Boolean as subtype of integer In languages oriented towards systems programming, digital logic and hardware design, it's common to treat boolean as a subtype of integer. In languages oriented towards mathematics and type theory, it's more usual to treat them as disjoint types.

Being from more of a systems programming background, it has always seemed to me appropriate to take the former approach. I would have thought it appropriate even from a mathematical viewpoint, since we already have a hierarchy of numeric types (natural numbers are a subtype of integer which is a subtype of rational etc.) so it would fit neatly at the start of that hierarchy; but then I'm not a real mathematician, so I may be missing something.

Apart from a little bit of extra error checking (catching the mistake if you pass a boolean to a function expecting an integer and you didn't mean to), is there any advantage to having them be disjoint types?
If you only think of Booleans as 0 and 1, then it is natural to include them as subtypes of natural numbers and integers. But as soon as you start applying operations to them, then things start to fall apart. 

Consider what happens when you define the + operation on Booleans. You have two choices:


Make + the same as OR, so 1+1=1. But then you have the problem that + on Booleans is no longer the same as + on Integers, which it should be.  More precisely, we have the following $(Integer)1_{Bool}+ (Integer)1_{Bool}\neq (Integer)(1_{Bool}+1_{Bool})$, where $1_{Bool}$ is true and $(Integer)\_$ is performing a cast from Bools to Integers. This means that there is a problem with coherence in the subtyping relation is we make this choice.
The second choice is to define + on Booleans so that 1+1=2. But now the operation takes you out of the world of Booleans. Semantically, this is not problematic ($\sqrt{-1}$ takes you out of the world of Real numbers), but it does suggest that you do not gain much by treating Booleans as numbers.


Generally, the way Booleans/Integers are treated in languages like C is that Integers can be used in places where Booleans are expected, which does not really follow the usual subtyping rules. 

Ultimately, the operations you apply to Booleans are not the same as the ones you apply to Naturals and Integers, so, from the perspective of coherence, you should not really consider them to be related by subtyping.

Edit: 
A third alternative, suggested by Peter Taylor in comments, is to make + be XOR. This results in the subtype relation boolean &lt; short &lt; integer &lt; long, where + is addition-modulo-overflow. That's fairly natural, programmatically.