
Polynomials Are Easier Than Integers - wglb
http://rjlipton.wordpress.com/2011/02/26/polynomials-are-easier-than-integers/
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vog
_> if I say that “we should definitely work hard on getting our proofs
correct,” consider the negation: “we should work hard on getting our proofs
incorrect.”_

There's a logical flaw in this example. The latter is not the negation of the
former. The correct negation is:

"we shouldn't work hard on getting our proofs correct."

~~~
trobertson
I believe you are talking about the inverse, as opposed to the contradiction.
Consider a * b

Contradiction: a * -b

Inverse: -a * -b

In the theorem that he provides us, H(X) = H(-X), it could be argued either
way as to whether he meant contradiction or inverse. However, his example of
"we should work harder on correct proofs; we should work harder on incorrect
proofs" very clearly uses two variables, and the form of the second is of a
contradiction.

EDIT: Wow did I screw my terms up. Changed Negation to Contradiction, and
Contrapositive to Inverse. Thank you barrkel for pointing this out.

~~~
vog
_> I screw my terms up. Changed Negation to Contradiction, and Contrapositive
to Inverse._

Even with those changes, your post doesn't make any sense to me.

The _negation_ of logical statements has a pretty well defined meaning [1]. It
seems to be neither what you call "inverse" not what you call "contradiction".

Also, I don't see what the " * " operator means in your example. It appears to
be some operator to combine two logical statements "a" and "b". However, the
original statement isn't a combination of smaller statements at all. (In
particular, the original statement is not an implication, i.e. a term of the
form "if a then b" or "from a follows b" or similar.)

Anyway, for whatever you mean by " * ", the negation is simply "- (a * b)".
The expressions you are proposing ("-a * b" and "a * -b") seem to be different
from that.

In summary, it is totally unclear what you mean by "a", "b", " * ", "inverse"
and "contradiction".

[1] The negation is true for exactly those instances for which the original
statement is false.

~~~
trobertson
Fair points. I was basing my terminology off of this table, after realizing
that I had used incorrect terms.

<http://en.wikipedia.org/wiki/Contraposition#Comparisons>

> If it is meant to be a synonym of what I'm calling negation [1], it would be
> "- (a * b)" for whatever you mean with " * ". However, this is by no means
> equal to "(-a) * (-b)".

This is absolutely correct, and was what originally prompted my comment. In
your original comment, you say that

> The correct negation is: "we shouldn't work hard on getting our proofs
> correct."

which, if we let a = "should work" and b = "getting proofs correct", would
mean that you said (-a) * (-b) is the negation, as opposed to -(a * b). By the
quoted text, the negation would be either (-a) * b, or a * (-b).

This distinction is what I was trying to communicate, though it seems I did a
poor job of that.

~~~
vog
Okay, so your " * " operator means "implication" [1], because the terms
"inverse", "contradiction" and "contrapositive" in your mentioned Wikipedia
article refer only to implications.

However, the original statement _isn't an implication_.

Also, neither your proposed part "should work" nor the other part "getting
proofs correct" are logical statements. Those are just parts of a sentence.

In other words, _your decomposition doesn't make any sense_.

[1] That is, "a * b" = "if a then b" = "from a follows b" = "a implies b"

