How do Introduction-to-Proof Textbooks Explain Conditionals and Implications?
Year: 2025
Type: article
Abstract: Conditionals are ubiquitous in mathematics: we routinely express theorems using universal conditionals of the form ‘for all $$x$$ x , if $$A(x)$$ A ( x ) then $$B(x)$$ B ( x ) ’. The logic of universal conditionals is underpinned by that of propositional conditionals , which take the form ‘if $$A({x}_{0})$$ A ( x 0 ) then $$B({x}_{0})$$ B ( x 0 ) ’, where $${x}_{0}$$ x 0 is a specific object. In mathematics, propositional conditionals are subject to a material conditional interpretation: they are true unless $$A({x}_{0})$$ A ( x 0 ) is true and $$B({x}_{0})$$ B ( x 0 ) is false. This, unfortunately, makes them peculiar in relation to natural language. Moreover, distinctions between propositional conditionals, universal conditionals, and implications are not always clear. How do introduction-to-proof textbooks deal with these issues? We address this question via a theoretically driven qualitative analysis of 17 texts commonly recommended at UK and US universities. We report on how these texts explain conditionals/implications, how they deal with the peculiarities of the material conditional, and how they discuss related language and reasoning. We then present a theoretical analysis of ambiguities that might leave a student confused, arguing that these arise due to the pragmatics of mathematical communication.
Authors: Lara Alcock, Rentuya Sa
Institutions: Loughborough University, University of Pittsburgh
Language: English
Subfield: Education
Field: Social Sciences
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $3,390
OpenAlex ID: W4408413558