Implications in Logic

Date: 10/29/2002 at 18:25:26
From: Kristy
Subject: Implications in Logic

I don't understand the first four rules to do with implications: 
Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive 
Syllogism. Can you explain them step by step, please?

Date: 10/29/2002 at 20:04:51
From: Doctor Achilles
Subject: Re: Implications in Logic

Hi Kristy,

Thanks for writing to Dr. Math.

First, just so we're on the same page, here are the symbols I use:

  (P -> Q)

Means: "If P, then Q." This sentence is true unless P is true and Q is 
false.

  (P ^ Q)

Means: "P and Q." This sentence is true if and only if P and Q are 
both true.

  (P v Q)

Means: "P or Q." This sentence is true if P is true, and it is also 
true if Q is true. The only way it can be false is if both P and Q are 
false.

  ~P

Means: "Not P." This is true if P is false.


Modus Ponens:

The rule for this is:

If you have:

  (P -> Q)

And you also have:

  P

Then you can conclude:

  Q

This follows directly from the definition of

  (P -> Q)

Which is "If P is true, then Q is true."

So, if I know that if P is true, then Q must be true, AND I know that 
P is true, then I can validly conclude that Q is true also.


Modus Tollens:

This is a little trickier. The rule here is:

If you have:

  (P -> Q)

And you have:

  ~Q

Then you can conclude:

  ~P

Here's why:

We know first of all that "If P is true, then Q is true." And we know 
that Q is false. With Q false, is it possible for P to be true? No!  
Because if P were true, then Q would have to be true. So if Q is 
false, then P has to be false also.


Hypothetical Syllogism:

The rule here is:

If you have:

  (P -> Q)

And you have:

  (Q -> R)

Then you can conclude:

  (P -> R)

Here's why:

We know that "if P is true, then Q is true." And we know that "if Q is 
true, then R is true." But we don't know ANYTHING about whether any of 
the letters are actually true or not.

Let's assume (or hypothesize) for a second that P is true. Then, by 
modus ponens, Q is true. And then by modus ponens again, R is true. 
So: IF we assume P is true, THEN we conclude R is true. Since we 
didn't KNOW P was true, we cannot take R home with us, but we can say 
that "If P was true, then R would be true." This is equivalent to 
saying "If P, then R" or (P -> R).


Disjunctive Syllogism:

The rule here is:

If you have:

  (P v Q)

And you have:

  ~P

Then you can conclude:

  Q

[This also works if you have (P v Q) and ~Q, you can conclude P.]

Here's why:

We know first of all that "P or Q is true." We also know that P is 
false. If P or Q is true, and P is false, then Q has no choice but to 
be true. So we can conclude that Q is true.

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/