Skip to main content

On the Importance of Set-Based Meanings for Categories and Connectives in Mathematical Logic


Based on data from a series of teaching experiments on standard tools of mathematical logic, this paper characterizes a range of student meanings for mathematical properties and logical connectives. Some observed meanings inhibited students’ adoption of logical structure, while others greatly facilitated it. Reasoning with predicates refers to students’ propensity to coordinate properties (e.g. “is a square” or “is not a square”) with the set of examples exhibiting the property (squares and non-squares). The negation/complement relation refers to students’ association of a negative property with the complement of the set associated with the corresponding positive property. These meanings afforded students efficient ways to reason about mathematical disjunctions in normative ways. The paper also provides accounts of how students who did not have these meanings reasoned about mathematical categories in ways that precluded normative logical structure. In particular, students frequently substituted positive categories for negative ones though they were not mathematically equivalent and overly relied in familiar categories learned in school, both forms of reasoning about properties. I conclude that proof-oriented instruction may need to help students develop set-based meanings and interpret negative claims in terms of set complements in order to appropriately interpret statements in ways compatible with mathematical logic.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Thompson et al. (2014) further distinguish whether a meaning is in-the-moment or stable based on whether the student is likely or not to assimilate other instances of an idea to the same meaning. Since we intend to set forth constructs for further study, we focus only on stable meanings we observed among our study participants.

  2. 2.

    All names are pseudonyms

  3. 3.

    Each turn is numbered for reference. “I” stands for the interviewer, who was the author. “…” denotes a pause while “[…]” denotes an omission.

  4. 4.

    According to formal logic, “∀x ∈ S , P(xor Q(x)” shares truth-values with “If not P(x) for x ∈ S, then Q(x).”

  5. 5.

    Horn and others dispute whether any principled distinction can be made between positive and negative claims, but this exceeds the scope of the present discussion.

  6. 6.

    I thank one anonymous reviewer for pointing out that this condition is necessary since “intersecting” and “parallel” are contraries in 3-dimensional space. As the reviewer noted, sometimes universal sets are implied rather than stated, leading to some ambiguity about negations and complements.

  7. 7.

    Teaching logic through guided reinvention was, at least in part, a research-oriented decision. By learning what meanings students need to construct to reinvent logic, one can better understand what learning all logic instruction should promote. I do not endorse that logic always be taught through guided reinvention, though I think there are some rich affordances there. In particular I think this lens forced me to consider “What is the mathematical activity that underlies logic?” I think future research on logic should attend more closely to grounding logic in students’ mathematical activity. I generally endorse teaching logic using meaningful mathematical language (see Dawkins and Cook 2017) rather than everyday language or pure logical syntax, but well-chosen everyday analogies can also be useful (Dawkins and Roh 2016).


  1. Alcock, L., & Simpson, A. (2002). Definitions: dealing with categories mathematically. For the Learning of Mathematics, 22(2), 28–34.

    Google Scholar 

  2. Antonini, S. (2001). Negation in mathematics: obstacles emerging from an exploratory study. In M van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Educatio (Vol. 2, pp. 49–56). Utrecht: PME.

  3. Barnard, T. (1995). The impact of ‘meaning’ on students' ability to negate statements. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th Conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 3–10). Recife: PME.

  4. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. H. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning (pp. 420–464). New York: Macmillan.

  5. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83–94.

  6. Dawkins, P. C., & Cook, J. P. (2017). Guiding reinvention of conventional tools of mathematical logic: Students’ reasoning about mathematical disjunctions. Educational Studies in Mathematics, 94(3), 241–256.

    Article  Google Scholar 

  7. Dawkins, P. C., & Roh, K. H. (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), 197–222.

    Article  Google Scholar 

  8. Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective. Educational Studies in Mathematics, 53(1), 5–34.

    Article  Google Scholar 

  9. Edwards, B., & Ward, M. (2008). The role of mathematical definitions in mathematics and in undergraduate mathematics courses. In M. Carlson & C. Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education MAA Notes #73 (pp. 223–232). Washington, DC: Mathematics Association of America.

  10. Epp, S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110, 886–899.

    Article  Google Scholar 

  11. Evans, J. (2005). Deductive reasoning. In K. J. Holyoak & R. G. Morrison (Eds.), Cambridge Handbook of Thinking and Reasoning (pp. 169–184). Cambridge: Cambridge University Press.

  12. Ferrari, P. L. (2004). Mathematical language and advanced mathematics learning. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 383–390). Bergen: PME.

  13. Freudenthal, H. (1973). Mathematics as an Educational task. Dordrecht: D. Reidel Publishing.

    Google Scholar 

  14. Freudenthal, H. (1991). Revisiting mathematics education: the China lectures. Dordrecht: Kluwer.

    Google Scholar 

  15. Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183–194.

    Article  Google Scholar 

  16. Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht: CD-β Press.

    Google Scholar 

  17. Horn, L. R. (1989). A natural history of negation. Chicago: University of Chicago Press.

    Google Scholar 

  18. Johnson-Laird, P. N. (1983). Mental models: Towards a cognitive science of language, inference, and consciousness. Cambridge: Cambridge University Press.

    Google Scholar 

  19. Johnson-Laird, P. N., & Byrne, R. M. J. (1991). Deduction. Hillsdale: Laurence Erlbaum Associates.

    Google Scholar 

  20. Johnson-Laird, P. N., & Byrne, R. M. J. (2002). Conditionals: a theory of meaning, pragmatics, and inference. Psychological Review, 109, 646–678.

    Article  Google Scholar 

  21. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

  22. Lockwood, E., Ellis, A. B., & Lynch, A. G. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. International Journal of Research in Undergraduate Mathematics Education, 2(2), 165–196.

    Article  Google Scholar 

  23. Murphy, G., & Hoffman, A. (2012). Concepts. In K. Frankish & W. Ramsey (Eds.), The Cambridge Handbook of Cognitive Science (pp. 151–170). New York: Cambridge University Press.

  24. Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63(3), 259–282.

    Article  Google Scholar 

  25. Piaget, J., & Garcia, R. (2011). Toward a logic of meanings. Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  26. Selden, A. (2012). Transitions and proof and proving at tertiary level. In G. Hanna & M. De Villiers (Eds.), Proof and Proving in Mathematics Education: The 19th ICMI Study (pp. 391–420). Dordrecht: Springer.

  27. Simon, M. A. (1996). Beyond inductive and deductive reasoning: the search for a sense of knowing. Educational Studies in Mathematics, 30(2), 197–210.

    Article  Google Scholar 

  28. Simon, M. A., Kara, M., Placa, N., & Sandir, H. (2016). Categorizing and promoting reversibility of mathematical concepts. Educational Studies in Mathematics, 93(2), 137–153.

  29. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research Design in Mathematics and Science Education (pp. 267–307). Hillsdale: Lawrence Erlbaum Associates.

  30. Stenning, K. (2002).Seeing reason: Image and language in learning to think. New York: Oxford University Press.

  31. Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate students' understanding of the contraposition equivalence rule in symbolic and verbal contexts. Educational Studies in Mathematics, 55, 133–162.

    Article  Google Scholar 

  32. Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: a hypothesis about foundational reasoning abilities in algebra. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.), Epistemic Algebraic Students: Emerging Models of Students' Algebraic Knowing (Vol. 4, pp. 1–24). Laramie: University of Wyoming.

  33. Von Glasersfeld, E. (1995). Radical constructivism: a way of knowing and learning. London: Falmer Press.

    Book  Google Scholar 

  34. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34–51.

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Paul Christian Dawkins.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dawkins, P.C. On the Importance of Set-Based Meanings for Categories and Connectives in Mathematical Logic. Int. J. Res. Undergrad. Math. Ed. 3, 496–522 (2017).

Download citation


  • Mathematical logic
  • Guided reinvention
  • Disjunctions
  • Quantification
  • Mathematical properties