Encyclopedia of Mathematics Education

2020 Edition
| Editors: Stephen Lerman

Logic in University Mathematics Education

  • Viviane Durand-GuerrierEmail author
  • Paul Christian Dawkins
Reference work entry
DOI: https://doi.org/10.1007/978-3-030-15789-0_100024


Mathematical logic generally addresses questions of reference and validity of inference. Questions of reference concern how statements in language refer to mathematical objects and under what conditions those statements are true or false.

Questions of inference concern how various statements in language are related to each other – such as equivalence, negation, and implication – and how statements in language may be verified by arguments, i.e., standards of proof.


Modern logic is in many ways the study of formal languages, which have been developed to deal with questions of reference and inference by taking in consideration the relationships between syntax and semantics. Part of the study of logic in undergraduate mathematics education should include students’ learning of such formal systems in model theory, proof theory, or computability. However, we are aware of no research-based evidence on the teaching and learning of such topics. Current research also...


University Mathematics Education Reference Inference Syntax Semantics Truth Validity 
This is a preview of subscription content, log in to check access.


  1. Barrier T (2011) Les pratiques langagières de validation des étudiants en analyse réelle. Recherches en Didactique des Mathématiques 31(3):259–290Google Scholar
  2. Dawkins PC, Cook JP (2017) Guiding reinvention of conventional tools of mathematical logic: students’ reasoning about mathematical disjunctions. Educ Stud Math 94(3):241–256.  https://doi.org/10.1007/s10649-016-9722-7CrossRefGoogle Scholar
  3. Dawkins PC, Roh KH (2016) Promoting metalinguistic and Metamathematical reasoning in proof-oriented mathematics courses: a method and a framework. International Journal of Research in Undergraduate Mathematics Education 2(2):197–222CrossRefGoogle Scholar
  4. Dubinsky E, Yiparaki O (2000) On students understanding of AE and EA quantification. Research in collegiate mathematics. Education IV. CBMS issues in mathematics education 8. American Mathematical Society, Providence, pp 239–289Google Scholar
  5. Durand-Guerrier V (2008) Truth versus validity in mathematical proof. ZDM 40(3):373–384CrossRefGoogle Scholar
  6. Durand-Guerrier V, Arsac G (2005) An epistemological and didactic study of a specific calculus reasoning rule. Educ Stud Math 60(2):149–172CrossRefGoogle Scholar
  7. Durand-Guerrier V, Boero P, Douek N, Epp S, Tanguay D (2012) Examining the role of logic in teaching proof. In: Hanna G, de Villiers M (eds) Proof and proving in mathematics education, vol 15, New ICMI study series. Springer, New York, p 369–389CrossRefGoogle Scholar
  8. Edwards B, Ward M (2008) The role of mathematical definitions in mathematics and in undergraduate mathematics courses. In: Carlson M, Rasmussen C (eds) Making the connection: research and teaching in undergraduate mathematics education MAA notes #73. Mathematics Association of America, Washington, DC, pp 223–232CrossRefGoogle Scholar
  9. Hawthorne C, Rasmussen C (2015) A framework for characterizing students’ thinking about logical statements and truth tables. Int J Math Educ Sci Technol 46(3):337–353CrossRefGoogle Scholar
  10. Hub A, Dawkins PC (2018) On the construction of set-based meanings for the truth of mathematical conditionals. J Math Behav 50:90–102CrossRefGoogle Scholar
  11. Schroyens W (2010) Logic and/in psychology: the paradoxes of material implication and psychologism in the cognitive science of human reasoning. In: Oaksford M, Chater N (eds) Cognition and conditionals: probability and logic in human thinking. Oxford University Press, Oxford, pp 69–84CrossRefGoogle Scholar
  12. Selden J, Selden A (1995) Unpacking the logic of mathematical statements. Educ Stud Math 29:123–151CrossRefGoogle Scholar
  13. Stenning K (2002) Seeing reason: image and language in learning to think. Oxford University Press, New YorkCrossRefGoogle Scholar
  14. Thurston WP (1994) On proof and progress in mathematics. Bull Am Math Soc 30(2):161–177CrossRefGoogle Scholar
  15. Weber K, Alcock L (2004) Semantic and syntactic proof production. Educ Stud Math 56:209–234CrossRefGoogle Scholar
  16. Wilkerson-Jerde MH, Wilensky UJ (2011) How do mathematicians learn math? Resources and acts for constructing and understanding mathematics. Educ Stud Math 78:21–43CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Viviane Durand-Guerrier
    • 1
    Email author
  • Paul Christian Dawkins
    • 2
  1. 1.IMAGUniv Montpellier, CNRSMontpellierFrance
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

Section editors and affiliations

  • Michèle Artigue
    • 1
  1. 1.Laboratoire de Didactique André Revuz (EA4434)Université Paris-DiderotParisFrance