Well, the title is a little bit misleading actually. Commutativity would not be desirable if we want to return one of the operands, because

("1" | 1) === "1"

but

(1 | "1") === 1

IOW, if we do want to return one of the operands, short-circuiting would make commutativity impossible to achieve.

For <a> & <b>

Here's one possible specification for it: it should be equivalent (or even rewrite) to

gensym := <a>; if gensym then <b> else gensym end

Note that <a> could have a side effect (e.g., appending an element to an array), so we need := so that the expression is evaluated only once, not twice. gensym refers to any new symbol that does not occur in the entire program.

Here's another possibility:

if <a> then (if <b> then true else false end) else false end

For <a> | <b>:

One possibility is:

gensym := <a>; if gensym then gensym else <b>

Another possibility is:

if <a> then true else (if <b> then true else false end) end

To make it compatible with other boolean operation, i.e., ^, I think the better way should be to cast the result to a boolean. This is also great if one want to implement via re-interpretation of <a> & <b> as if <a> then (if <b> then true else false end) else false end.

Looking at this again, I'm for the "return a boolean" option. This seems more predictable, and possibly more useful (or at least what people might expect) in the context of abuse filters. I guess the change itself wouldn't be super-hard, the problem here is understanding what filters would be affected by this change, and I don't think this is really doable...

Most programming languages I know which allow non-strict coding (i.e. variables or operations with mixed data types) seem to try to convert anything to Boolean if there is a Boolean involved (and in cases not involving a Boolean, they will try to convert everything to a number if a number is involved). While my knowledge and experience is limited, I agree that "convert everything to a Boolean if a Boolean is involved" makes sense.

Indeed, the results are a bit inconsistent currently…

I would also opt for returning a boolean in all of the above cases.