Skip to content

Research at St Andrews

Geometrisation of first-order logic

Research output: Contribution to journalArticle


Open Access permissions



That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.


Original languageEnglish
Pages (from-to)123–163
Number of pages41
JournalBulletin of Symbolic Logic
Issue number2
Early online date4 Jun 2015
StatePublished - Jun 2015

    Research areas

  • Coherent implication, Coherent logic, Geometric logic, Weakly positive formula

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Intuitionistic decision procedures since Gentzen

    Dyckhoff, R. 5 May 2016 Advances in Proof Theory. Kahle, R., Strahm, T. & Studer, T. (eds.). Birkhäuser Basel, p. 245-267 (Progress in Computer Science and Applied Logic; vol. 28)

    Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

  2. POSIX lexing with derivatives of regular expressions (proof pearl)

    Ausaf, F., Dyckhoff, R. & Urban, C. 2016 Interactive Theorem Proving: 7th International Conference, ITP 2016, Nancy, France, August 22-25, 2016, Proceedings. Blanchette, J. C. & Merz, S. (eds.). Springer, p. 69-86 18 p. (Lecture Notes in Computer Science; vol. 9807)

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

  3. Some remarks on proof-theoretic semantics

    Dyckhoff, R. 2016 Advances in Proof-Theoretic Semantics. Piecha, T. & Schroeder-Heister, P. (eds.). Springer, p. 79-93 15 p. (Trends in Logic; vol. 43)

    Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

  4. Cut-elimination, substitution and normalisation

    Dyckhoff, R. 2015 Dag Prawitz on Proofs and Meaning. Wansing, H. (ed.). Springer, p. 163-187 (Outstanding Contributions to Logic; vol. 7)

    Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

  5. Algebra, proof theory and applications for an intuitionistic logic of propositions, actions and adjoint modal operators

    Dyckhoff, R., Sadrzadeh, M. & Truffaut, J. Nov 2013 In : ACM Transactions on Computational Logic. 14, 4, 38 p., 34

    Research output: Contribution to journalArticle

ID: 181704500