic, to appear.Google Scholar
[36] Shelah, Saharon and Väänänen, Jouko, Δ(l(a1)) is not finitely generated, assuming CH, to appear.Google Scholar
[37] Shelah, Saharon, A new infinitary logic, to appear.Google Scholar
[38] Vaught, Robert L., The completeness of logic with the added quantifier “there are uncountably many”, Fundamenta Mathematicæ, vol. 54 (1964), pp. 303–304.Google Scholar