Pedantic: the axioms of Boolean algebra don’t assign any natural numbers to the elements “top” and “bottom” of the set it operates on. The notation is usually “1” and “0” but it doesn’t have to be. It’s a convenience that many computer languages have named those elements “true” and “false”, and yes, it’s totally valid that in some representations, top = 0 = true and bottom = 1 = false.

Technically true, in practice using 0 for the bottom element and 1 for the top is a pretty strong convention; justified, for example, by the connection to probability measures and isomorphism with the Boolean ring ℤ/2ℤ.

If you want to make an argument for something else being the representations of boolean variables than 0 for false and 1 for true, one could make the case for true being all bits set.

That would make it slightly easier to do things like memset()'ing a vector of boolean, or a struct containing a boolean like in this case. Backwards compatibility with pre-_Bool boolean expressions in C99 probably made that a non starter in any case.

A 1-bit integer can be interpreted as either a signed integer or as an unsigned integer, exactly like an integer number of any other size.

Converting a 1-bit integer to a byte-sized or word-sized integer, by using the same extension rules as for any other size (i.e. by using either sign extension or zero extension), yields as the converted value for "true" either "1" for the unsigned integer interpretation or the value with all ones (i.e. "-1") for the signed integer interpretation.

So you could have "unsigned bool" and "signed bool", exactly like you have "unsigned char" and "signed char", to choose between the 2 possible representations.

> one could make the case for true being all bits set

Historical note: this was the case in QBasic, where true was defined as -1.

There, apparently, were quite a number of ISAs where checking the sign bit was more convenient (performant?) than checking (in)equality with zero.

Some Fortran compilers also did this. MS Powerstation Fortran at least, IIRC.

Including the words "top" and "bottom". In my language, if(x) is the same as if(x==bottom), and 1<2 resolves to bottom. Take that.

That is right, and you could map the Boolean values to other numbers, e.g. mapping them to +1 and -1 corresponds better to many of the hardware implementation techniques for logic circuits.

However when the use of Boolean algebra is embedded in some bigger theories, there are cases when the mapping to 0 and 1 becomes mandatory, e.g. in relationship with the theory of probabilities or with the theory of binary polynomials, where the logical operations can be mapped to arithmetic or algebraic operations.

The mapping to 0 and 1 is fully exploited in APL and its derivatives, where it enables the concise writing of many kinds of conditional expressions (in a similar manner to how mask registers are used in GPUs and in AVX-512).

> Pedantic: the axioms of Boolean algebra don’t assign any natural numbers to the elements “top” and “bottom” of the set it operates on.

Yes? That's precisely what I meant when I said that the traditional presentation of mathematical logic get it wrong: it assigns 0 to FALSE and 1 to TRUE, but it can be done other way around.

Zero as the absence of something definitely makes more sense as false than true. It is the case for these common C idioms:

    while(p) {
        // do something with p
        ...
        p = p->next;
    }

    if(err=foo()) {
        printf("Error %d occurred\n", err);
        ...
    }