[eeps] Proposal for /\ and \/ operators

[Richard O'Keefe]

On 26 Feb 2009, at 10:35 pm, Raimo Niskanen wrote:
> Can you explain to a non-mathematician (me) why you have defined∩
> (E1 /\ E2) as min(E1, E2) and
> (E1 \/ E2) as max(E1, E2).


Because those are the only standard mathematical signs
for the job and have been for at least fifty years.

> To me it seems counterintuitive
> since /\ looks like a mountain hence max
> and \/ looks like a valley hence min.

This is covered in another message.
Basically, you are reading them upside done.
The result is at the *top*.

> And I do not want to fully understand Lattice theory∩

You don't need to.  You understand < without fully
understanding lattice theory; its logical and visual
relative /\ is no harder to understand.

Lattice theory is a generalisation of (parts of)
arithmetic, (parts of) Boolean algebra, and (parts of)
set theory.  Any set can be considered as a lattice,
and
	X \union Y = X \/ Y
	X \inter Y = X /\ Y
I should have listed that as another analogy:
union and join are the same except for round vs pointy;
intersection and meet are the same except for round vs pointy.

>
> Since (1.0 == 1) and (1.0 >= 1) and (1 >= 1.0),
> could you also for /\ and \/ elaborate on
> 	max(A, B) when A >= B -> A;
> 	max(A, B) -> B.
> vs
> 	max(A, B) when A > B -> A;
> 	max(A, B) -> B.

>
> Which may be most useful in the light of operator
> precedence and binding?

I can't make sense of that last sentence.  The question
of which operand you get when the two compare equal has
nothing to do with operator precedence or binding; it's
just as much a problem for functional syntax, as you've
shown yourself.

Over the years I must have agonised over which way to
define max and min dozens of times.  The one thing I've
always insisted on is that
	X \/ Y ==> Y  implies that X /\ Y ==> X
and conversely, so that [X,Y] => [X/\Y,X\/y] is a
permutation.

All that agonising was wasted.  If you ever find yourself
caring whether X\/Y is X or Y when X == Y, you are doing
something wrong.  You cannot expect /\ or \/ to make
distinctions that < and > do not make.

Prolog distinguishes between X < Y (numeric order) and
X @< Y (term order) precisely because identifying integers
and floats gets you into endless trouble.

I provided precise definitions which I think are workable.

By the way, I first met the /\ and \/ operators _as_
programming language operators in the PDP-11 UNIX V7
C compiler.  I happily started using them and found a
useful improvement in the clarity of my code.  I was
deeply unhappy when the VAX C compiler didn't support
them.

As for the person who found the standard symbols
"ugly and unintuitive", I just mentioned this to a
colleague who fell about laughing at the idea.