## Unless

Mark Meckes recently wrote (private communication):

I’m teaching a fairly new transition course at Case this term, which involves explicitly teaching students the basics of mathematical English along with the obvious things like logic and proof techniques.  I had a student recently ask about how to interpret “A unless B”.  After a fairly lively discussion in class today, we couldn’t agree on the truth table for this statement, and concluded in the end that “unless” is best avoided in mathematical writing.  I checked the Handbook of Mathematical Discourse to see if you had anything to say about it there, but there isn’t an entry for it.  So, are you aware of a standard interpretation of “unless” in mathematical English?

I did not consider  “unless” while writing HMD.   What should be done to approach a subject like this is to

• think up examples  (preferably in a bull session with other mathematicians) and try to understand what they mean logically, then
• do an extensive research of the mathematical literature to see if you can find examples that do and do not correspond  with your tentative understanding.  (Usually you find other uses besides the one you thought of, and sometimes you will discover that what you came up with is completely wrong.)

What follows is an example of this process.

I can think of three possible meanings for “P unless Q”:

1.  “P if and only if not Q”,
2.  “not Q implies P”
3.  “not P implies Q”.

An example that satisfies (1) is “$latex x^2-x$ is positive unless $latex 0 \leq x \leq 1$”.  I have said that specific thing to my classes — calculus students tend not to remember that the parabola is below the line $latex y=x$ on that interval. (And that’s the way you should show them — draw a picture, don’t merely lecture.  Indeed, make them draw a picture.)

An example of (2) that is not an example of (1) is “$latex x^2-x$ is positive unless $latex x = 1/2$”.  I don’t think anyone would say that, but they might say “$latex x^2-x$ is positive unless, for example, $latex x = 1/2$”.  I would say that is a correct statement in mathematical English.  I guess the phrase “for example” translates into telling you that this is a statement of form “Q implies not P”, where Q is now “x = 1/2″.   Using the contrapositive, that is equivalent to “P implies not Q”, but that is neither (2) nor (3).

An example of (3) that is not an example of (1) is “$latex x^2-x$ is positive unless $latex -1 < x < 1$”.  I think that any who said that (among math people) would be told that they are wrong, because for example $latex (\frac{-1}{2})^2-\frac{-1}{2} = \frac{3}{4}$.  That reaction amounts to saying that (3) is not a correct interpretation of “P unless Q”.

Because of examples like these, my conjecture is that “P unless Q” means “P if and only if not Q”.  But to settle this point requires searching for “unless” in the math literature and seeing if you can find instances where “P unless Q” is not equivalent to “P if and only if not Q”.  (You could also see what happens with searching for “unless” and “example” close together.)

Having a discussion such as the above where you think up examples can give you a clue, but you really need to search the literature.  What I did with the Handbook is to search JStor, available online at Case.  I have to say I had definite opinions about several usages that were overturned during the literature search. (What “brackets” means is an example.)

My proxy server at Case isn’t working right now but when I get it repaired I will look into this question.