How do Introduction-to-Proof Textbooks Explain Conditionals and Implications?

Abstract Conditionals are ubiquitous in mathematics: we routinely express theorems using universal conditionals of the form ‘for all $$x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> , if $$A(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> then $$B(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ’. The logic of universal conditionals is underpinned by that of propositional conditionals , which take the form ‘if $$A({x}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> then $$B({x}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ’, where $${x}_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is a specific object. In mathematics, propositional conditionals are subject to a material conditional interpretation: they are true unless $$A({x}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is true and $$B({x}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.