I think you are trying to say a "but" operator is equivalent "not and" operator and that !(A->B) is equivalent to "but". I am assuming that "!" symbolizes the negation operator and "->" symbolized the if-then ( or implies ) operator.

Yet ...

P	Q	P->Q
T	T	T
T	F	F
F	T	T
F	F	T

P	Q	P and Q	!(P and Q)
T	T	T	F
T	F	F	T
F	T	F	T
F	F	F	T

So I would say "A and !B"is not equivalent to "!(A -> B)" and I wouldn't say "A but B" <=> "A and !B", but I could just be misunderstanding your notation. :)

As far a what a but operator is I would say it should be equivalent to the "Boolean And" operator. And is not needed in a programming language. The word "but" is used in commonly to emphasis that an assumption is false. Example ...

If a student where to errorously assume that multiplaction is 
the same as addition the student might state ...


1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 and 1 * 1 = 2


... which we know to be false.  The student's teacher would say ...


1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 but 1 * 1 != 2


... to correct the student.


This is the same as


1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 and 1 * 1 != 2


    T and T and T and T is TRUE


It is not equal to


1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 and not (1 * 1 != 2)



     T and T and T and F is FALSE

But merely gives emphasis to the fact that the student's assumption is false.

Janitored by davido: removed excess spaces within pre tags that caused horizontal scrolling.

You forgot a couple ...

P and not Q	!(P -> Q)
F	F
T	T
F	F
F	F

Hence, the relation holds.

Being right, does not endow the right to be rude; politeness costs nothing.
Being unknowing, is not the same as being stupid.
Expressing a contrary opinion, whether to the individual or the group, is more often a sign of deeper thought than of cantankerous belligerence.
Do not mistake your goals as the only goals; your opinion as the only opinion; your confidence as correctness. Saying you know better is not the same as explaining you know better.

I shouldn't have to say this, but any code, unless otherwise stated, is untested

Ahh! I see. I don't know why I was thinking you meant nand !(P and Q). I misunderstood your notation I guess.

The student's teacher would say ...

1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 but 1 * 1 != 2

... which could be written as ...

1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 and not (1 * 1 = 2)

I get it now.



Plankton: 1% Evil, 99% Hot Gas.

Since "A but B" is really a conjunctive inversion, discursively it should be equivalent to "A and not B".

And since "A and not B" is equivalent to "not (if A then B)", or in other words, "not (B if A)", couldn't this be said thusly, to be perly, "B unless A"?

This is where predicate logic doesn't work as well as set theory. "A but B" is actually more akin to "A minus B", where A and B are both sets. Or, in Perl,

my @A = ( ... );
my @B = ( ... );

my %B = map { $_ => !!1 } @B;

# "A but B"
grep { !$B{$_} } @A;
[download]

Which is very close to the mixin concept that Perl6 will apply to the term "but".

Being right, does not endow the right to be rude; politeness costs nothing.
Being unknowing, is not the same as being stupid.
Expressing a contrary opinion, whether to the individual or the group, is more often a sign of deeper thought than of cantankerous belligerence.
Do not mistake your goals as the only goals; your opinion as the only opinion; your confidence as correctness. Saying you know better is not the same as explaining you know better.

I shouldn't have to say this, but any code, unless otherwise stated, is untested