Java’s Broken Booleans

  • Thread starter Lawrence D'Oliveiro
  • Start date
L

Lew

Andreas said:
I see another equivalence relation to {Foo, Bar}, FWIW. (namely nothing)
My statement was about a natural ordering in the set {Nothing, Universe}.

There is none such. Just because you call it so doesn't make it "natural",
only "natural for Andreas". There are many well-established philosophical
systems with millions of adherents in which "Nothing == Universe". You are
not God or any other arbiter of "naturalness" in such arbitrary schemes.
It looks entirely natural to me, that you'd come up with the reverse
as being natural (no matter whether or not influenced by Mayan illusions).

So if you easily and ahead of time predict that someone else has a different
"natural" order than yours, it argues strongly that "natural" for {Nothing,
Universe}" ordering is idiosyncratic, not universal, and entirely arbitrary.

You define your own personal notion of "natural" as though the rest of us
should adhere to it on your ukase. Just because you find it "natural" doesn't
make it so.

There's no universal, "natural" order to 'true' and 'false', nor to 'nothing'
and 'Universe'. It's arbitrary and conventional only.
 
L

Lew

Jukka Lahtinen said:
Before Universe, there was Nothing.
Not even yet a timeline on which to place the Nothing.
And after Universe, there will again be Nothing.
No longer even a timeline on which to place the Nothing.

Btw., will the interval of Universe's existence in time be a
[closed] or ]open[ one? Or maybe ]semi]-[closed[?
So, clearly no obvious unambiguous natural ordering there.

Oh, and then, there surely isn't a natural ordering on natural numbers,
either, since when talking about rankings, obviously 1 becomes superior
to all the others...

True, if disingenuous. You're comparing apples and oranges. (Which is
greater, btw?)

The "natural" ordering on natural numbers is the one arbitrarily *defined* for
natural numbers. You've taken maths, surely? Then you know that it's all by
definition, nothing "natural" about any of it.

If you haven't taken any maths courses (algebra or greater) then you'll have
to take my word for it or google for it.
 
L

Lew

Jukka Lahtinen said:
Probably it's just me, but I see an even more obvious natural
ordering in {Nothing, Universe}, than even in {false, true}.
Before Universe, there was Nothing.
Not even yet a timeline on which to place the Nothing.
And after Universe, there will again be Nothing.
No longer even a timeline on which to place the Nothing.

Btw., will the interval of Universe's existence in time be a
[closed] or ]open[ one? Or maybe ]semi]-[closed[?
So, clearly no obvious unambiguous natural ordering there.

Oh, and then, there surely isn't a natural ordering on natural numbers,
either, since when talking about rankings, obviously 1 becomes superior
to all the others...
But there most definitely _are_ intuitive ordering*s* on natural
numbers, which is the point. Non-strict or strict, we've got<,<=,>
and>=.

Whereas with boolean any ordering must be completely artificial.

I'm pretty sure at this point that we're never going to see Andreas move off
his position on this one. He's already down to, "I would have guessed *you*
would say that" style of response, indicating that he's not letting logic or
truth reach him.
 
A

Arved Sandstrom

Is that the sound of the goalposts moving? You said “find me a mathematician
who statesâ€, and I did.


So you deny that he was using 1 and 0 to represent true and false?

You want some fries as well?

Why don't you try this? Why don't you just read "An
Investigation of the Laws of Thought"? Chapters II and III in particular
should set you straight. I recommend in particular reading the material
that leads up to logical Equation 2 in II.9, and then keep that in mind
(without skipping the good intervening stuff) for when you arrive at
Proposition II in III.13.

You'll see - assuming you bother reading any of this - that Boole refers
to 0 and 1 as _symbols_ in the system of _Logic_ (they are *not*
numbers), and both logical 0 and logical 1 represent *classes*. Logical
0 is Nothing, the empty or null set, and logical 1 is the Universe, the
set of all elements under discourse.

I made that reference to x and 1-x on purpose. For Boole, x is a class.
For example, x might be "all people who are programmers". In Boole's
terminology, 1-x (where '1' is very much the logical symbol 1, the
Universe) means the complement, i.e. all people who are not programmers,
where '1' would then represent the universe of discourse, "all people".

Furthermore, "true" and "false" only enter into the discussion, with
Boole, when he has already defined logical 0 and logical 1 and his signs
and his fundamental algebra. For example, the _expression_ 1-x contains
the logical symbol 1, but there is precisely zero notion of truth or
falsity implied by that expression, because there is as yet no
_proposition_. In fact it's nonsensical to associate 1 with True in that
context.

As an example, you might have it that x is one class, y is another
class, and z yet another, and express a positive proposition that all
elements which belong to x but not to y must be precisely those which
belong to z as

x(1-y)=z

or x(1-y)-z=0

But here 0 still does not mean False; it means the empty set.

Now, there is a very important statement by Boole, on page 70 of Chapter
V, which states that "We may in fact lay aside the logical
interpretation of the symbols in the given equation; convert them into
quantitative symbols, susceptible only of the values 0 and 1; perform
upon them as such all the requisite processes of solution; and finally
restore to them their logical interpretation".

That's all well and good if you actually even are aware of the logical
interpretation - everything I talked about prior. Everything you've said
indicates that you don't have a clue about anything that Boole developed
leading up to this statement. Your arithmetic tricks in various
programming languages demonstrate the shallowness of your understanding.
You're fixated on 0 and 1 as numbers, and incapable of getting that that
is a concrete representation.

Furthermore, you'd still be making a mistake if you thought of the
quantitative symbols 0 and 1 as identically True and False. After all,
True and False are _logical_ interpretations. Hie you forth and read
Chapter VI.

You should in fact read the cautions in Chapter V, after this above
statement of Boole's, to get a better grip.

AHS
 
A

Andreas Leitgeb

Arved Sandstrom said:
But there most definitely _are_ intuitive ordering*s* on natural
numbers, which is the point. Non-strict or strict, we've got <, <=, >
and >=.

Whereas with boolean any ordering must be completely artificial.

Now that's really well argued. Congrats!
This "must be" is really convincing.
 
A

Andreas Leitgeb

Lew said:
You are not God ...
blasphemy!!! ;)
So if you easily and ahead of time predict that someone else ...

Long time ago, back in school there was some lecture about psychology and
language. The book had two drawings: a curve and sequence of line segments.
it was up to the pupils to name one of them "Maluma" and the other one
"Takete" (both artificial - especially not German - words). It was easily
predictable for one particular classmate that he would vividly claim it to
be just as natural to call the curve "Takete" and the other one "Maluma".
You remind me of him. Maybe you even sympathise with him, based on this
story.
You define your own personal notion of "natural" as though [...].
Just because you find it "natural" doesn't make it so.

I never intended to "make" it so. I've only written *my perception*
of what already is. My perception may be wrong, but you haven't yet
added anything but plain unfounded contradiction... like as in Monty
Pythons "I want an argument"-sketch.
 
A

Andreas Leitgeb

Lew said:
Jukka Lahtinen said:
Probably it's just me, but I see an even more obvious natural
ordering in {Nothing, Universe}, than even in {false, true}.
Before Universe, there was Nothing.
Not even yet a timeline on which to place the Nothing.
And after Universe, there will again be Nothing.
No longer even a timeline on which to place the Nothing.
Btw., will the interval of Universe's existence in time be a
[closed] or ]open[ one? Or maybe ]semi]-[closed[?

A pity you didn't feel like answering that. Jukka, perhaps?
True, if disingenuous. You're comparing apples and oranges.

But in the end, it's all fruit!
The "natural" ordering on natural numbers is the one arbitrarily *defined* for
natural numbers. You've taken maths, surely? Then you know that it's all by
definition, nothing "natural" about any of it.

How would you define "natural"ity of any sort of comparatives on any set?
Perhaps, my definition of "natural" just isn't compatible with yours and
some others'. Arved changed use of "natural" to use of "intuitive" - is
that a goal post move or meant to be synonymous?
 
A

Arved Sandstrom

Now that's really well argued. Congrats!
This "must be" is really convincing.

Your proposed "natural" orderings for booleans have so far been based on
ethics and morality, Andreas. I'm trying to stick to mathematics.

AHS
 
A

Andreas Leitgeb

Arved Sandstrom said:
Your proposed "natural" orderings for booleans have so far been based on
ethics and morality, Andreas. I'm trying to stick to mathematics.

Mathematics itself has no concept of "natural". It's just those
areas where it models something natural(*). The so-called "natural"
numbers existed long before calculations on them were formalized.
IIRC, Aristotle had some definition of truth and untruth long before
Boole formalized calculations on "be"s and "not be"s. Aristotle surely
didn't invent these words (nor those in his own language) either.

What lets you accept numbers as being natural, but not logic?

(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely
more obviously a natural thing to do. Distinguishing yes and no,
true and false, or nothing and all appears to me just as natural.
 
A

Arved Sandstrom

Mathematics itself has no concept of "natural". It's just those
areas where it models something natural(*). The so-called "natural"
numbers existed long before calculations on them were formalized.
IIRC, Aristotle had some definition of truth and untruth long before
Boole formalized calculations on "be"s and "not be"s. Aristotle surely
didn't invent these words (nor those in his own language) either.

What lets you accept numbers as being natural, but not logic?

(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely
more obviously a natural thing to do. Distinguishing yes and no,
true and false, or nothing and all appears to me just as natural.

In the same vein, it appears to me to be just as natural that not all
sets need have an ordering at all. The elements can be distinguished -
otherwise they would not be elements in a set - but there is no partial
or total ordering that one can devise.

At least with counting numbers, say, one can state that 6 represents
more things than 5 does, and less than 7 does. There is an unambiguous
sense of "greater than" and "less than". Although to express the
concepts requires an intelligence, it's readily apparent that the
concept itself is not _of us_.

I cannot myself conceive of a way of dispassionately stating that True
is greater than False, however. Different yes, but greater or less than, no.

I don't even see the overriding need to impose an order on boolean. Not
in mathematics. Programming languages, OTOH, have enough mathematical
vagueness in them that creating an artificial boolean ordering when
required for purely technical reasons (sorting a list) is perfectly
OK...but it doesn't have any deep meaning to it.

AHS
 
P

Paul Cager

blasphemy!!!  ;)

That's exactly how Nazi Germany started.
You define your own personal notion of "natural" as though [...].
Just because you find it "natural" doesn't make it so.

I never intended to "make" it so. I've only written *my perception*
of what already is.  My perception may be wrong, but you haven't yet
added anything but plain unfounded contradiction...  like as in Monty
Pythons "I want an argument"-sketch.

No it isn't.
 
A

Andreas Leitgeb

Arved Sandstrom said:
In the same vein, it appears to me to be just as natural that not all
sets need have an ordering at all.

Indeed, there's no requirement for any set to have an ordering - it
just happens that any set of at least two elements does allow for
at least a semi-ordering plus its reverse. Even in a set {Foo, Bar}
one element can be "Foo"er or "Bar"er than the other.
Whether one of these comparators happens to be more interesting
than its reverse depends on what real-world(==natural) concept
it models and whether that real-world model prefers one of these
comparators. (It does, if the Comparator's outcome is typically
positively related to an individual's success and well-being.
Nature is sooo selfish!)

Birds can count&compare. When they see two spots with fodder they
typically approach the one with "more". But before they even get
to count pieces, they've first made a decision: "it is something
to eat", or "it isn't something to eat", and I bet they have a
clear preference as to which outcome they are more excited about. :)
That's what I mean with nature not caring for the theoretic symmetry
(as layed out by de Morgan's laws) between boolean values.
I cannot myself conceive of a way of dispassionately stating that True
is greater than False, however. Different yes, but greater or less than, no.

That's just because you limit yourself to a particular comparator.
How would you compare timespans? "Our light bulbs will last greater
time than our competitors'." Doesn't make sense. ;-)
I don't even see the overriding need to impose an order on boolean.

That's the original discussion of which we diverged by going into whether
there is a naturally preferred comparator for logic. If there's one, then
it makes sense to model it. I, not surprisingly, think that it would make
in mathematics. Programming languages, OTOH, have enough mathematical
vagueness in them that creating an artificial boolean ordering when
required for purely technical reasons (sorting a list) is perfectly
OK...

And convenience for programmers rolling their own .compareTo based on
(lower-case) boolean fields of their class obviously has been dogmatized
(by most participants of this thread, fwiw) not to count as a reason...
 
A

Andreas Leitgeb

Paul Cager said:
That's exactly how Nazi Germany started.

I'm now feeling slightly guilty for declaring all
the falses to be minor values than the trues.
You define your own personal notion of "natural" as though [...].
Just because you find it "natural" doesn't make it so.
I never intended to "make" it so. I've only written *my perception*
of what already is.  My perception may be wrong, but you haven't yet
added anything but plain unfounded contradiction...  like as in Monty
Pythons "I want an argument"-sketch.
No it isn't.
Sure is!
 
J

John B. Matthews

Andreas Leitgeb said:
(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely more
obviously a natural thing to do. Distinguishing yes and no, true and
false, or nothing and all appears to me just as natural.

OK, I'm confused: Is AC analog current, for which complex numbers are
useful [1], or axiom of choice, for which well-ordering is key [2]?

[1]:<http://en.wikipedia.org/wiki/Electrical_impedance>
[2]:<http://en.wikipedia.org/wiki/Axiom_of_choice>
 
L

Lew

OK, I'm confused: Is AC analog current, for which complex numbers are
useful [1], or axiom of choice, for which well-ordering is key [2]?

[1]:<http://en.wikipedia.org/wiki/Electrical_impedance>
[2]:<http://en.wikipedia.org/wiki/Axiom_of_choice>

Which is greater, (4 + 3i) or (3 + 4i)?

I can't even imagine why Andreas brought up complex numbers in a
discussion about inappropriate application of the adjective "natural"
to an ordering, unless he wants to make the point that there are sets
besides that of boolean values for which there is no "natural" order.
 
A

Andreas Leitgeb

John B. Matthews said:
(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely more
obviously a natural thing to do. Distinguishing yes and no, true and
false, or nothing and all appears to me just as natural.
OK, I'm confused: Is AC analog current, for which complex numbers are
useful [1], or axiom of choice, for which well-ordering is key [2]?

I was thinking of [1]. I've yet to see a physical effect of [2].

PS: What's yellow and equivalent to the Axiom of choice?


Zorn's lemon.
 
A

Andreas Leitgeb

Lew said:
John said:
Andreas said:
(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely more
obviously a natural thing to do. Distinguishing yes and no, true and
false, or nothing and all appears to me just as natural.
OK, I'm confused: Is AC analog current, for which complex numbers are
useful [1], or axiom of choice, for which well-ordering is key [2]?
[1]:<http://en.wikipedia.org/wiki/Electrical_impedance>
[2]:<http://en.wikipedia.org/wiki/Axiom_of_choice>
Which is greater, (4 + 3i) or (3 + 4i)?
I can't even imagine why Andreas brought up complex numbers in a
discussion about ...

The discussion had already shifted to definition of "natural" in
a more general sense than just orderings. If you mean it wasn't,
then, well, lets pretend to agree on that I shifted it for the
sake of that sidenote.
to an ordering, unless he wants to make the point that there are sets
besides that of boolean values for which there is no "natural" order.

You *can* define total orderings on the *set* of complex numbers(1), but
they won't be consistent with either or both of the *field* operations,
so they are generally not much of interest.
Booleans with "and", "or" and "not" operations don't have field structure,
in the first place, and there is nothing an ordering on them could be
inconsistent with.

(1) after all, they are of equal cardinality as the real numbers, so
there exist bijective functions reversibly mapping between C and R.
Any such bijective function induces an ordering on the *set* of C.
 
L

Lew

Lew said:
John said:
Andreas Leitgeb wrote:
(*) nowadays, it seems like even complex numbers would be natural
(e.g. ask a physician about AC), but counting things is surely more
obviously a natural thing to do. Distinguishing yes and no, true and
false, or nothing and all appears to me just as natural.
OK, I'm confused: Is AC analog current, for which complex numbers are
useful [1], or axiom of choice, for which well-ordering is key [2]?
[1]:<http://en.wikipedia.org/wiki/Electrical_impedance>
[2]:<http://en.wikipedia.org/wiki/Axiom_of_choice>
Which is greater, (4 + 3i) or (3 + 4i)?
I can't even imagine why Andreas brought up complex numbers in a
discussion about ...

The discussion had already shifted to definition of "natural" in
a more general sense than just orderings. If you mean it wasn't,
then, well, lets pretend to agree on that I shifted it for the
sake of that sidenote.
to an ordering, unless he wants to make the point that there are sets
besides that of boolean values for which there is no "natural" order.

You *can* define total orderings on the *set* of complex numbers(1), but

Which ones are "natural"?
they won't be consistent with either or both of the *field* operations,
so they are generally not much of interest.

Nor very "natural".
Booleans with "and", "or" and "not" operations don't have field structure,
in the first place, and there is nothing an ordering on them could be
inconsistent with.

Other than "naturalness".
 
D

Daniele Futtorovic

Thank you. Can we now invoke Godwin's Law and declare this thread
officially dead, please?

I second that motion. And then third and fourth it, for good measure.
 

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