Skip to main content


  • ROY DYCKHOFF (a1) and SARA NEGRI (a2)

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.

Hide All
[1]Antonius W., Théories cohérentes et prétopos. Thèse de Maitrise ès Sciences (Mathématiques), Université de Montréal, Montreal, 1975.
[2]Avigad J., Dean E., and Mumma J., A formal system for Euclid’s elements. Review of Symbolic Logic, vol. 2 (2009), pp. 700767.
[3]Avron A., Gentzen-type systems, resolution and tableaux. Journal of Automated Reasoning, vol. 10 (1993), pp. 265281.
[4]Barr M., Toposes without points. Journal of Pure and Applied Algebra, vol. 5 (1974), pp. 265280.
[5]Bezem M., Final Report: Automating Coherent Logic—ACL., 2013.
[6]Bezem M. and Coquand T., Automating Coherent Logic, Proceedings of LPAR 2005, LNCS 3835, Springer, Berlin, 2005, pp. 246260.
[7]Bezem M., Coquand T., and Waaler A., Research Proposal: Automating Coherent Logic., 2006.
[8]Bezem M. and Hendricks T., On the mechanization of the proof of Hessenberg’s theorem in coherent logic. Journal of Automated Reasoning, vol. 40 (2008), pp. 6185.
[9]Blass A., Topoi and computation. Bulletin of the EATCS, vol. 36 (1988), pp. 5765.
[10]Blass A., Does Quantifier-Elimination Imply Decidability?, 2012.
[11]Castellini C. and Smaill A., A systematic presentation of quantified modal logics. Logic Journal of the IGPL, vol. 10 (2002), pp. 571599.
[12]Chagrov A. and Zakharyaschev M., Modal logic, Oxford University Press, Oxford, 1997.
[13]Ciabattoni A., Maffezioli P., and Spandier L., Hypersequent and Labelled Calculi for Intermediate Logics, In Tableaux 2013 Proceedings, LNCS 8123, Springer, Heidelberg, 2013, pp. 8196.
[14]van Dalen D, Logic and Structure, third ed., Springer, Berlin, 1997.
[15]Dyckhoff R., Contraction-free calculi for intuitionistic logic, this Journal, vol. 57 (1992), pp. 795807.
[16]Dyckhoff R., Implementations of Coherentisations of First-order Logic,, School of Computer Science, University of St Andrews, 2014.
[17]Dyckhoff R. and Lengrand S., LJQ: A Strongly Focused Calculus For Intuitionistic Logic, In CiE 2006 Proceedings, LNCS 3988, Springer, 2006, pp. 173185.
[18]Dyckhoff R. and Negri S., Proof analysis in intermediate logics. Archive for Mathematical Logic, vol. 51 (2012), pp. 7192.
[19]Dyckhoff R. and Negri S., An idempotent coherentisation algorithm. MS. In preparation, 2015.
[20]Fisher J., CoFOL Report and User Guide,∼jrfisher/colog2012/reports/coFOL.pdf, 20 April 2012.
[21]Fisher J. and Bezem M., Query Completeness of Skolem Machine Computations, Machines, Computations and Universality, 2007 Proceedings, LNCS 4664, Springer, 2007, pp. 182192.
[22]Fisher J. and Bezem M., Skolem machines, Fundamenta Informaticae, vol. 91 (2009), pp. 79103.
[23]Freyd P., Aspects of topoi. Bulletin of the Australian Mathematical Society, vol. 7 (1972), pp. 176.
[24]Harrison J., Handbook of Practical Logic and Automated Reasoning, Cambridge University Press, New York, NY, 2009.
[25]van Heijenoort J, From Frege to Gödel, Harvard University Press, Cambridge, MA, 1967.
[26]Hilbert D. and Bernays P., Foundations of mathematics I, Translated from German “Grundlagen der Mathematik I”, second ed., (1968) (C.-P. Wirth, editor), College Publications, London, 2011.
[27]Hodges W., Model Theory, Cambridge University Press, Cambridge, 1993.
[28]Holen B., Hovland D., and Giese M., Efficient Rule-Matching for Hyper-Tableaux, 9th International Workshop on Implementation of Logics Proceedings, EasyChair Proceedings in Computing Series, vol. 22, EasyChair, 2013, pp. 417.
[29]Jervell H., Thoralf Skolem: Pioneer of computational logic. Nordic Journal of Philosophical Logic, vol. 1 (1996), pp. 107117.
[30]Johnstone P., Stone Spaces, Cambridge University Press, Cambridge, 1982.
[31]Johnstone P., Sketches of an Elephant: A Topos Theory Companion, I and II, Oxford Logic Guides, vol. 43, 44, Oxford University Press, Oxford, 2002.
[32]López-Escobar E. G. K., An interpolation theorem for denumerably long formulae. Fundamenta Mathematicae, vol. 57 (1965), pp. 253272.
[33]Mac Lane S. and Moerdijk I., Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, New York, 1992.
[34]Maffezioli P., Naibo A., and Negri S., The Church-Fitch knowability paradox in the light of structural proof theory. Synthese, vol. 190 (2013), pp. 26772716.
[35]Makkai M. and Reyes G. E., First-Order Categorical Logic, Lecture Notes in Mathematics, vol. 611, Springer, Berlin, 1977.
[36]Marquis J. P. and Reyes G. E., The History of Categorical Logic: 1963–1977, Handbook of the History of Logic, vol. 6 (Sets and Extensions in the Twentieth Century), 2012, pp. 689800.
[37]Minker J., Overview of disjunctive logic programming. Annals of Mathematics and Artificial Intelligence, vol. 12 (1994), pp. 124.
[38]Mints G., Classical and Intuitionistic Geometric Logic. Talk at Conference on Philosophy, Mathematics, Linguistics: Aspects of Interaction 2012,, 2012.
[39]Negri S., Contraction-free sequent calculi for geometric theories, with an application to Barr’s theorem. Archive for Mathematical Logic, vol. 42 (2003), pp. 389401.
[40]Negri S., Proof analysis in modal logic. Journal of Philosophical Logic, vol. 34 (2005), pp. 507544.
[41]Negri S., Proof analysis beyond geometric theories: from rule systems to systems of rules. Journal of Logic and Computation,, 2014.
[42]Negri S., Proofs and countermodels in non-classical logics. Logica Universalis, vol. 8 (2014), pp. 2560.
[43]Negri S. and von Plato J., Structural Proof Theory, Cambridge University Press, Cambridge, 2001.
[44]Negri S. and von Plato J., Proof Analysis, Cambridge University Press, Cambridge, 2011.
[45]de Nivelle H and Meng J., Geometric Resolution: a Proof Procedure Based on Finite Model Search, Proceedings of IJCAR 2006, LNAI 4130, Springer, 2006, pp. 303317.
[46]OCaml: An industrial strength programming language supporting functional, imperative and object-oriented styles,
[47]Orevkov V. P., Glivenko’s Sequence Classes, Logical and logico-mathematical calculi 1, Proceedings of the Steklov Institute of Mathematics, vol. 98 (pp. 131–154 in Russian original), 1968, pp. 147–173.
[48]Palmgren E., An intuitionistic axiomatisation of real closed fields. Mathematical Logic Quarterly, vol. 48 (2002), pp. 297299.
[49]von Plato J, In the shadows of the Löwenheim-Skolem theorem: Early combinatorial analyses of mathematical proofs. Bulletin of Symbolic Logic, vol. 13 (2005), pp. 189225.
[50]Polonsky A., Proofs, types and lambda calculus. PhD thesis, University of Bergen, Bergen, 2011.
[51]Rathjen M., Notes on proof theory, Leeds University, Leeds, 2014. Unpublished MS.
[52]Reyes G. E., Sheaves and concepts: a model-theoretic interpretation of Grothendieck topoi. Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol. 18 (1977), pp. 105137.
[53]Rothmaler P., Introduction to Model Theory, Taylor and Francis, New York, 2000.
[54]Sacks G., Saturated Model Theory, W. A. Benjamin Inc, Reading MA, 1972 (Second ed., World Scientific, Singapore, 2009).
[55]Simpson A., The proof theory and semantics of intuitionistic modal logic. PhD thesis, Edinburgh University, Edinburgh, 1994.
[56]Skolem T., Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen, Skrifter I, vol. 4, Det Norske Videnskaps-Akademi, 1920, pp. 1–36. Also in [57, pp. 103–136]. Also (so far as §1 is concerned) in translation in [25, pp. 254–263].
[57]Skolem T., Selected Works in Logic (Fenstad J. E., editor), Universitetsforlaget, Oslo, 1970.
[58]Smullyan R. M., First-Order Logic. Corrected reprint of 1968 original. Dover Pubs. Inc., New York, 1995.
[59]Stojanović S., Pavlović V., and Janic̆ić P., A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs, Proceedings of Automated Deduction in Geometry 2010, LNAI 6877, Springer, Heidelberg, 2011, pp. 201220.
[60]Troelstra A. S. and Schwichtenberg H., Basic Proof Theory, second ed., Cambridge University Press, Cambridge, 2001.
[61]Wraith G., Generic Galois Theory of Local Rings, Proceedings of Applications of Sheaves, Durham 1977, LNM 753, Springer, New York, 1979, pp 739767.
[62]Wraith G., Intuitionistic Algebra: Some Recent Developments in Topos Theory, Proceedings of International Congress of Mathematics, Helsinki, 1978, pp. 331337.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 152 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 11th December 2017. This data will be updated every 24 hours.