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8th Grade Logic


Date: Thu, 15 Dec 94 09:52:21 EST
Subject: Questions from the honors 8th grade at MDS
From: Anonymous

Dear Dr. Math,

        Here are our questions:

        1) Logic - Why in a conditional statement if the "p" in the 
hypothesis is false, then the entire statement is true?  Why isn't it 
undecided?

        2) Why do you use the symbols p and q for logic statements?

        Thank you for your time.


Date: Sat, 17 Dec 1994 15:29:46 -0500 (EST)
From: Dr. Sydney
Subject: Re: Questions from the honors 8th grade at MDS

Greetings!

        Thanks for the question about logic. I am not normally a Dr. of
Math, but my friend called me in for this question, so I am making a 
guest appearance as Dr. 

        In order to answer your question, you must first understand the
property of "or statements." In order for an or statement to be true, 
one of the atomic sentences which composes it must be true. For instance, 
if you have the statement, "P or Q," it will be true if either P or Q is true. 
Of course, if both sentences are true, the sentence will be true as well.

        OK, on to the conditionals..............

Another way of thinking about conditional sentences is as or sentences. 
If you have a conditional of the form P==>Q, another way of writing it is 
"not P or Q" where the not applies only to the sentence P. This statement, 
then, will be true any time the sentence "not P" is true.

        Something to remember about conditionals is that they cannot be
evaluated merely by looking at the truth values of the atomic sentences 
that compose them. Therefore, if you are thinking of the sentence, in 
English, as "if P, then Q" it will seem illogical for the statement to be true 
if P is false. Because of this confusion, it is a good idea to think of the 
sentence as "not P or Q." In fact, computer programs that are written to 
evaluate the truth values of logic statements are programmed to approach 
conditional statements in this way.

        Just as x, y, and z are the common variables in math, P and Q are
the common letters chosen to represent atomic sentences in logic. We do 
not really know why P and Q were chosen for this, but if one of the other 
math doctors knows why, he or she will write. 

        I hope that this answer helps you. Thanks for writing the question,
because it gave me a chance to make this guest appearance on Dr. Math.

Good Luck,

        Megin "Really a doctor of English and Philosophy" Charner


Date: Tue, 3 Jan 1995 12:55:18 -0500 (EST)
From: Dr. Ken
Subject: Re: Questions from the honors 8th grade at MDS

Hello there!

I thought I'd add to my esteemed colleague Megin.  Sometimes it's not so
clear just what we mean by the concept of a conditional statement.  It's
true that we can translate each conditional statement into an "or" statement:
(P) => (Q) means exactly the same thing as the statement (Not P) or (Q).

Here's what this means.  A conditional statement means "whenever (P) is
true, (Q) is true too."  So if we can show that (Q) is always true, then
we're all set; whenever (P) is true, (Q) will be true, since (Q) is always
true.  Likewise, if we can show that (P) is never true, which is the same
thing as showing that (Not P) is true, then we're all set again; whenever
(P) is true, (Q) will be true, since (P) is never true.

Here are a couple of examples that I hope will help illustrate what's going
on.  Let's think about the conditional statement 

If pigs can fly then Ken eats his hat.       Or if you prefer,
(Pigs can fly) => (Ken eats his hat)

Now, I'm not going to have to eat my hat.  You see, pigs can't fly.  I'm
just making sure we have some of the basics down here.  Moreover, not 
only will I not have to chow cap, I can make this claim with complete 
moral impunity.  No, I'm not going to eat my hat, but if pigs flew, I would 
do it.  

Here's another example.
If 1=0 then 938493849384938=7         or
(1=0) => (938493849384938=7)

And here's a proof.  Start out with the equality 7=7, and then keep adding
the equation 1=0 to it.  You'll get 8=7, 9=7, 10=7, and so on, as high as we
want.  Pretty soon, we'll get the desired result, that 938493849384938=7.

In fact, since (1=0) => (Q) no matter what (Q) is, I should be able to prove
ANYTHING IN THE WORLD if I get to start with the premise that 1=0.  
I can prove that 8x5=9, that Elvis is still alive, and that your teeth are 
green. So there you have it.  I hope this helps some.

-Ken "Dr." Math
    
Associated Topics:
Middle School Logic

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