Logical Implications as Actions or Objects
Students can treat logical implications as actions or as objects. As actions, logical implications consist of three separate components: the hypothesis (P), the conclusion (Q), and the inference (arrow) connecting them. As objects, logical implications comprise a single logical structure. Students' treatment of logical implications, as actions or as objects, hold implications for how they might quantify, negate, or evaluate a logical implication.
Treating Logical Implications as Actions
Students who treat logical implications as actions might think about the statement “If P, then Q” as a command to operate: verify that P is true, and if so, conclude that Q is true (e.g., modus ponens).
Students who treat logical implications as actions might attempt to distribute quantifications and negations within the implication, applying the quantification/negation to the hypothesis or conclusion (see image below).
Treating Logical Implications as Objects
Students with an object conception have interiorized this way of operating, so they might meaningfully transform a logical implication by into its converse, contrapositive, or negation (e.g., modus tollens).
Students who treat logical implications as objects should be able to apply quantifications and negations to the implication as a whole; the quantification/negation applies to the implication itself--not to the hypothesis or conclusion.
Related Research Findings
- Students who treat logical implications as actions have a tendency to negate logical implications by distributing the negation to the conclusion (Norton et al., 2025).
- Students who treat logical implications as actions have a tendency to quantify logical implications by quantifying the hypothesis (Norton et al., 2025).
- Students who treat logical implications as actions have a tendency to assume the truth of the hypothesis in evaluating the truth of the implications, ignoring vacuous cases in whch the implication might be true because the hypothesis is false (Norton et al., 2025).
- Students who treat logical implications as objects can more readily address challenges accociated with the principle of mathematical induction (Arnold & Norton, 2017; Arnold et al., in press; Kokushkin et al., 2020; Norton & Arnold, 2017; Norton et al., 2022). Note that this finding fits a conjecture put forth by Dubinsky (1991).
- Students who treat logical implications as objects can further act on these objects by transforming them intro other forms (e.g., the contrapositive) and composing them to form new logical implications (Antonides et al., in review; Norton et al., 2025).
- Teachers might support students' construction of logical implications as objects by encouraging them to use Euler diagrams, wherein spatial transformations serve as proxies for logical transformations (Antonides et al., 2024).
- Students who treat logical implications as objects can more readily address epistemological obstacles such as the following (Norton et al., 2025):
- Treating logical implications as biconditional
- Trouble transforming logical implications into their converses, contrapoisitves, and negations
- Overlooking hidden quantification
- Diffuculty in representing logical implications with Euler diagrams.
If you're interested in reading more about students' treatment of logical implications and how this treatment relates to the epistemological obstacles they experience, please read our journal article linked below.