Conditionals are ubiquitous in mathematics: we routinely express theorems using universal conditionals of the form ‘for all \(x\) , if \(A(x)\) then \(B(x)\) ’. The logic of universal conditionals is underpinned by that of propositional conditionals, which take the form ‘if \(A({x}_{0})\) then \(B({x}_{0})\) ’, where \({x}_{0}\) is a specific object. In mathematics, propositional conditionals are subject to a material conditional interpretation: they are true unless \(A({x}_{0})\) is true and \(B({x}_{0})\) 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.
