I had the good fortune as an undergraduate to have a “bridge class” in my math curriculum. We learned basic logic, set manipulation, formal functions and relations, proof structure and induction. In that class our instructor had us read an article by Reuben Hirsch called “Math Lingo vs. Plain English: Double Entendre” (published in the American Mathematical Monthly and available from Hirsch’s publication page, “in the classroom” section). Much more recently I came across a followup column, which gives extra examples and references. As a bit of a language geek, after I was made aware of the key subtle differences, I tried to suss out the core and articulate it for myself – and teach my students about it as well.

Inclusive or is a low hurdle. In plain English, when someone asks “Should we take our vacation to New York or Boston?” the assumption is that the answer will be “New York” or “Boston” (or “I don’t care” or “neither”). The geeky joke – sometimes serious – answer of “yes” is totally unhelpful. However, it’s not too hard to get used to inclusive or, and we do have examples in natural language. One of the best is “Would you like sugar or cream in your coffee?” Of course, even then “yes” isn’t a useful answer, since there are three possible coffee fixings that would lead to it.

Implication is a much higher bar; there’s still a part of me, even, that doesn’t think A implies B means much of anything when A is false. Implications where there is clearly no causal relationship between A and B can be helpful, since they rarely appear in plain English (outside of statistical correlations, I suppose) and thus resist natural language intuition. For teaching, in addition to that, I settled on the approach of “an implication is true unless proven false.” You can only prove that it’s false by having A be true and B be false, so in the A-false situations the implication is therefore true. This is basically the conversion of “A implies B” into the disjunction “B or not-A,” but hopefully in a way that doesn’t just shift the confusion to a different location.

I think the base of that confusion is a disconnect between allowed truth values. In plain English, a sentence can be true, false, or nonsensical. The third option is not permitted in mathematics (except in the sense of ill-formed formulas), and it is confusing that many implications that seem nonsensical or are constructed from false clauses (“if the moon is made of green cheese, then fish swim in the sea;” “if the moon is made of green cheese, then carriages turn into pumpkins at midnight”) are logically true.

In the fifteen-plus years since my bridge class, I have found only one plain English example of an implication considered true but constructed with false clauses, and in general my students were unfamiliar with it: the adage “If wishes were horses, beggars would ride.” I would love another, even though nowadays it would be purely for my own interest.