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### Tautology and Substitution Principles

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Date: 02/28/2001 at 09:32:06
From: Sara
Subject: Tautology and Substitution principles

I don't understand the difference between the tautology principle and
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Date: 02/28/2001 at 10:48:59
From: Doctor Achilles
Subject: Re: Tautology and Substitution principles

Hi Sara,

Thanks for writing to Dr. Math.

Just to make sure we're on the same page, these are the logical
symbols I will use:

~:  This means "not". The sentence ~p is true if and only if
p is false.

^:  This means "and". The sentence (p^q) is true if and only if
p is true and q is true.

v:  This means "or". The sentence (pvq) is true if p is true,
or if q is true, or if both are true.

->:  This means "if...then" or "entails". The sentence (p->q)
has this truth table:

p         q            (p->q)
T         T              T
T         F              F
F         T              T
F         F              T

=: This means "if and only if" or "logically equivilant".
The sentence (p = q) is true if and only if p has the same truth
value as q. They can both be true or they can both be false.

Now that that's out of the way, what is a tautology? A tautology is a
sentence that is always true. A classic example of a tautology is the
sentence (pv~p).

Here's why it's a tautology:

p is either true or false, but we're not sure which

IF p were true, then the sentence (pv~p) would be true

IF p were false, then the sentence (pv~p) would be true

So before we even know if p is true, we know (pv~p) is true.

A few other examples of tautologies are: (q->q), (r=r), ~(p&~p),
((p&q)->q), (((p&q)v(p&~q))=p).

That last one is a little tricky. Try substituting in different truth
values for p and for q and see what you get.

What about the substitution rule? There may be more than one
substitution rule in logic, but this is the one I'm familiar with:

2) Pick any sentence in logic that you like.  This can be a
tautology or not, whatever makes you happy. For example
((r&s)->t).

3) Replace (substitute) every letter in the tautology from step 1
with your new sentence. Let's try replacing q. NOTE: it is VERY
important at this step that you get EVERY occurrence of the
letter you are replacing. This gives us:

((p&((r&s)->t)->((r&s)->t))

We just proved a new tautology. In other words, we just found a new
logical truth. What this says is: if p is true, and r and s together
entail t, then r and s together entail t. Not too exciting, I know,
but logic works in small steps.

any other math topics, please write back.

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Logic

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