Interchanging Variables in Logic

Date: 04/10/2003 at 20:32:04
From: Jordan
Subject: 
logic

What are the four rules of logic? Can the variables (p and q) be 
interchanged? Why or why not?

Date: 04/10/2003 at 20:50:57
From: Doctor Achilles
Subject: Re: Logic

Hi Jordan,

Thanks for writing to Dr. Math.

For an introduction to the rules of logic, you should go to the Dr. 
Math Crash Course in Symbolic Logic:

   http://www.mathforum.com/dr.math/faq/symbolic_logic.html 

More specifically to your question: the variables (P, Q, R, etc.) are 
arbitrarily assigned to represent simple English sentences. It doesn't 
matter how we assign them as long as we keep straight what each 
variable means.

For example, I could assign the three letters P, Q, and R as 
follows:

  P: The dog is black.
  Q: It is raining.
  R: Telephones can ring.

I can then do logic on them. For example, I could prove the sentence:

  ((P ^ Q) -> P)

[If the dog is black and it is raining, then the dog is black.]

Not a very remarkable sentence, but I could prove it using these 
steps:

            {
    1)      (P ^ Q)                        [assumption]
    2)      P                              [^elim on 1]
            }
    3)  ((P^Q) -> P)                   [->intro on 1-2]

The fact that I proved that sentence does not mean that I have also 
proved the related sentence:

  ((Q ^ P) -> Q)

I would have to go through a similar proof before I could claim to 
have proven this new sentence true.

Logic by its very nature is tedious and moves in tiny steps, to make 
sure that we aren't misled by something that 'looks right' but has 
some small flaw that we didn't think about. You can come up with 
shortcuts, but you have to be very careful not to apply them beyond 
the scope of what you originally proved.

In one sense, you are right that the letters are interchangeable. The 
rules of logic don't care which letter you use for each step as long 
as you are consistent. But once you start a proof, you cannot 
interchange the letters.

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/