Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T15:27:00.469Z Has data issue: false hasContentIssue false
  • English
  • Français

ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION

Published online by Cambridge University Press:  21 June 2017

DAVIDE RINALDI ,
PETER SCHUSTER  and
DANIEL WESSEL
DAVIDE RINALDI
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI VERONA STRADA LE GRAZIE, 15 37134VERONA, ITALYE-mail: davide.rinaldi@univr.it
PETER SCHUSTER
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI VERONA STRADA LE GRAZIE, 15 37134VERONA, ITALYE-mail: peter.schuster@univr.it
DANIEL WESSEL
Affiliation:
DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DEGLI STUDI DI TRENTO VIA SOMMARIVE, 14 38123POVO (TN), ITALYE-mail: daniel.wessel@unitn.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.

To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.

Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.

Keywords

03F0303E2503F65syntactical conservationaxiom of choiceHilbert’s Programmedisjunction eliminationentailment relation
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 2 , June 2017 , pp. 181 - 200
Copyright
Copyright © The Association for Symbolic Logic 2017 

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

Aczel, P., Ishihara, H., Nemoto, T., and Sangu, Y., Generalized geometric theories and set-generated classes . Mathematical Structures in Computer Science, vol. 25 (2015), no. 7, pp. 1466–1483.CrossRefGoogle Scholar
Aczel, P. and Rathjen, M., Notes on constructive set theory , Technical report, Institut Mittag–Leffler, 2000/01, Report No. 40.Google Scholar
Aczel, P. and Rathjen, M., Constructive set theory , Book draft, 2010.Google Scholar
Artin, E., Über die Zerlegung definiter Funktionen in Quadrate . Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 5 (1927), no. 1, pp. 100–115.CrossRefGoogle Scholar
Artin, E. and Schreier, O., Algebraische Konstruktion reeller Körper . Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 5 (1927), no. 1, pp. 85–99.CrossRefGoogle Scholar
Avron, A., Simple consequence relations . Information and Computation, vol. 92 (1991), pp. 105–139.CrossRefGoogle Scholar
Basu, S., Pollack, R., and Roy, M.-F., Algorithms in Real Algebraic Geometry, Springer, Berlin, 2003.CrossRefGoogle Scholar
Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.Google Scholar
Bishop, E. and Bridges, D., Constructive Analysis, Springer, Berlin and Heidelberg, 1985.CrossRefGoogle Scholar
Bochnak, J., Coste, M., and Roy, M.-F., Real Algebraic Geometry, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
Cederquist, J. and Coquand, T., Entailment relations and distributive lattices , Logic Colloquium ’98. (Buss, S. R., Hájek, P., and Pudlák, P., editors), Lecture Notes Logic, vol. 13, A. K. Peters, Natick, MA, 2000, pp. 127–139.Google Scholar
Cederquist, J., Coquand, T., and Negri, S., The Hahn-Banach theorem in type theory , Twenty-five Years of Constructive Type Theory (Venice, 1995) (Sambin, G. and Smith, J. M., editors), Oxford Logic Guides, vol. 36, Oxford University Press, New York, 1998, pp. 57–72.Google Scholar
Ciraulo, F., Rinaldi, D., and Schuster, P., Lindenbaum’s lemma via open induction , Advances in Proof Theory (Kahle, R., Strahm, T., and Studer, T., editors), Progress in Computer Science and Applied Logic, vol. 28, Birkhäuser, Basel, 2016, pp. 65–77.CrossRefGoogle Scholar
Ciraulo, F. and Sambin, G., Finitary formal topologies and Stone’s representation theorem . Theoretical Computer Science, vol. 405 (2008), no. 1–2, pp. 11–23.CrossRefGoogle Scholar
Coquand, T., Two applications of Boolean models . Archive for Mathematical Logic, vol. 37 (1998), no. 3, pp. 143–147.CrossRefGoogle Scholar
Coquand, T., A direct proof of the localic Hahn-Banach theorem, 2000. Manuscript, available from author’s webpage, http://www.cse.chalmers.se/∼coquand/formal.html.Google Scholar
Coquand, T., Lewis Carroll, Gentzen and entailment relations, 2000. Manuscript, available from the author’s website, http://www.cse.chalmers.se/∼coquand/formal.html.Google Scholar
Coquand, T., Geometric Hahn-Banach theorem . Mathematical Proceedings of the Cambridge Philosophical Society, vol. 140 (2006), pp. 313–315.CrossRefGoogle Scholar
Coquand, T., Space of valuations . Annals of Pure and Applied Logic, vol. 157 (2009), pp. 97–109.CrossRefGoogle Scholar
Coquand, T. and Lombardi, H., Hidden constructions in abstract algebra (3): Krull dimension of distributive lattices and commutative rings , Commutative Ring Theory and Applications (Fontana, M., Kabbaj, S.-E., and Wiegand, S., editors), Lecture Notes in Pure and Applied Mathematics, vol. 231, Marcel Dekker, New York, 2002, pp. 477–499.Google Scholar
Coquand, T. and Lombardi, H., A logical approach to abstract algebra.Mathematical Structures in Computer Science, vol. 16 (2006), pp. 885–900.CrossRefGoogle Scholar
Coquand, T., Lombardi, H., and Neuwirth, S., Lorenzen’s theory of divisibility , preprint, 2016.Google Scholar
Coquand, T., Lombardi, H., and Neuwirth, S., Lattice-ordered groups generated by ordered groups and regular systems of ideals , preprint, 2017, https://arxiv.org/pdf/1701.05115.pdf.Google Scholar
Coquand, T. and Persson, H., Valuations and Dedekind’s Prague theorem . Journal of Pure and Applied Algebra, vol. 155 (2001), no. 2–3, pp. 121–129.CrossRefGoogle Scholar
Coquand, T. and Zhang, G.-Q., Sequents, frames, and completeness , Computer science logic 2000 (Clote, P. G. and Scwichtenberg, H., editors), Lecture Notes in Computer Science, vol. 1862, Springer, Berlin, 2000, pp. 277–291.Google Scholar
Coste, M., Lombardi, H., and Roy, M.-F., Dynamical method in algebra: Effective Nullstellensätze . Annals of Pure and Applied Logic, vol. 111 (2001), no. 3, pp. 203–256.CrossRefGoogle Scholar
Delzell, C. N., Kreisel’s Unwinding of Artin’s Proof , Kreiseliana. (Odifreddi, P., editor), A K Peters, Wellesley, MA, 1996, pp. 113–246.Google Scholar
Delzell, C. N., González-Vega, L., and Lombardi, H., A continuous and rational solution to Hilbert’s 17th problem and several cases of the Positivstellensatz , Computational Algebraic Geometry (Eyssette, F. and Galligo, A., editors), Birkhäuser, Boston, MA, 1993, pp. 61–75.CrossRefGoogle Scholar
Došen, K., On passing from singular to plural consequences , Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa (Orlowska, E., editor), Studies in Fuzziness and Soft Computing, vol. 24, Physica, Heidelberg, 1999, pp. 533–547.Google Scholar
Gabbay, D. M., Semantical Investigations in Heyting’s Intuitionistic Logic , Synthese Library, vol. 148, D. Reidel Publishing Co., Dordrecht-Boston, MA, 1981.Google Scholar
Gentzen, G., Untersuchungen über das logische Schließen I. Mathematische Zeitschrift, vol. 39 (1934), pp. 176–210.CrossRefGoogle Scholar
Gentzen, G., Untersuchungen über das logische Schließen II. Mathematische Zeitschrift, vol. 39 (1934), pp. 405–431.CrossRefGoogle Scholar
Hansson, B., Choice structures and preference relations . Synthese, vol. 18 (1968), no. 4, pp. 443–458.CrossRefGoogle Scholar
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme. I. Teil. Sätze ersten Grades . Mathematische Annalen, vol. 87 (1922), no. 3, pp. 246–269.CrossRefGoogle Scholar
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme. II. Teil. Sätze höheren Grades . Mathematische Annalen, vol. 89 (1923), no. 1, pp. 76–102.CrossRefGoogle Scholar
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme . Mathematische Annalen, vol. 101 (1929), no. 1, pp. 457–514.CrossRefGoogle Scholar
Humberstone, L., On a conservative extension argument of Dana Scott . Logic Journal of the IGPL, vol. 19 (2011), pp. 241–288.CrossRefGoogle Scholar
Humberstone, L., Dana Scott’s work with generalized consequence relations , Universal Logic: An Anthology - From Paul Hertz to Dov Gabbay (Béziau, J.-Y.), Studies in Universal Logic, Birkhäuser, Basel, 2012, pp. 263–279.CrossRefGoogle Scholar
Ishihara, H. and Nemoto, T., Non-deterministic inductive definitions and fullness , Concepts of Proof in Mathematics, Philosophy, and Computer Science (Probst, D. and Schuster, P., editors), Ontos Mathematical Logic, vol. 6, Walter de Gruyter, Berlin, 2016, pp. 163–170.CrossRefGoogle Scholar
Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium, vol. 1, Oxford Logic Guides, vol. 43, The Clarendon Press Oxford University Press, New York, 2002.Google Scholar
Krull, W., Idealtheorie in Ringen ohne Endlichkeitsbedingung . Annals of Mathematics, vol. 101 (1929), pp. 729–744.CrossRefGoogle Scholar
Lombardi, H., Le contenu constructif d’un principe local-global avec une application à la structure d’un module projectif de type fini , Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres, 1997, Fascicule 94–95 & 95–96.Google Scholar
Lombardi, H., Relecture constructive de la théorie d’Artin-Schreier . Annals of Pure and Applied Logic, vol. 91 (1998), pp. 59–92.CrossRefGoogle Scholar
Lombardi, H., Hidden constructions in abstract algebra. I. Integral dependance . Journal of Pure and Applied Algebra, vol. 167 (2002), pp. 259–267.CrossRefGoogle Scholar
Lombardi, H., Algèbre dynamique, espaces topologiques sans points et programme de Hilbert . Annals of Pure and Applied Logic, vol. 137 (2006), pp. 256–290.CrossRefGoogle Scholar
Lombardi, H. and Quitté, C., Commutative Algebra: Constructive Methods: Finite Projective Modules, Springer, Netherlands, Dordrecht, 2015.CrossRefGoogle Scholar
Lorenzen, P., Über halbgeordnete Gruppen . Mathematische Zeitschrift, vol. 52 (1950), no. 1, pp. 483–526.CrossRefGoogle Scholar
Lorenzen, P., Algebraische und logistische Untersuchungen über freie Verbände . The Journal of Symbolic Logic, vol. 16 (1951), no. 2, pp. 81–106.CrossRefGoogle Scholar
Lorenzen, P., Teilbarkeitstheorie in Bereichen . Mathematische Zeitschrift, vol. 55 (1952), no. 3, pp. 269–275.CrossRefGoogle Scholar
Lorenzen, P., Die Erweiterung halbgeordneter Gruppen zu Verbandsgruppen . Mathematische Zeitschrift, vol. 58 (1953), no. 1, pp. 15–24.CrossRefGoogle Scholar
McKubre-Jordens, M., Material implications over minimal logic (joint work with Hannes Diener) , Conference Presentation, May 2016, Mathematics for Computation, Benediktinerabtei Niederaltaich, Germany, 8–13 May 2016.Google Scholar
Mines, R., Richman, F., and Ruitenburg, W., A Course in Constructive Algebra, Universitext, Springer, New York, 1988.CrossRefGoogle Scholar
Mulvey, C. J. and Wick-Pelletier, J., The dual locale of a seminormed space . Cahiers de topologie et géométrie différentielle catégoriques, vol. 23 (1982), no. 1, pp. 73–92.Google Scholar
Mulvey, C. J. and Wick-Pelletier, J., A globalization of the Hahn-Banach theorem . Advances in Mathematics, vol. 89 (1991), pp. 1–59.CrossRefGoogle Scholar
Negri, S., Stone bases alias the constructive content of Stone representation , Logic and Algebra. (Ursini, A. and Aglianò, P., editors), Lecture Notes in Pure and Applied Mathematics, vol. 180, Marcel Dekker, New York, 1996, pp. 617–636.Google Scholar
Negri, S., Continuous domains as formal spaces . Mathematical Structures in Computer Science, vol. 12 (2002), no. 1, pp. 19–52.CrossRefGoogle Scholar
Negri, S. and von Plato, J., Proof Analysis, Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Negri, S., von Plato, J., and Coquand, T., Proof-theoretical analysis of order relations . Archive for Mathematical Logic, vol. 43 (2004), pp. 297–309.CrossRefGoogle Scholar
Payette, G. and Schotch, P. K., Remarks on the Scott–Lindenbaum theorem . Studia Logica, vol. 102 (2014), no. 5, pp. 1003–1020.CrossRefGoogle Scholar
Prestel, A. and Delzell, C. N., Positive Polynomials. From Hilbert’s 17th Problem to Real Algebra, Springer-Verlag, Berlin, 2001.Google Scholar
Raoult, J.-C., Proving open properties by induction . Information Processing Letters, vol. 29 (1988), no. 1, pp. 19–23.CrossRefGoogle Scholar
Rinaldi, D., Formal Methods in the Theories of Rings and Domains, Doctoral dissertation, Universität München, 2014.Google Scholar
Rinaldi, D. and Schuster, P., A universal Krull-Lindenbaum theorem . Journal of Pure and Applied Algebra, vol. 220 (2016), pp. 3207–3232.CrossRefGoogle Scholar
Sambin, G., Intuitionistic formal spaces—a first communication , Mathematical Logic and its Applications (Skordev, D., editor), Plenum, New York, 1987, pp. 187–204.CrossRefGoogle Scholar
Schuster, P., Induction in algebra: A first case study , 27th Annual ACM/IEEE Symposium on Logic in Computer Science, IEEE Computer Society Publications, 2012, Proceedings, LICS 2012, Dubrovnik, Croatia, pp. 581–585.Google Scholar
Schuster, P., Induction in algebra: A first case study . Logical Methods in Computer Science, vol. 9 (2013), no. 3, p. 20.Google Scholar
Schwichtenberg, H. and Senjak, C., Minimal from classical proofs . Annals of Pure and Applied Logic, vol. 164 (2013), pp. 740–748.CrossRefGoogle Scholar
Scott, D., On engendering an illusion of understanding . Journal of Philosophy, vol. 68 (1971), pp. 787–807.CrossRefGoogle Scholar
Scott, D., Completeness and axiomatizability in many-valued logic , Proceedings of the Tarski Symposium (Henkin, L., Addison, J., Chang, C. C., Craig, W., Scott, D., and Vaught, R., editors), American Mathematical Society, Providence, RI, 1974, pp. 411–435.CrossRefGoogle Scholar
Scott, D. S., Background to formalization , Truth, Syntax and Modality (Leblanc, H., editor), Studies in Logic and the Foundations of Mathematics, vol. 68, North-Holland, Amsterdam, 1973, pp. 244–273.CrossRefGoogle Scholar
Shoenfield, J. R., Mathematical Logic, Addison-Wesley, Reading, MA, 1967.Google Scholar
Shoesmith, D. J. and Smiley, T. J., Multiple-Conclusion Logic, Cambridge University Press, Cambridge, 1978.CrossRefGoogle Scholar
Szpilrajn, E., Sur l’extension de l’ordre partiel . Fundamenta Mathematicae, vol. 16 (1930), pp. 368–389.CrossRefGoogle Scholar
Tarski, A., Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften . I. Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 361–404.CrossRefGoogle Scholar
Troelstra, A. S. and Schwichtenberg, H., Basic Proof Theory, second ed., Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar
van den Berg, B., Non-deterministic inductive definitions . Archive for Mathematical Logic, vol. 52 (2013), no. 1–2, pp. 113–135.CrossRefGoogle Scholar
Wójcicki, R., Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library, vol. 199, Kluwer Academic Publishers Group, Dordrecht, 1988.CrossRefGoogle Scholar
Yengui, I., Constructive Commutative Algebra. Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases, Lecture Notes in Mathematics, vol. 2138, Springer International Publishing, Switzerland, 2015.CrossRefGoogle Scholar