How do introduction-to-proof textbooks explain conditionals and implications?

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(𝑥₀) then 𝐵(𝑥₀)’, where 𝑥₀ is a specific object. In mathematics, propositional conditionals are subject to a material conditional interpretation: they are true unless 𝐴(𝑥₀) is true and 𝐵(𝑥₀) 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.