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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

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

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