# Negation of a statment.

• Sep 10th 2010, 05:41 AM
berachia
Negation of a statment.
1.
Write down the negations of the following assertions (where m; n; a; b N):
(i) if Coke is not worse than Pepsi then nothing Mandelson says can be trusted.

Is the following the generally correct and complete?

If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted

mandelson can be trusted so coke is worse than pepsi

Can anyone recommend any texts where logic of this kind (mathematical I guess), suitable for self study or even unsoluble excepting serious effort, is introduced and developed fully (into whatever it develops)?

I have been eyeing up Introduction to Mathematical Logic by Elliott Mendelson, is this suitable, or not, what is the best text out there?

I have been, and will be, studying Aristotelian Logic in a different context, how do these two compare in terms of similarity, difference, overlap etc.

• Sep 10th 2010, 06:21 AM
Plato
Quote:

Originally Posted by berachia
1.
[LEFT]Write down the negations of the following assertions (where m; n; a; b N):
(i) if Coke is not worse than Pepsi then nothing Mandelson says can be trusted.
Is the following the generally correct and complete?
If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted
mandelson can be trusted so coke is worse than pepsi
Can anyone recommend any texts where logic of this kind (mathematical I guess), suitable for self study or even unsoluble excepting serious effort, is introduced and developed fully (into whatever it develops)?
I have been eyeing up Introduction to Mathematical Logic by Elliott Mendelson, is this suitable, or not, what is the best text out there?

The correct negation of “if Coke is not worse than Pepsi then nothing Mandelson says can be trusted” is:
Coke is not worse than Pepsi and something Mandelson says can be trusted.

Mendelson's textbook is a standard. I like logic textbooks by Copi.

BTW: When posting please use plain text fonts. That makes it much easier to read what is written.
• Sep 10th 2010, 06:24 AM
undefined
Quote:

Originally Posted by berachia
1.
Write down the negations of the following assertions (where m; n; a; b N):
(i) if Coke is not worse than Pepsi then nothing Mandelson says can be trusted.

Is the following the generally correct and complete?

If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted

mandelson can be trusted so coke is worse than pepsi

Well the simplest way to negate is:

It is not true that if Coke is not worse than Pepsi then nothing Mandelson says can be trusted.

More sophisticated is to write what Plato did since

$\displaystyle \neg (p\to q)\equiv p \land \neg q$
• Sep 11th 2010, 09:54 AM
berachia
So this, my original attempt, is in essence, fully correct?

"If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted"
• Sep 11th 2010, 10:01 AM
undefined
Quote:

Originally Posted by berachia
So this, my original attempt, is in essence, fully correct?

"If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted"

But that's not even a proper English sentence.

Let p = "coke is worse than pepsi"
Let q = "something mendelson says can be trusted"

How do you write "If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted" using p and q?

Well it can't be right anyway since you completely got rid of the "says" part.

Actually it also can't be right because the right side of the implication is a tautology, thus the entire sentence is also a tautology (which it shouldn't be).
• Sep 11th 2010, 10:03 AM
Plato
Quote:

Originally Posted by berachia
So this, my original attempt, is in essence, fully correct?
"If coke (not) is not worse than pepsi than madnelson can be trusted or cannot be trusted"

Quote:

Originally Posted by Plato
The correct negation of “if Coke is not worse than Pepsi then nothing Mandelson says can be trusted” is:
Coke is not worse than Pepsi and something Mandelson says can be trusted.
Mendelson's textbook is a standard. I like logic textbooks by Copi.
BTW: When posting please use plain text fonts. That makes it much easier to read what is written.

The negation of an "If...then... is a conjunction(and).
Look again at my negation.
• Sep 11th 2010, 10:31 AM
berachia
Plato, you wrote,

"Coke is not worse than Pepsi and something Mandelson says can be trusted." (why has the "if" disappeared?)

which is equivalent to (or is it?)

"Coke is not worse than Pepsi and something Mandelson says cannot be trusted."

so we can say (or can't we?)

"Coke is not worse than Pepsi and something Mandelson says can or cannot be trusted"

What is meant here by "something"? Is it "whatever", "anything"? Which doesn't seem right or is it "some of the things"?

One of my istakes previously was that I was illegally negating the premise, right (or am I?), by saying

"If coke is not, not worse.."

To Undefined, what is a tautology? (The phrase directly above this line is correct English, or is it not?)
• Sep 11th 2010, 10:43 AM
undefined
Quote:

Originally Posted by berachia
Plato, you wrote,

"Coke is not worse than Pepsi and something Mandelson says can be trusted." (why has the "if" disappeared?)

which is equivalent to (or is it?)

"Coke is not worse than Pepsi and something Mandelson says cannot be trusted."

so we can say (or can't we?)

"Coke is not worse than Pepsi and something Mandelson says can or cannot be trusted"

What is meant here by "something"? Is it "whatever", "anything"? Which doesn't seem right or is it "some of the things"?

One of my istakes previously was that I was illegally negating the premise, right (or am I?), by saying

"If coke is not, not worse.."

To Undefined, what is a tautology? (The phrase directly above this line is correct English, or is it not?)

Umm, well you ask a bunch of questions, but the general blanket response is:

Two statements that involve the same logical values are logically equivalent if and only if their truth tables are identical.

When you look at truth tables you'll see why the "if" disappears and why your methodology is incorrect. Also, while you can have double negatives in English (such as "It is not uncommon for blah blah blah"), on the whole we avoid those things and what you wrote ("If coke is not, not worse") would be considered incorrect... also that comma would be incorrect regardless.

Here's a link that explains some things in more detail including tautology

Truth Tables, Tautologies, and Logical Equivalence
• Sep 11th 2010, 11:41 AM
Plato
@berachia
You need to study the square of opposition.
A statement “no P is Q” an E proposition, universal negative.
Its negation is “some P is Q” an I proposition, existential positive.

The statement “nothing Mandelson says can be trusted” is an E proposition
Its negation is “something Mandelson says can be trusted” is an I proposition.