[rhplus]

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.

[pwdisswordfishy]

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.

[jabl]

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.

[adrian_b]

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).

[Joker_vD]

> 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.

[direwolf20]

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.