Skip to main content
Log in

The road to two theorems of logic

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Beth E.W. (1953). On Padoa’s method in the theory of definitions. Indagationes Mathematicae 15: 330–339

    Google Scholar 

  • Büchi R.J. and Craig W. (1956). Notes on the family PC Δ of sets of models. Abstract. Journal of Symbolic Logic 21: 221–223

    Google Scholar 

  • Craig, W. (1951). A theorem about first order functional calculus with identity. Ph.D. thesis, Harvard, 242 pp.

  • Craig W. (1953). On axiomatizability within a system. Journal of Symbolic Logic 18: 30–32

    Article  Google Scholar 

  • Craig W. (1956). Review of E. W. Beth, On Padoa’s method in the theory of definitions. Journal of Symbolic Logic 21: 194–195

    Google Scholar 

  • Craig W. (1957a). Linear reasoning. A new form of the Herbrand–Gentzen Theorem. Journal of Symbolic Logic 22: 250–268

    Article  Google Scholar 

  • Craig W. (1957b). Three uses of the Herbrand–Gentzen Theorem in relating proof theory and model theory. Journal of Symbolic Logic 22: 269–285

    Article  Google Scholar 

  • Craig W. (1958). Replacement of auxiliary expressions. Philosophical Review 67: 38–55

    Google Scholar 

  • Craig W. (1960). Bases for first-order theories and subtheories. Journal of Symbolic Logic 25: 97–142

    Article  Google Scholar 

  • Craig W. and Vaught R. (1958). Finite axiomatizability using additional predicates. Journal of Symbolic Logic 25: 289–308

    Google Scholar 

  • Ebbinghaus, A. D. (1985). Extended logics. The general framework. In J. Barwise & S. Feferman (Eds.), Model-theoretic logics (p. x+893). Springer-Verlag.

  • Gentzen, G. (1934–1935). Untersuchungen über das logische Schliessen. Mathematicsche Zeitschrift, 39, 179–210, 405–431.

    Google Scholar 

  • Herbrand, J. (1930) Recherches sur la théorie de la démonstration. Travaux de la Société des Sciences et Lettres de Varsovie III, 128 pp.

  • Kleene S.C. (1952). Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Memoirs of the American Mathematical Society 10: 27–68

    Google Scholar 

  • Robinson A. (1956). A result ones consistency and its application to the theory of definitions. Indagationes Math 18: 17–58

    Google Scholar 

  • Turing A. (1936). On computable numbers with an application to the Entscheidungs problem. Proceedings of the London Mathematical Society ser. 2 42: 230–265

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to William Craig.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Craig, W. The road to two theorems of logic. Synthese 164, 333–339 (2008). https://doi.org/10.1007/s11229-008-9353-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-008-9353-3

Keywords

Navigation