PROCEEDINGS OF THE 18TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2015
Pittsburgh, Pennsylvania
Semantic and logical negation: Students’ interpretations of negative predicates
Page: 435
During exploratory teaching experiments intended to guide students to reinvent basic truthfunc- tional definitions for basic logical connectives, it unexpectedly emerged that undergraduate stu- dents reasoned about negation and negative properties in ways incompatible with the conventions and norms of mathematical practice. Specifically, our study participants often unpacked negative properties (not a rectangle) in terms of positive properties (is a parallelogram), which we call semantic negation. In this way, students did not readily adopt the mathematical assumption that the logical negation of a predicate designates the complement of the set of examples that satisfy that predicate. Students especially understood geometric sets of objects as being partitioned by familiar categories rather than those stipulated in a given statement. Semantic negation inhibited students systematizing activity regarding linguistic interpretation because their reasoning about disjunctions in various mathematical contexts depended intrinsically upon their understanding of particular topics so as to preclude abstractions approximating normative logical tools.