A type theory mnemonic for boolean operator precedence

IN COMPUTER languages which support boolean operators like && (and) and || (or), && typically has higher precedence than ||. In other words, the following happens:

p && q || r == (p && q) || r
p || q && r == p || (q && r)

You can take advantage of this to construct simpler expressions which don't need so many parentheses. Eg, let's look at Cloudflare's expression language:

http.host eq "app.foo.com" and http.request.uri.path eq "/api" or http.host eq "api.foo.com" and http.request.uri.path eq "/app"

The above is equivalent to:

(http.host eq "app.foo.com" and http.request.uri.path eq "/api") or (http.host eq "api.foo.com" and http.request.uri.path eq "/app")

Let's leave aside for a moment the argument that you should use parentheses for explicit grouping and readability; readability is a very subjective question. The bigger problem might be that, unless you regularly design or use such logical expressions, it might be difficult to remember this operator precedence.

Math operator precedence

However, one precedence rule that most people remember pretty easily is that of math operators like + and * (yes, I'm using the programming language symbol * instead of the math symbol ✖️). We of course know, probably since childhood, that multiplication has higher precedence than addition. So,

a * b + c == (a * b) + c
a + b * c == a + (b * c)

Look familiar? These are the same precedence rules as for the logical operators shown above, where:

&& is equivalent to *
|| is equivalent to +

The type theory connection

Of course, type theory enthusiasts know pretty well that the symbols + and * are used to represent sum types and product types. The name is no coincidence: a sum type like a + b has the same number of values as the sum of the numbers of values of a and b. And a product type like a * b has the same number of values as the product of the numbers of values of a and b.

Try working through an example where type a = bool (2 possible values) and type b = unit (1 possible value). How many values does the type a + b have, and how many does a * b have?

In type theory, sum types represent alternatives, so a + b means 'a value that can be a or b', while a * b means 'a value that contains a and b' (commonly known as a tuple or pair). So in the type theory world,

* == &&
+ == ||

And very conveniently, they have the same precedence rules! So, if you are ever trying to remember the precedence, just remind yourself that 'and is times, or is plus'.