[Haskell-cafe] What puts False before True?
Scott Brickner
scottb at brickner.net
Mon Jun 4 15:56:38 EDT 2007
All these smart math guys hanging around and nobody's given a really
decent answer to this?
It's actually not arbitrary. There's a strong connection between
predicates (functions that return boolean values) and sets. Any
predicate uniquely determines a set - the set of values for which it
returns true. Similarly, any set determines a predicate - one that
returns true exactly when the argument is a member of the set.
It's natural to define a partial order among sets from inclusion: A ≤ B
iff A ⊆ B. Viewing the sets as predicates, the corresponding
relationship between the predicates is implication. A ⊆ B iff (x ∊ A) ⇒
(x ∊ B) - so predicates are naturally ordered by implication. Viewed as
sets, the predicate that always returns False is equivalent to ∅ - the
empty set, while the predicate that always returns True is equivalent to
U - the universal set that contains everything (in naive set theory,
anyway - in axiomatic theories it gets a little more complicated).
So, since "subset" gives a "natural" order to sets, "implication" gives
a natural order to predicates, thus it's "natural" to say that the
Haskell expressions (const False) and (const True), which are
predicates, are ordered: (const False) < (const True). And therefore,
it's natural to say that False < True.
Anyway, that's my explanation for it. :)
Derek Elkins wrote:
> Henning Thielemann wrote:
>
>> On Thu, 31 May 2007, Paul Hudak wrote:
>>
>>> PR Stanley wrote:
>>>
>>>>> I think so, too. In Boolean algebra (which predates computers,
>>>>> much less
>>>>> C), FALSE has traditionally been associated with 0, and TRUE with
>>>>> 1. And
>>>>> since 1 > 0, TRUE > FALSE.
>>>>
>>>> The question, however, still remains: why False = 0 and True 1? I
>>>> appreciate that it's so in boolean algebra but why? Why not True = 0
>>>> and False = 1?
>>>
>>> Because if you take (&&) to be (*), and (||) to be (+), you get a
>>> homomorphism between the two resulting algebras (assuming 1+1 = 1).
>>
>>
>> It seems however, that if the number representations of False and True
>> are flipped, then we only need to flip the interpretations of (&&) and
>> (||).
>> For me the choice fromEnum False == 0 && fromEnum True == 1 seems rather
>> arbitrary.
>
>
> It -is- arbitrary, in boolean algebra as well. not is an automorphism
> between the two. However, we tend to think of 0 as associated with
> nothing/the empty set and that maps naturally to {x | False}. There
> are intuitive reasons why this choice was chosen, but no formal
> reasons. Obviously, there are pragmatic reasons to use this choice in
> programming languages, e.g. many languages use 0 to represent false
> and non-zero or 1 to represent true and this is consistent with that.
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