Pronouncing "≤" constructively

Posted on May 13, 2016

A type XX with a relation :XX𝒫\le : X \to X \to \mathcal{P} is partially ordered if it obeys the following rules (reflexivity, transitivity, and antisymmetry):

xx\frac{ }{x \le x}

xyyzxz\frac{x \le y \qquad y \le z}{x \le z}

xyyxx=y\frac{x \le y \qquad y \le x}{x = y}

How are we to pronounce the symbol \le? In most contexts, I hear it pronounced as “less than or equal to”; that is, xyx \le y is read as “xx is less than or equal to yy”. Another alternative that I’ve heard is “not greater than”, which is a bit more concise.

Neither are particular concise for such an important concept, but more egregiously, from a constructive viewpoint, they each mean something different from \le! That is, we have the implications

x<yx=yxy¬(x>y),x < y \vee x = y \quad \Longrightarrow \quad x \le y \quad \Longrightarrow \quad \neg (x > y),

but not necessarily the other way around.
That is, if xx is less than or equal to yy, then certainly xyx \le y, and if xyx \le y, then xx is not greater than yy. But constructively, each of these implications is potentially strict. For instance, each of these implications is strict for the partial ordering on the real numbers \mathbb{R} as interpreted in Brouwer’s intuitionism.

Another way to read xyx \le y, perhaps in rarer use than the previously mentioned ways, is as “xx is at most yy”. I am surprised it has not caught on, given its concision. Moreover, it doesn’t have the classical bias of the other ones. Particularly, if you are working constructively, I strongly recommend saying “xx is at most yy” rather than the alternatives to avoid the cognitive dissonance of the other phrases to a constructive audience.

There is a preponderance of code abbreviating the relation of a partial order (that is, \le) as “le”, including LaTeX, Coq, and many other software tools and programming languages. But we need not perpetuate the mistake! The simple two-letter abbreviation “am”, for “as most”, is a fitting replacement, and its lexicographic neighbor, “al” (“at least”), can be used for \ge.